CINXE.COM
Eulersche Winkel – Wikipedia
<!DOCTYPE html> <html class="client-nojs" lang="de" dir="ltr"> <head> <meta charset="UTF-8"> <title>Eulersche Winkel – Wikipedia</title> <script>(function(){var className="client-js";var cookie=document.cookie.match(/(?:^|; )dewikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t.",".\t,"],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","Januar","Februar","März","April","Mai","Juni","Juli","August","September","Oktober","November","Dezember"],"wgRequestId":"764f69cf-6617-4aaf-8f94-7494f8749e84","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Eulersche_Winkel","wgTitle":"Eulersche Winkel","wgCurRevisionId":243597841,"wgRevisionId":243597841,"wgArticleId":242830,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"], "wgCategories":["Wikipedia:Vorlagenfehler/Vorlage:Cite journal/temporär","Analytische Geometrie","Kristallographie","Klassische Mechanik","Winkel","Leonhard Euler als Namensgeber","Koordinatensystem"],"wgPageViewLanguage":"de","wgPageContentLanguage":"de","wgPageContentModel":"wikitext","wgRelevantPageName":"Eulersche_Winkel","wgRelevantArticleId":242830,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":1}}},"wgStableRevisionId":243597841,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"de","pageLanguageDir":"ltr","pageVariantFallbacks":"de"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":40000,"wgRelatedArticlesCompat":[], "wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":true,"wgVector2022LanguageInHeader":false,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q751290","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.citeRef":"ready","ext.gadget.defaultPlainlinks":"ready","ext.gadget.dewikiCommonHide":"ready","ext.gadget.dewikiCommonLayout":"ready","ext.gadget.dewikiCommonStyle":"ready","ext.gadget.NavFrame":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.styles.legacy":"ready", "ext.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","codex-search-styles":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.legacy.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.createNewSection","ext.gadget.WikiMiniAtlas","ext.gadget.OpenStreetMap","ext.gadget.CommonsDirekt","ext.gadget.donateLink","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.compactlinks","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.checkUser.clientHints","ext.quicksurveys.init", "ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=de&modules=codex-search-styles%7Cext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cmediawiki.codex.messagebox.styles%7Cskins.vector.styles.legacy%7Cwikibase.client.init&only=styles&skin=vector"> <script async="" src="/w/load.php?lang=de&modules=startup&only=scripts&raw=1&skin=vector"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=de&modules=ext.gadget.NavFrame%2CciteRef%2CdefaultPlainlinks%2CdewikiCommonHide%2CdewikiCommonLayout%2CdewikiCommonStyle&only=styles&skin=vector"> <link rel="stylesheet" href="/w/load.php?lang=de&modules=site.styles&only=styles&skin=vector"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/8/85/Euler2a.gif"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1135"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/8/85/Euler2a.gif"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="757"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="605"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Eulersche Winkel – Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//de.m.wikipedia.org/wiki/Eulersche_Winkel"> <link rel="alternate" type="application/x-wiki" title="Seite bearbeiten" href="/w/index.php?title=Eulersche_Winkel&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (de)"> <link rel="EditURI" type="application/rsd+xml" href="//de.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://de.wikipedia.org/wiki/Eulersche_Winkel"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de"> <link rel="alternate" type="application/atom+xml" title="Atom-Feed für „Wikipedia“" href="/w/index.php?title=Spezial:Letzte_%C3%84nderungen&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin-vector-legacy mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Eulersche_Winkel rootpage-Eulersche_Winkel skin-vector action-view"><div id="mw-page-base" class="noprint"></div> <div id="mw-head-base" class="noprint"></div> <div id="content" class="mw-body" role="main"> <a id="top"></a> <div id="siteNotice"><!-- CentralNotice --></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Eulersche Winkel</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Euler2a.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Euler2a.gif/220px-Euler2a.gif" decoding="async" width="220" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Euler2a.gif/330px-Euler2a.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/8/85/Euler2a.gif 2x" data-file-width="368" data-file-height="348" /></a><figcaption>Drehung eines Körpers als Folge von drei einzelnen Drehungen um seine Körperachsen z,x',z".<br /> Eigenes Koordinatensystem: <span style="color:red">rot</span><br /> festes Referenzsystem:<span style="color:blue">blau</span></figcaption></figure> <p>Die <b>eulerschen Winkel</b> (oder <b>Euler-Winkel</b>), benannt nach dem Schweizer <a href="/wiki/Mathematiker" title="Mathematiker">Mathematiker</a> <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, sind ein Satz von drei <a href="/wiki/Winkel" title="Winkel">Winkeln</a>, mit denen die Orientierung (Drehlage) eines festen Körpers im dreidimensionalen <a href="/wiki/Euklidischer_Raum" title="Euklidischer Raum">euklidischen Raum</a> beschrieben werden kann.<sup id="cite_ref-goldstein_1-0" class="reference"><a href="#cite_note-goldstein-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Sie werden üblicherweise mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta ,\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>,</mo> <mi>β<!-- β --></mi> <mo>,</mo> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta ,\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301cc1b37ba8f0fb0c9bedee5efa5e0b5bc9e791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.15ex; height:2.676ex;" alt="{\displaystyle \alpha ,\beta ,\gamma }"></span> oder mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ,\theta ,\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>,</mo> <mi>θ<!-- θ --></mi> <mo>,</mo> <mi>ψ<!-- ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ,\theta ,\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4d3dd32ddeb948dfcec2c817b231fb2c88db48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.192ex; height:2.676ex;" alt="{\displaystyle \varphi ,\theta ,\psi }"></span> bezeichnet. Der Körper kann zum Beispiel ein Kreisel sein (in der theoretischen Physik) oder ein Fahrzeug, ein Schiff oder ein Flugzeug. In der Astronomie kann der „Körper“ auch die Bahnellipse eines Himmelskörpers sein. </p><p>Anstatt der Drehlage eines Körpers können eulersche Winkel auch die Lage eines <a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">kartesischen Koordinatensystems</a> in Bezug auf ein anderes kartesisches Koordinatensystem beschreiben und werden deshalb für <a href="/wiki/Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformationen</a> verwendet. Oft ist das gedrehte Koordinatensystem an einen gedrehten Körper „angeheftet“. Man spricht dann vom <i>körperfesten</i> Koordinatensystem und nennt das ursprüngliche Koordinatensystem <i>raumfest</i>. </p><p>Die Drehlage wird erzeugt, indem der Körper aus seiner Ursprungslage heraus nacheinander um die drei Eulerwinkel um Koordinatenachsen <a href="/wiki/Drehung" title="Drehung">gedreht</a> wird. Für die Wahl der Achsen gibt es verschiedene Konventionen: </p> <ul><li><i>Eigentliche Eulerwinkel:</i> Die erste und die dritte Drehung finden um die gleiche Koordinatenachse statt (z. B. Drehung um z-Achse, x-Achse, z-Achse).</li> <li><b>Kardanwinkel</b> oder <i>Tait-Bryan-Winkel</i>: Alle drei Drehungen werden um verschiedene Koordinatenachsen gedreht (z. B. in der Reihenfolge x-Achse, y-Achse, z-Achse).</li></ul> <p>Dabei wird entweder bei der zweiten und dritten Drehung um die zuvor gedrehten Koordinatenachsen gedreht (<i>intrinsische</i> Drehungen) oder immer um die ursprünglichen Koordinatenachsen (<i>extrinsische</i> Drehungen). </p><p>Die aus den drei Einzeldrehungen zusammengesetzte Drehung kann durch eine <a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrix</a> beschrieben werden, die sich entsprechend als Produkt von drei elementaren <a href="/wiki/Drehmatrix" title="Drehmatrix">Drehmatrizen</a> darstellen lässt. Je nach Anwendungszweck betrachtet man verschiedene Matrizen: </p> <ul><li><a href="/wiki/Transformationsmatrix" class="mw-redirect" title="Transformationsmatrix">Transformationsmatrix</a> für die Koordinatentransformation vom gedrehten (körperfesten) ins ursprüngliche (raumfeste) Koordinatensystem,</li> <li>Transformationsmatrix für die Koordinatentransformation vom raumfesten ins körperfeste Koordinatensystem,</li> <li><a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a> der Drehung bezüglich des raumfesten Koordinatensystems.</li></ul> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Geschichte"><span class="tocnumber">1</span> <span class="toctext">Geschichte</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Eigentliche_Eulerwinkel"><span class="tocnumber">2</span> <span class="toctext">Eigentliche Eulerwinkel</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Geometrische_Beschreibung"><span class="tocnumber">2.1</span> <span class="toctext">Geometrische Beschreibung</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Beschreibung_durch_intrinsische_Drehungen"><span class="tocnumber">2.2</span> <span class="toctext">Beschreibung durch intrinsische Drehungen</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Beschreibung_durch_extrinsische_Drehungen"><span class="tocnumber">2.3</span> <span class="toctext">Beschreibung durch extrinsische Drehungen</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Beschreibung_durch_Matrizen"><span class="tocnumber">2.4</span> <span class="toctext">Beschreibung durch Matrizen</span></a> <ul> <li class="toclevel-3 tocsection-7"><a href="#Abbildungsmatrix_(aktive_Drehung)"><span class="tocnumber">2.4.1</span> <span class="toctext">Abbildungsmatrix (aktive Drehung)</span></a></li> <li class="toclevel-3 tocsection-8"><a href="#Transformationsmatrix"><span class="tocnumber">2.4.2</span> <span class="toctext">Transformationsmatrix</span></a></li> </ul> </li> <li class="toclevel-2 tocsection-9"><a href="#Konventionen"><span class="tocnumber">2.5</span> <span class="toctext">Konventionen</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-10"><a href="#Kardan-Winkel"><span class="tocnumber">3</span> <span class="toctext">Kardan-Winkel</span></a> <ul> <li class="toclevel-2 tocsection-11"><a href="#Roll-,_Nick-_und_Gierwinkel:_z-y′-x″-Konvention"><span class="tocnumber">3.1</span> <span class="toctext">Roll-, Nick- und Gierwinkel: z-y′-x″-Konvention</span></a> <ul> <li class="toclevel-3 tocsection-12"><a href="#Beschreibung"><span class="tocnumber">3.1.1</span> <span class="toctext">Beschreibung</span></a></li> <li class="toclevel-3 tocsection-13"><a href="#Transformationsmatrizen"><span class="tocnumber">3.1.2</span> <span class="toctext">Transformationsmatrizen</span></a> <ul> <li class="toclevel-4 tocsection-14"><a href="#Anwendungsbeispiel"><span class="tocnumber">3.1.2.1</span> <span class="toctext">Anwendungsbeispiel</span></a></li> </ul> </li> </ul> </li> </ul> </li> <li class="toclevel-1 tocsection-15"><a href="#Matrix-Herleitung_im_allgemeinen_Fall"><span class="tocnumber">4</span> <span class="toctext">Matrix-Herleitung im allgemeinen Fall</span></a></li> <li class="toclevel-1 tocsection-16"><a href="#Ergebnis,_Interpretation"><span class="tocnumber">5</span> <span class="toctext">Ergebnis, Interpretation</span></a></li> <li class="toclevel-1 tocsection-17"><a href="#Mathematische_Eigenschaften"><span class="tocnumber">6</span> <span class="toctext">Mathematische Eigenschaften</span></a></li> <li class="toclevel-1 tocsection-18"><a href="#Nachteile,_Alternativen"><span class="tocnumber">7</span> <span class="toctext">Nachteile, Alternativen</span></a></li> <li class="toclevel-1 tocsection-19"><a href="#Anwendungen"><span class="tocnumber">8</span> <span class="toctext">Anwendungen</span></a></li> <li class="toclevel-1 tocsection-20"><a href="#Literatur"><span class="tocnumber">9</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-21"><a href="#Weblinks"><span class="tocnumber">10</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-22"><a href="#Einzelnachweise"><span class="tocnumber">11</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=1" title="Abschnitt bearbeiten: Geschichte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Geschichte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Euler_1778.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Euler_1778.jpg/220px-Euler_1778.jpg" decoding="async" width="220" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Euler_1778.jpg/330px-Euler_1778.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Euler_1778.jpg/440px-Euler_1778.jpg 2x" data-file-width="1100" data-file-height="1300" /></a><figcaption>Euler (1778)</figcaption></figure> <p>Drehungen wurden spätestens seit etwa 1600 durch drei Winkel beschrieben. So bestimmte <a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Johannes Kepler</a> in der <i><a href="/wiki/Johannes_Kepler#Astronomia_Nova" title="Johannes Kepler">Astronomia nova</a></i> die Orientierung der Marsbahn in Bezug auf die Ekliptik durch drei Winkel. Eine algebraische Beschreibung, mit der die Drehlage von beliebigen Punkten berechnet werden konnte, wurde aber erst ab 1775 von <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> in zunehmender Tiefe formuliert.<sup id="cite_ref-ChengGupta89_2-0" class="reference"><a href="#cite_note-ChengGupta89-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In der ersten Arbeit<sup id="cite_ref-E478_3-0" class="reference"><a href="#cite_note-E478-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> zeigte er, dass die neun, den Elementen der Abbildungsmatrix entsprechenden, Koeffizienten wegen der <a href="/wiki/Isometrie" title="Isometrie">Längentreue</a> einer <a href="/wiki/Bewegung_(Mathematik)" title="Bewegung (Mathematik)">Bewegung</a> nicht unabhängig voneinander sind, sondern durch nur drei voneinander unabhängige Winkel festgelegt werden. Es handelt sich bei diesen aber nicht um die hier behandelten eulerschen Winkel, sondern um reine Rechengrößen ohne geometrische Bedeutung. Bekannt ist diese Arbeit heute besonders, weil er in einem Zusatz das heute nach ihm benannte <a href="/wiki/Satz_vom_Fu%C3%9Fball" title="Satz vom Fußball">Rotationstheorem</a> bewies, nach dem jede <a href="/wiki/Bewegung_(Mathematik)" title="Bewegung (Mathematik)">eigentliche Bewegung</a> mit einem Fixpunkt eine Drehung um eine Achse ist. Die aus diesem Ergebnis resultierenden Abbildungsgleichungen, in denen eine Drehung durch die <a href="/wiki/Richtungskosinus" title="Richtungskosinus">Richtungskosinus</a> der Drehachse und den Drehwinkel parametrisiert wird, fand er in einer kurz darauf folgenden zweiten Arbeit.<sup id="cite_ref-E479_4-0" class="reference"><a href="#cite_note-E479-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Euler_angles_(Euler%27s_paper_E825).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Euler_angles_%28Euler%27s_paper_E825%29.png/220px-Euler_angles_%28Euler%27s_paper_E825%29.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/Euler_angles_%28Euler%27s_paper_E825%29.png/330px-Euler_angles_%28Euler%27s_paper_E825%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/Euler_angles_%28Euler%27s_paper_E825%29.png/440px-Euler_angles_%28Euler%27s_paper_E825%29.png 2x" data-file-width="1400" data-file-height="1400" /></a><figcaption>Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> aus der Arbeit<sup id="cite_ref-E825_5-0" class="reference"><a href="#cite_note-E825-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> von Euler</figcaption></figure> <p>In einer dritten, erst <a href="/wiki/Postum" class="mw-redirect" title="Postum">postum</a> erschienenen Arbeit<sup id="cite_ref-E825_5-1" class="reference"><a href="#cite_note-E825-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> führte er schließlich drei Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> ein, mit denen er die Transformation von körperfesten in raumfeste Koordinaten beschrieb und die bis auf Vorzeichen und additive Konstanten mit den heute nach ihm benannten Winkeln übereinstimmen. Sein Vorgehen unterschied sich dabei deutlich vom heute gängigen Verfahren, bei dem das eine Koordinatensystem durch drei aufeinanderfolgende Drehungen um die Koordinatenachsen in das andere Koordinatensystem überführt wird. Euler argumentiere ähnlich wie bei der ersten Arbeit, kommt aber zu einem günstigeren Ansatz<sup id="cite_ref-E825_5-2" class="reference"><a href="#cite_note-E825-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference">:S. 50</sup> mit den drei Winkeln <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, weil er hier – in moderner Sprechweise – nicht nur die Orthonormalität der Zeilen, sondern auch die der Spalten der Transformationsmatrix verwendete. Zur Klärung ihrer geometrischen Bedeutung betrachtete er die Schnittpunkte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> des raumfesten Koordinatensystems mit der Einheitskugel und die entsprechenden Schnittpunkte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> des körperfesten Systems und zeigte, dass die Kosinusse der Bögen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d5ef9a4c8d9cf3a91c6e55730f191f2a92a0fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.973ex; height:2.176ex;" alt="{\displaystyle aA}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418f1f3a1290ca104ec3df87f599619f258cc0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.994ex; height:2.176ex;" alt="{\displaystyle aB}"></span>, …, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046df371f91e52392ad93f55247e15a153dd4cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.773ex; height:2.176ex;" alt="{\displaystyle cC}"></span> gerade die Koeffizienten der Transformationsgleichungen sind. Es ist also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=aA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mi>a</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=aA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc30326062efd092105282b7eaf002188973624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.33ex; height:2.509ex;" alt="{\displaystyle p=aA}"></span> der Winkel zwischen der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>- und der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Achse. Außerdem zeigt sich mittels <a href="/wiki/Sph%C3%A4rische_Trigonometrie" title="Sphärische Trigonometrie">sphärischer Trigonometrie</a>, dass <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> der Winkel bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> im Kugeldreieck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ABa}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABa}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/637c028f6131bb8514d14f6dd9bb037feae06dda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.737ex; height:2.176ex;" alt="{\displaystyle ABa}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> der Winkel bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> im Kugeldreieck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle abA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle abA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85d4e5b1dfabb9660452674c28bd3416da2f5063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.97ex; height:2.176ex;" alt="{\displaystyle abA}"></span> ist. Heute werden die entsprechenden Winkel nicht von der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xX}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894465fc22c358d6b85c82297d75197b4a54365f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.31ex; height:2.176ex;" alt="{\displaystyle xX}"></span>-Ebene, sondern von der zu ihr senkrechten Knotenlinie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> aus gemessen; der Zusammenhang mit den heute meist verwendeten Eulerwinkeln für die Drehfolge x-y-x ist gegeben durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =r+90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mi>r</mi> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘<!-- ∘ --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =r+90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27adbe1cd75f7169203eaf3bce38f48891c035f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.854ex; height:2.509ex;" alt="{\displaystyle \alpha =r+90^{\circ }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/495bf232a840cd6526249b28979b5d96e3679f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.6ex; height:2.509ex;" alt="{\displaystyle \beta =p}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =-q+90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> <mo>=</mo> <mo>−<!-- − --></mo> <mi>q</mi> <mo>+</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘<!-- ∘ --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =-q+90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d98e1d94891876c563703bd3935b72bc56b322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.458ex; height:2.843ex;" alt="{\displaystyle \gamma =-q+90^{\circ }}"></span>. </p><p><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a> brachte in der 1788 erschienenen <i>Mécanique Analytique</i><sup id="cite_ref-Lagrange_6-0" class="reference"><a href="#cite_note-Lagrange-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> zwei Ableitungen der Transformationsgleichungen. Die erste<sup id="cite_ref-Lagrange_6-1" class="reference"><a href="#cite_note-Lagrange-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference">:S. 381–388</sup> stimmt bis auf die Namen der Winkel (bei ihm heißen sie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> <mo>=</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfb234773176ce5782f19e919f9e032c1ab874e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.623ex; height:2.509ex;" alt="{\displaystyle \lambda =p}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ<!-- μ --></mi> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5d813c8ff3125394d50d23a41d06f68904bb35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.549ex; height:2.176ex;" alt="{\displaystyle \mu =r}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν<!-- ν --></mi> <mo>=</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc98bdee18ccca1acc1566b667104d1d40a94d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.4ex; height:2.009ex;" alt="{\displaystyle \nu =q}"></span>) im Wesentlichen mit der von Euler überein. Die zweite<sup id="cite_ref-Lagrange_6-2" class="reference"><a href="#cite_note-Lagrange-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup class="reference">:S. 398–401</sup> deckt sich mit der modernen, unten ausführlich behandelten Darstellung für die z-x-z-Drehfolge – wiederum mit anderen Winkelnamen (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> <mo>=</mo> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92fa30c8d0384ef454ffd19bdd96deb59227d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.881ex; height:2.176ex;" alt="{\displaystyle \varphi =\gamma }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi =\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> <mo>=</mo> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi =\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b287ed627bea7bd39922184201ee3e3efc647e1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.099ex; height:2.509ex;" alt="{\displaystyle \psi =\alpha }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω<!-- ω --></mi> <mo>=</mo> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7867367edd681ab6c30f7f6f0000e6773404ec67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.876ex; height:2.509ex;" alt="{\displaystyle \omega =\beta }"></span>). </p> <div class="mw-heading mw-heading2"><h2 id="Eigentliche_Eulerwinkel">Eigentliche Eulerwinkel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=2" title="Abschnitt bearbeiten: Eigentliche Eulerwinkel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Eigentliche Eulerwinkel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Eulerangles.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Eulerangles.svg/220px-Eulerangles.svg.png" decoding="async" width="220" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Eulerangles.svg/330px-Eulerangles.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Eulerangles.svg/440px-Eulerangles.svg.png 2x" data-file-width="688" data-file-height="775" /></a><figcaption>Koordinatentransformation, Drehfolge in Standard-x-Konvention<br /> <span style="color:#00BFF0">blau</span>: Koordinatensystem in Ausgangslage<br /> <span style="color:green">grün</span>: Schnittgerade der xy-Ebenen, Zwischenlage der x-Achse<br /> <span style="color:red">rot</span>: Koordinatensystem in Ziellage</figcaption></figure> <p>Im Folgenden werden wie in der nebenstehenden Grafik die Achsen des Koordinatensystems in Ausgangslage (in der Grafik blau) mit den Kleinbuchstaben <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>, die Achsen in Ziellage (in der Grafik rot) mit den entsprechenden Großbuchstaben <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span> bezeichnet. </p> <div class="mw-heading mw-heading3"><h3 id="Geometrische_Beschreibung">Geometrische Beschreibung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=3" title="Abschnitt bearbeiten: Geometrische Beschreibung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Geometrische Beschreibung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-Ebene und die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>-Ebene schneiden sich in einer Geraden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> (Knotenlinie). Diese steht senkrecht auf der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse und auf der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>-Achse. </p> <ul><li>Der erste Euler-Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> (auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>) ist der Winkel zwischen der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-Achse und der Geraden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> (gemessen in Richtung der y-Achse).</li> <li>Der zweite Euler-Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> (auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>) ist der Winkel zwischen der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse und der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>-Achse.</li> <li>Der dritte Euler-Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> (auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span>) ist der Winkel zwischen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> und der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Achse.</li></ul> <p>Die hier beschriebene Version der eulerschen Winkel, bei der der Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> von der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-Achse aus zur Knotenlinie und der Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> von der Knotenlinie zur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Achse gemessen wird, nennt man die <i>Standard-x-Konvention</i>. Entsprechend werden bei der <i>Standard-y-Konvention</i> die Winkel von der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-Achse zur Knotenlinie und von der Knotenlinie zur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>-Achse gemessen. </p><p>In der klassischen Physik wird meist die Standard-x-Konvention verwendet. Statt wie hier mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> werden die Winkel meist mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> bezeichnet. In der Quantenmechanik hat sich hingegen die Standard-y-Konvention durchgesetzt. Insbesondere werden die Darstellungen der Drehgruppe SO3 durch sogenannte <a href="/wiki/Wignersche_D-Matrix" title="Wignersche D-Matrix">Wignersche D-Matrizen</a> entsprechend parametrisiert. </p> <div class="mw-heading mw-heading3"><h3 id="Beschreibung_durch_intrinsische_Drehungen">Beschreibung durch intrinsische Drehungen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=4" title="Abschnitt bearbeiten: Beschreibung durch intrinsische Drehungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Beschreibung durch intrinsische Drehungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Drehung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, welche das <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xyz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3957b3fbfe29dfdb2f7cdad4e3d7fc2ea7be84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.573ex; height:2.009ex;" alt="{\displaystyle xyz}"></span>-System in das <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XYZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XYZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea07eba7954100995bf73897619ce2df498349e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.434ex; height:2.176ex;" alt="{\displaystyle XYZ}"></span>-System dreht, kann in drei Drehungen aufgeteilt werden. Bei der Standard-x-Konvention sind das: </p> <ul><li>Zunächst die Drehung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{z}(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{z}(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ac34164a0fd4f1eed4b83e3f706f637dbf2e86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.347ex; height:2.843ex;" alt="{\displaystyle r_{z}(\alpha )}"></span> um den Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse,</li> <li>dann die Drehung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{N}(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{N}(\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a3d4e4a0bd3edefb9ac8cd7cca13a004283c255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.881ex; height:2.843ex;" alt="{\displaystyle r_{N}(\beta )}"></span> um den Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> um die Knotenlinie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>,</li> <li>zuletzt die Drehung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{Z}(\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{Z}(\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68c4dd048d304078fdcc41f631bef8a8a2787a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.541ex; height:2.843ex;" alt="{\displaystyle r_{Z}(\gamma )}"></span> um den Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>-Achse.</li></ul> <p>Bei diesen Drehungen entstehen nacheinander neue Koordinatensysteme: </p> <ul><li>ursprüngliches Koordinatensystem: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>- und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse</li> <li>nach der ersten Drehung: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span>- und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c017ef8616d0c836e4b2a88c40931333c38dd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle z'}"></span>-Achse</li> <li>nach der zweiten Drehung: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span>-, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147ec60507e44a6d376237c0a16132cf7461cd62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.843ex;" alt="{\displaystyle y''}"></span>- und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span>-Achse</li> <li>nach der dritten Drehung: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>‴</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ebb943445ec15d1e8456f36a8b40728735c26e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.919ex; height:2.509ex;" alt="{\displaystyle x'''}"></span>-, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>‴</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2985fa78214cb25116c5b6b34d92c0ce75d8484a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.843ex;" alt="{\displaystyle y'''}"></span>- und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>‴</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z'''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4b0d22d7ca2bdaf4c0681c06ce888f7019ed0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.68ex; height:2.509ex;" alt="{\displaystyle z'''}"></span>-Achse (bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>- und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>-Achse)</li></ul> <p>Die Drehung um <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> ist also eine Drehung um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-Achse, die Drehung um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>-Achse eine Drehung um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span>-Achse. Die Gesamtdrehung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> setzt sich also aus den Drehungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{z}(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{z}(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ac34164a0fd4f1eed4b83e3f706f637dbf2e86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.347ex; height:2.843ex;" alt="{\displaystyle r_{z}(\alpha )}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{x'}(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{x'}(\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0040127e43f972d48df92ec1cc4cc9c32de445f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.894ex; height:2.843ex;" alt="{\displaystyle r_{x'}(\beta )}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{z''}(\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{z''}(\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/015d7fa6b7b0364c0e2df989030599b682762da6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.022ex; height:2.843ex;" alt="{\displaystyle r_{z''}(\gamma )}"></span> zusammen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{z''}(\gamma )\circ r_{x'}(\beta )\circ r_{z}(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> <mo>∘<!-- ∘ --></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>∘<!-- ∘ --></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{z''}(\gamma )\circ r_{x'}(\beta )\circ r_{z}(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2cfbcb678a6494024af85c937ca9424aeaba762" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.8ex; height:2.843ex;" alt="{\displaystyle r=r_{z''}(\gamma )\circ r_{x'}(\beta )\circ r_{z}(\alpha )}"></span></dd></dl> <p>Die Reihenfolge der Drehachsen ist also: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse → <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-Achse → <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span>-Achse oder kurz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span>. </p><p>Eine solche Zerlegung in Drehungen, bei denen jeweils um die mitgedrehten Koordinatenachsen gedreht wird, nennt man <i>intrinsische</i> Drehfolge. </p> <div class="mw-heading mw-heading3"><h3 id="Beschreibung_durch_extrinsische_Drehungen">Beschreibung durch extrinsische Drehungen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=5" title="Abschnitt bearbeiten: Beschreibung durch extrinsische Drehungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Beschreibung durch extrinsische Drehungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Euler_angles_zxz_ext%2Baxes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Euler_angles_zxz_ext%2Baxes.png/220px-Euler_angles_zxz_ext%2Baxes.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Euler_angles_zxz_ext%2Baxes.png/330px-Euler_angles_zxz_ext%2Baxes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Euler_angles_zxz_ext%2Baxes.png/440px-Euler_angles_zxz_ext%2Baxes.png 2x" data-file-width="1400" data-file-height="1400" /></a><figcaption>Eulerwinkel bei extrinsischen Einzeldrehungen in der Reihenfolge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> mit den Winkeln <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. <span style="color:#0033CC">blau</span>: Ausgangslage des Koordinatensystems, <span style="color:#FF0000">rot</span>: Ziellage, <span style="color:#339900">grün</span>: Schnittgerade der beiden xy-Ebenen.</figcaption></figure> <p>Dieselbe Drehung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> kann auch durch drei Einzeldrehungen um die ursprünglichen Koordinatenachsen beschrieben werden. Dabei bleiben die Winkel gleich, aber die Reihenfolge der Drehungen kehrt sich um, und die Zwischenlagen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'y'z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'y'z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a6d6757b2072b6461ea7faa7c62a6316ab79a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.635ex; height:2.843ex;" alt="{\displaystyle x'y'z'}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''y''z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> <msup> <mi>y</mi> <mo>″</mo> </msup> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''y''z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f736440b1d4605a22d4e6dd00844de1943f01fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.992ex; height:2.843ex;" alt="{\displaystyle x''y''z''}"></span> sind andere als bei der intrinsischen Drehung: Zuerst wird der Körper um den Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse gedreht, dann um <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β<!-- β --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-Achse (der Winkel zwischen der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>- und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span>-Achse ist derselbe wie der zwischen der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>- und der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-Achse, nämlich <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>) und zuletzt um den Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse (dabei wird die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-Achse in die Knotenlinie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> gedreht und der Winkel zwischen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> und der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Achse ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ<!-- γ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span>). Eine algebraische Begründung findet sich weiter unten im Abschnitt <a href="#Matrix-Herleitung_im_allgemeinen_Fall">Matrix-Herleitung im allgemeinen Fall</a>. Es ist also </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{z}(\alpha )\circ r_{x}(\beta )\circ r_{z}(\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>∘<!-- ∘ --></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>∘<!-- ∘ --></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{z}(\alpha )\circ r_{x}(\beta )\circ r_{z}(\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd7dc49f7d8beaa7d9f41ab2d4bfdac51b245d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.368ex; height:2.843ex;" alt="{\displaystyle r=r_{z}(\alpha )\circ r_{x}(\beta )\circ r_{z}(\gamma )}"></span>.</dd></dl> <p>Eine solche Drehfolge, bei der immer um die ursprünglichen Koordinatenachsen gedreht wird, heißt <i>extrinsische</i> Drehfolge. </p><p>Die Beschreibungen durch intrinsische und durch extrinsische Drehungen sind also äquivalent. Die Beschreibung durch intrinsische Drehungen ist jedoch anschaulicher, während die Beschreibung durch extrinsische Drehungen mathematisch leichter zugänglich ist. </p> <div class="mw-heading mw-heading3"><h3 id="Beschreibung_durch_Matrizen">Beschreibung durch Matrizen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=6" title="Abschnitt bearbeiten: Beschreibung durch Matrizen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Beschreibung durch Matrizen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Drehungen um die Eulerschen Winkel können mit Hilfe von <a href="/wiki/Drehmatrix" title="Drehmatrix">Drehmatrizen</a>, deren Einträge <a href="/wiki/Sinus_und_Kosinus" title="Sinus und Kosinus">Sinus- und Kosinus</a>-Werte der Euler-Winkel sind, beschrieben werden. Dabei unterscheidet man zwischen <a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrizen</a> und <a href="/wiki/Transformationsmatrix" class="mw-redirect" title="Transformationsmatrix">Koordinatentransformationsmatrizen</a>. Im Folgenden werden diese Matrizen für die Standard-x-Konvention angegeben. Die Matrizen für die Standard-y-Konvention erhält man analog, indem man statt der elementaren Drehmatrix für die Drehung um die x-Achse die Drehmatrix für die Drehung um die y-Achse verwendet. </p> <div class="mw-heading mw-heading4"><h4 id="Abbildungsmatrix_(aktive_Drehung)"><span id="Abbildungsmatrix_.28aktive_Drehung.29"></span>Abbildungsmatrix (aktive Drehung)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=7" title="Abschnitt bearbeiten: Abbildungsmatrix (aktive Drehung)" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Abbildungsmatrix (aktive Drehung)"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei einer <i>aktiven Drehung</i> (<i>Alibi-Drehung</i>) werden die Punkte und <a href="/wiki/Vektor" title="Vektor">Vektoren</a> des Raums gedreht. Das Koordinatensystem wird festgehalten. Die Drehmatrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> ist die <a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrix</a> dieser <a href="/wiki/Lineare_Abbildung" title="Lineare Abbildung">Abbildung</a>. Die Koordinaten des gedrehten Vektors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {w}}=r({\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}=r({\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db27e0fcab74da9eff39f0c60980dd4fb3f84248" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.796ex; height:2.843ex;" alt="{\displaystyle {\vec {w}}=r({\vec {v}})}"></span> ergeben sich aus den Koordinaten des ursprünglichen Punkts <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> durch <a href="/wiki/Matrix-Vektor-Produkt" title="Matrix-Vektor-Produkt">Multiplikation</a> mit der Drehmatrix: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}w_{x}\\w_{y}\\w_{z}\\\end{pmatrix}}=R{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\\\end{pmatrix}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}w_{x}\\w_{y}\\w_{z}\\\end{pmatrix}}=R{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\\\end{pmatrix}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3127455a1fc8473a7f869980cc09ac60a004431d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:20.67ex; height:9.509ex;" alt="{\displaystyle {\begin{pmatrix}w_{x}\\w_{y}\\w_{z}\\\end{pmatrix}}=R{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\\\end{pmatrix}}\,.}"></span></dd></dl> <p>Die <a href="/wiki/Abbildungsmatrix" title="Abbildungsmatrix">Abbildungsmatrizen</a> für Drehungen um die Koordinatenachsen (elementare Drehmatrizen) lauten: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}},\quad R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}},\quad R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}},\quad R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}},\quad R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e857ddee2adbf7734a6e76612a002a3d9fcdcb6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:102.027ex; height:9.176ex;" alt="{\displaystyle R_{x}(\alpha )={\begin{pmatrix}1&0&0\\0&\cos \alpha &-\sin \alpha \\0&\sin \alpha &\cos \alpha \end{pmatrix}},\quad R_{y}(\alpha )={\begin{pmatrix}\cos \alpha &0&\sin \alpha \\0&1&0\\-\sin \alpha &0&\cos \alpha \end{pmatrix}},\quad R_{z}(\alpha )={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}}"></span></dd></dl> <p>für die Drehung um den Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-Achse, die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-Achse und die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse. </p><p>Die Drehmatrix der zusammengesetzten Drehung erhält man durch <a href="/wiki/Matrizenmultiplikation" title="Matrizenmultiplikation">Matrixmultiplikation</a> aus den Matrizen der einzelnen Drehungen. Da die elementaren Drehmatrizen die Drehungen um die ursprünglichen Koordinatenachsen beschreiben, verwendet man die extrinsische Drehfolge </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{z}(\alpha )\circ r_{x}(\beta )\circ r_{z}(\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>∘<!-- ∘ --></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>∘<!-- ∘ --></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{z}(\alpha )\circ r_{x}(\beta )\circ r_{z}(\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd7dc49f7d8beaa7d9f41ab2d4bfdac51b245d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.368ex; height:2.843ex;" alt="{\displaystyle r=r_{z}(\alpha )\circ r_{x}(\beta )\circ r_{z}(\gamma )}"></span></dd></dl> <p>und erhält die Abbildungsmatrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R_{zxz}&=R_{z}(\alpha )\,R_{x}(\beta )\,R_{z}(\gamma )\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &\sin \alpha \sin \beta \\\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &-\cos \alpha \sin \beta \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{zxz}&=R_{z}(\alpha )\,R_{x}(\beta )\,R_{z}(\gamma )\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &\sin \alpha \sin \beta \\\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &-\cos \alpha \sin \beta \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da79c664ea390ded4afdfc3f7be2f20efac25e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:87.754ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}R_{zxz}&=R_{z}(\alpha )\,R_{x}(\beta )\,R_{z}(\gamma )\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &\sin \alpha \sin \beta \\\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &-\cos \alpha \sin \beta \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{pmatrix}}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Transformationsmatrix">Transformationsmatrix</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=8" title="Abschnitt bearbeiten: Transformationsmatrix" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Transformationsmatrix"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Transformationsmatrizen beschreiben Koordinatentransformationen vom ursprünglichen (raumfesten) Koordinatensystem ins gedrehte (körperfeste) oder umgekehrt. Die Transformationsmatrix für die Koordinatentransformation vom körperfesten Koordinatensystem ins raumfeste stimmt mit der oben beschriebenen Abbildungsmatrix überein, die Matrix für die umgekehrte Transformation ist die Transponierte dieser Matrix. Hat der Vektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> im raumfesten Koordinatensystem die Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x},v_{y},v_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{x},v_{y},v_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97eda4a2961e5de309a88c1debd01602dc763f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.674ex; height:2.343ex;" alt="{\displaystyle v_{x},v_{y},v_{z}}"></span> und im körperfesten die Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{X},v_{Y},v_{Z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{X},v_{Y},v_{Z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8d0316c0b1f476ec7cbf898006be77a0852d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.99ex; height:2.009ex;" alt="{\displaystyle v_{X},v_{Y},v_{Z}}"></span>, so gilt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}&=R_{zxz}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&=R_{z}(\alpha )\,R_{x}(\beta )\,R_{z}(\gamma ){\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &\sin \alpha \sin \beta \\\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &-\cos \alpha \sin \beta \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}&=R_{zxz}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&=R_{z}(\alpha )\,R_{x}(\beta )\,R_{z}(\gamma ){\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &\sin \alpha \sin \beta \\\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &-\cos \alpha \sin \beta \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be7fdf5aca11f9e8d07cf3de5719f7fc407d2116" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.505ex; width:97.976ex; height:38.176ex;" alt="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}&=R_{zxz}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&=R_{z}(\alpha )\,R_{x}(\beta )\,R_{z}(\gamma ){\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha &-\sin \alpha &0\\\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &-\sin \beta \\0&\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \gamma &-\sin \gamma &0\\\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &\sin \alpha \sin \beta \\\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &-\cos \alpha \sin \beta \\\sin \beta \sin \gamma &\sin \beta \cos \gamma &\cos \beta \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\end{aligned}}}"></span></dd></dl> <p>und </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}&=R_{zxz}^{T}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&=R_{z}(\gamma )^{T}\,R_{x}(\beta )^{T}\,R_{z}(\alpha )^{T}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \gamma &\sin \gamma &0\\-\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &\sin \beta \\0&-\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &\sin \beta \sin \gamma \\-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &\sin \beta \cos \gamma \\\sin \alpha \sin \beta &-\cos \alpha \sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>γ<!-- γ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>α<!-- α --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>γ<!-- γ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>α<!-- α --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>β<!-- β --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}&=R_{zxz}^{T}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&=R_{z}(\gamma )^{T}\,R_{x}(\beta )^{T}\,R_{z}(\alpha )^{T}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \gamma &\sin \gamma &0\\-\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &\sin \beta \\0&-\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &\sin \beta \sin \gamma \\-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &\sin \beta \cos \gamma \\\sin \alpha \sin \beta &-\cos \alpha \sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a8b176e534c2d3dcf596ce5d7f11e9d7fb82ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.838ex; width:98.397ex; height:38.843ex;" alt="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}&=R_{zxz}^{T}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&=R_{z}(\gamma )^{T}\,R_{x}(\beta )^{T}\,R_{z}(\alpha )^{T}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \gamma &\sin \gamma &0\\-\sin \gamma &\cos \gamma &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \beta &\sin \beta \\0&-\sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}\cos \alpha &\sin \alpha &0\\-\sin \alpha &\cos \alpha &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \alpha \cos \gamma -\sin \alpha \cos \beta \sin \gamma &\sin \alpha \cos \gamma +\cos \alpha \cos \beta \sin \gamma &\sin \beta \sin \gamma \\-\cos \alpha \sin \gamma -\sin \alpha \cos \beta \cos \gamma &-\sin \alpha \sin \gamma +\cos \alpha \cos \beta \cos \gamma &\sin \beta \cos \gamma \\\sin \alpha \sin \beta &-\cos \alpha \sin \beta &\cos \beta \end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Konventionen">Konventionen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=9" title="Abschnitt bearbeiten: Konventionen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Konventionen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Es gibt sechs verschiedene Möglichkeiten, die Achsen für eigentliche Eulerwinkel zu wählen. Bei allen ist die erste und die dritte Achse die gleiche. Die sechs Möglichkeiten sind: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> (extrinsisch): Standard-x-Konvention</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> (extrinsisch): Standard-y-Konvention</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c017ef8616d0c836e4b2a88c40931333c38dd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle z'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147ec60507e44a6d376237c0a16132cf7461cd62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.843ex;" alt="{\displaystyle y''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147ec60507e44a6d376237c0a16132cf7461cd62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.843ex;" alt="{\displaystyle y''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c017ef8616d0c836e4b2a88c40931333c38dd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle z'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (extrinsisch)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Kardan-Winkel">Kardan-Winkel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=10" title="Abschnitt bearbeiten: Kardan-Winkel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Kardan-Winkel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei den <i>Kardan-Winkeln</i> (nach <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>) oder auch <i>Tait-Bryan-Winkeln</i> (benannt nach <a href="/wiki/Peter_Guthrie_Tait" title="Peter Guthrie Tait">Peter Guthrie Tait</a> und <a href="/wiki/George_Hartley_Bryan" title="George Hartley Bryan">George Hartley Bryan</a>) erfolgen die drei Drehungen um drei verschiedene Achsen. Wie bei den eigentlichen Eulerwinkeln gibt es sechs mögliche Drehfolgen: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147ec60507e44a6d376237c0a16132cf7461cd62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.843ex;" alt="{\displaystyle y''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c017ef8616d0c836e4b2a88c40931333c38dd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle z'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc413768e660ba75642aaf6f2981fe7429fca586" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.228ex; height:2.509ex;" alt="{\displaystyle z''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (extrinsisch)</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c017ef8616d0c836e4b2a88c40931333c38dd19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.775ex; height:2.509ex;" alt="{\displaystyle z'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/147ec60507e44a6d376237c0a16132cf7461cd62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.843ex;" alt="{\displaystyle y''}"></span> (intrinsisch) bzw. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (extrinsisch)</li></ul> <div class="mw-heading mw-heading3"><h3 id="Roll-,_Nick-_und_Gierwinkel:_z-y′-x″-Konvention"><span id="Roll-.2C_Nick-_und_Gierwinkel:_z-y.E2.80.B2-x.E2.80.B3-Konvention"></span>Roll-, Nick- und Gierwinkel: z-y′-x″-Konvention</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=11" title="Abschnitt bearbeiten: Roll-, Nick- und Gierwinkel: z-y′-x″-Konvention" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Roll-, Nick- und Gierwinkel: z-y′-x″-Konvention"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Beschreibung">Beschreibung</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=12" title="Abschnitt bearbeiten: Beschreibung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Beschreibung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Taitbrianzyx.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taitbrianzyx.svg/220px-Taitbrianzyx.svg.png" decoding="async" width="220" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taitbrianzyx.svg/330px-Taitbrianzyx.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Taitbrianzyx.svg/440px-Taitbrianzyx.svg.png 2x" data-file-width="680" data-file-height="665" /></a><figcaption>Drehfolge z, y′, x″ (Gier-Nick-Roll)<br /> <span style="color:#00BFF0">blau</span>: raumfestes Koordinatensystem<br /> <span style="color:green">grün</span>: y'-Achse = Knotenlinie N(y′)<br /> <span style="color:red">rot</span>: körperfestes Koordinatensystem<br /> Anmerkung: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> ist hier negativ.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Plane.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Plane.svg/220px-Plane.svg.png" decoding="async" width="220" height="248" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Plane.svg/330px-Plane.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Plane.svg/440px-Plane.svg.png 2x" data-file-width="687" data-file-height="775" /></a><figcaption>Gier-, Nick- und Rollwinkel<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\psi ,\theta ,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ψ<!-- ψ --></mi> <mo>,</mo> <mi>θ<!-- θ --></mi> <mo>,</mo> <mi>φ<!-- φ --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\psi ,\theta ,\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad2f779c69e70e60614169474d1f8b6333e6f125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.001ex; height:2.843ex;" alt="{\displaystyle (\psi ,\theta ,\varphi )}"></span> als Lagewinkel eines Flugzeugs</figcaption></figure> <div class="sieheauch" role="navigation" style="font-style:italic;"><span class="sieheauch-text">Siehe auch</span>: <a href="/wiki/Roll-Nick-Gier-Winkel" title="Roll-Nick-Gier-Winkel">Roll-Nick-Gier-Winkel</a></div> <p>Die in der Luftfahrt, Schifffahrt und dem Automobilbau angewendeten und genormten (Luftfahrt: <a href="/wiki/DIN-Norm" title="DIN-Norm">DIN</a> 9300; Automobilbau: <a href="/wiki/DIN-Norm" title="DIN-Norm">DIN ISO</a> 8855) Drehfolgen gehören in die Gruppe der Tait-Bryan-Drehungen. In den Normen sind die Namen <a href="/wiki/Roll-Nick-Gier-Winkel" title="Roll-Nick-Gier-Winkel">Gier-, Nick- und Roll-Winkel</a> (engl. <i>yaw</i>, <i>pitch</i> and <i>roll angle</i>) für die drei Euler-Winkel vorgeschrieben. Durch die drei Drehungen wird das erdfeste <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xyz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3957b3fbfe29dfdb2f7cdad4e3d7fc2ea7be84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.573ex; height:2.009ex;" alt="{\displaystyle xyz}"></span>-System (engl. <i>world frame</i>) in das körperfeste <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle XYZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XYZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea07eba7954100995bf73897619ce2df498349e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.434ex; height:2.176ex;" alt="{\displaystyle XYZ}"></span>-Koordinatensystem (engl. <i>body frame</i>) gedreht. </p><p>Intrinsische Reihenfolge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.845ex; height:2.843ex;" alt="{\displaystyle y'}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c56431607b889d0f2fff3c7120a466db5aa2e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.509ex;" alt="{\displaystyle x''}"></span> (Gier-Nick-Roll-Winkel): </p> <ul><li>Mit dem im erdfesten System gemessenen <a href="/wiki/Gierwinkel" class="mw-redirect" title="Gierwinkel">Gierwinkel</a> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span></b> (auch Steuerkurs oder Azimut genannt) wird um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse gedreht. Die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-Achse wird zur Knotenachse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(y')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(y')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7864c87ad9296ac8065f82bfd86bac510995b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.718ex; height:3.009ex;" alt="{\displaystyle N(y')}"></span>.<br />Hauptwertebereich: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\psi \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mi>π<!-- π --></mi> <mo><</mo> <mi>ψ<!-- ψ --></mi> <mo>≤<!-- ≤ --></mo> <mi>π<!-- π --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\psi \leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9246950f79b6fd8ced4462ce6f38b7c231bb7e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.182ex; height:2.509ex;" alt="{\displaystyle -\pi <\psi \leq \pi }"></span>. Die Drehrichtung ist mathematisch positiv (gegen den Uhrzeigersinn)</li> <li>Mit dem gegen die Erdoberfläche (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-Ebene) gemessenen <a href="/wiki/Nickwinkel" class="mw-redirect" title="Nickwinkel">Nickwinkel</a> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span></b> wird um die Knotenachse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(y')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(y')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7864c87ad9296ac8065f82bfd86bac510995b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.718ex; height:3.009ex;" alt="{\displaystyle N(y')}"></span> gedreht. Es entsteht die fahrzeugfeste <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Achse.<br />Hauptwertebereich: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> <mo>≤<!-- ≤ --></mo> <mi>θ<!-- θ --></mi> <mo>≤<!-- ≤ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/563cd29aeaa853396000d286fa2f5f89dcba4f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.432ex; height:4.676ex;" alt="{\displaystyle -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}}"></span>. Die Drehrichtung ist mathematisch positiv.</li></ul> <ul><li>Der <a href="/wiki/Rollwinkel" class="mw-redirect" title="Rollwinkel">Rollwinkel</a> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span></b> (auch Wankwinkel genannt) beschreibt die Drehung um die fahrzeugfeste <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Achse. Es entstehen die fahrzeugfesten Achsen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>.<br />Hauptwertebereich: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\pi <\varphi \leq \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mi>π<!-- π --></mi> <mo><</mo> <mi>φ<!-- φ --></mi> <mo>≤<!-- ≤ --></mo> <mi>π<!-- π --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\pi <\varphi \leq \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27eb4930922e649015e37b8d782b6b1ad7d55b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.189ex; height:2.509ex;" alt="{\displaystyle -\pi <\varphi \leq \pi }"></span>. Die Drehrichtung ist mathematisch positiv.</li></ul> <p>Extrinsisch entspricht dies der Reihenfolge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> (Roll-Nick-Gier-Winkel). </p><p>Statt der Kleinbuchstaben <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ<!-- φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> werden auch die entsprechenden Großbuchstaben <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ψ<!-- Ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5471531a3fe80741a839bc98d49fae862a6439a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Θ<!-- Θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc927b19f46d005b4720db7a0f96cd5b6f1a0d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Theta }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ<!-- Φ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> verwendet. </p> <div class="mw-heading mw-heading4"><h4 id="Transformationsmatrizen">Transformationsmatrizen</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=13" title="Abschnitt bearbeiten: Transformationsmatrizen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Transformationsmatrizen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Koordinatentransformation vom körperfesten ins raumfeste Koordinatensystem wird durch die Matrix </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R_{GNR}&=R_{z}(\psi )\,R_{y}(\theta )\,R_{x}(\varphi )\\&={\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \\\cos \theta \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \\-\sin \theta &\sin \varphi \cos \theta &\cos \varphi \cos \theta \end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>N</mi> <mi>R</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ψ<!-- ψ --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ<!-- φ --></mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{GNR}&=R_{z}(\psi )\,R_{y}(\theta )\,R_{x}(\varphi )\\&={\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \\\cos \theta \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \\-\sin \theta &\sin \varphi \cos \theta &\cos \varphi \cos \theta \end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701a215b6e946c60fd92541cf85273044165f49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.325ex; margin-bottom: -0.18ex; width:85.062ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}R_{GNR}&=R_{z}(\psi )\,R_{y}(\theta )\,R_{x}(\varphi )\\&={\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \\\cos \theta \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \\-\sin \theta &\sin \varphi \cos \theta &\cos \varphi \cos \theta \end{pmatrix}}\end{aligned}}}"></span></dd></dl> <p>beschrieben. Die umgekehrte Transformation vom raumfesten ins körperfeste Koordinatensystem wird durch die Transponierte dieser Matrix beschrieben. (Eigentlich die Inverse, aber bei Drehmatrizen stimmt die inverse mit der transponierten Matrix überein.) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R_{GNR}^{T}&=R_{x}(\varphi )^{T}\,R_{y}(\theta )^{T}\,R_{z}(\psi )^{T}\\&={\begin{pmatrix}1&0&0\\0&\cos \varphi &\sin \varphi \\0&-\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &\sin \psi &0\\-\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>N</mi> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>φ<!-- φ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>θ<!-- θ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ψ<!-- ψ --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{GNR}^{T}&=R_{x}(\varphi )^{T}\,R_{y}(\theta )^{T}\,R_{z}(\psi )^{T}\\&={\begin{pmatrix}1&0&0\\0&\cos \varphi &\sin \varphi \\0&-\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &\sin \psi &0\\-\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34a91692e8a757a07bd301afb7f3d1ffd0dade3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:85.069ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}R_{GNR}^{T}&=R_{x}(\varphi )^{T}\,R_{y}(\theta )^{T}\,R_{z}(\psi )^{T}\\&={\begin{pmatrix}1&0&0\\0&\cos \varphi &\sin \varphi \\0&-\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &\sin \psi &0\\-\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}\end{aligned}}}"></span></dd></dl> <p>Das bedeutet: Hat der Vektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> im raumfesten System die Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704b7ad1ece77840fde455daa6d2e51e64282b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.009ex;" alt="{\displaystyle v_{x}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/922aa64dad09633a401e14be9b4389795835cd8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.177ex; height:2.343ex;" alt="{\displaystyle v_{y}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b553b9ecda9fc7f5c9da6f14a870e219080984e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle v_{z}}"></span> und im körperfesten System die Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9c7e7d9c4a791e93c24b375504a3e72b558d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.76ex; height:2.009ex;" alt="{\displaystyle v_{X}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0879295ac9f1b5878e804a085b5e50e58c6635c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.614ex; height:2.009ex;" alt="{\displaystyle v_{Y}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{Z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{Z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af56e473b93e8469fa8072b3a018cb26f91e34c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle v_{Z}}"></span>, so gilt </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}&={\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \\\cos \theta \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \\-\sin \theta &\sin \varphi \cos \theta &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}&={\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \\\cos \theta \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \\-\sin \theta &\sin \varphi \cos \theta &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12cb1182c9af9281f6b6ce6300d6b07306ec1cdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.771ex; margin-bottom: -0.234ex; width:93.764ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}&={\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\cos \theta &0&\sin \theta \\0&1&0\\-\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \varphi &-\sin \varphi \\0&\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi \\\cos \theta \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi \\-\sin \theta &\sin \varphi \cos \theta &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}\end{aligned}}}"></span></dd></dl> <p>und </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}&={\begin{pmatrix}1&0&0\\0&\cos \varphi &\sin \varphi \\0&-\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &\sin \psi &0\\-\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}&={\begin{pmatrix}1&0&0\\0&\cos \varphi &\sin \varphi \\0&-\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &\sin \psi &0\\-\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97752361ab1bf98c5ca3051c516e3d0285d86d9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:93.771ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}{\begin{pmatrix}v_{X}\\v_{Y}\\v_{Z}\end{pmatrix}}&={\begin{pmatrix}1&0&0\\0&\cos \varphi &\sin \varphi \\0&-\sin \varphi &\cos \varphi \end{pmatrix}}{\begin{pmatrix}\cos \theta &0&-\sin \theta \\0&1&0\\\sin \theta &0&\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &\sin \psi &0\\-\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}v_{x}\\v_{y}\\v_{z}\end{pmatrix}}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Anwendungsbeispiel">Anwendungsbeispiel</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=14" title="Abschnitt bearbeiten: Anwendungsbeispiel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Anwendungsbeispiel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Der Gewichtsvektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241bb14fd31373ea9f4a602cc59e4ec1da27b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:3.009ex;" alt="{\displaystyle {\vec {G}}}"></span> hat im erdfesten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xyz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3957b3fbfe29dfdb2f7cdad4e3d7fc2ea7be84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.573ex; height:2.009ex;" alt="{\displaystyle xyz}"></span>-Koordinatensystem nur eine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Komponente (in Richtung Erdmittelpunkt): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {G}}_{E}={\begin{pmatrix}0\\0\\mg\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <mi>g</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {G}}_{E}={\begin{pmatrix}0\\0\\mg\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4864cf25760633cbc1c68aa25344754f830dcb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:14.388ex; height:9.509ex;" alt="{\displaystyle {\vec {G}}_{E}={\begin{pmatrix}0\\0\\mg\end{pmatrix}}}"></span></dd></dl> <p>Die Transformation ins flugzeugfeste Koordinatensystem geschieht dann durch Multiplikation des erdfesten Gewichtsvektors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {G}}_{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {G}}_{E}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18dec010d59c8bfd1b56dc24e6e588fa9937bd25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.315ex; height:3.343ex;" alt="{\displaystyle {\vec {G}}_{E}}"></span> mit der Transformationsmatrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{GNR}^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>N</mi> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{GNR}^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ebaf46dde49dd639adb7be9e9feff3ee0ac06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.994ex; height:3.176ex;" alt="{\displaystyle R_{GNR}^{T}}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\vec {G}}_{F}&=R_{GNR}^{T}\,{\vec {G}}_{E}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}0\\0\\mg\end{pmatrix}}\\&={\begin{pmatrix}\ldots &\ldots &-\sin \theta \\\ldots &\ldots &\sin \varphi \cos \theta \\\ldots &\ldots &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}0\\0\\mg\end{pmatrix}}={\begin{pmatrix}-\sin \theta \\\sin \varphi \cos \theta \\\cos \varphi \cos \theta \end{pmatrix}}mg\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mi>N</mi> <mi>R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>+</mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>ψ<!-- ψ --></mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <mi>g</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>…<!-- … --></mo> </mtd> <mtd> <mo>…<!-- … --></mo> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mo>…<!-- … --></mo> </mtd> <mtd> <mo>…<!-- … --></mo> </mtd> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mo>…<!-- … --></mo> </mtd> <mtd> <mo>…<!-- … --></mo> </mtd> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>m</mi> <mi>g</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−<!-- − --></mo> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>φ<!-- φ --></mi> <mi>cos</mi> <mo>⁡<!-- --></mo> <mi>θ<!-- θ --></mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mi>m</mi> <mi>g</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\vec {G}}_{F}&=R_{GNR}^{T}\,{\vec {G}}_{E}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}0\\0\\mg\end{pmatrix}}\\&={\begin{pmatrix}\ldots &\ldots &-\sin \theta \\\ldots &\ldots &\sin \varphi \cos \theta \\\ldots &\ldots &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}0\\0\\mg\end{pmatrix}}={\begin{pmatrix}-\sin \theta \\\sin \varphi \cos \theta \\\cos \varphi \cos \theta \end{pmatrix}}mg\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91dd73f6e3014e12c5a7776ef1f2bb85ef7019de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:90.339ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}{\vec {G}}_{F}&=R_{GNR}^{T}\,{\vec {G}}_{E}\\&={\begin{pmatrix}\cos \theta \cos \psi &\cos \theta \sin \psi &-\sin \theta \\\sin \varphi \sin \theta \cos \psi -\cos \varphi \sin \psi &\sin \varphi \sin \theta \sin \psi +\cos \varphi \cos \psi &\sin \varphi \cos \theta \\\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi &\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}0\\0\\mg\end{pmatrix}}\\&={\begin{pmatrix}\ldots &\ldots &-\sin \theta \\\ldots &\ldots &\sin \varphi \cos \theta \\\ldots &\ldots &\cos \varphi \cos \theta \end{pmatrix}}{\begin{pmatrix}0\\0\\mg\end{pmatrix}}={\begin{pmatrix}-\sin \theta \\\sin \varphi \cos \theta \\\cos \varphi \cos \theta \end{pmatrix}}mg\end{aligned}}}"></span></dd></dl> <p>Physikalisch richtig wirkt die Gewichtskraft <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">→<!-- → --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/241bb14fd31373ea9f4a602cc59e4ec1da27b10b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:3.009ex;" alt="{\displaystyle {\vec {G}}}"></span> bei vorhandenem Nickwinkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> im Flugzeug beispielsweise auch nach hinten (in negative <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>-Richtung). </p> <div class="mw-heading mw-heading2"><h2 id="Matrix-Herleitung_im_allgemeinen_Fall">Matrix-Herleitung im allgemeinen Fall</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=15" title="Abschnitt bearbeiten: Matrix-Herleitung im allgemeinen Fall" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Matrix-Herleitung im allgemeinen Fall"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Für eine beliebige Wahl der Drehachsenreihenfolge kann die sich ergebende Drehmatrix durch die Zuhilfenahme des folgenden Zusammenhangs einfach hergeleitet werden (aktive Drehungen):<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Die Drehmatrizen um die globalen Achsen sind bekannt. Wenn nun um eine bereits gedrehte Achse erneut gedreht werden soll, dann entspricht das der Drehmatrix um die entsprechende globale Achse, allerdings in einer transformierten <a href="/wiki/Basis_(Vektorraum)" title="Basis (Vektorraum)">Vektorbasis</a>. Die Transformationsmatrix (<a href="/wiki/Basiswechselmatrix" class="mw-redirect" title="Basiswechselmatrix">Basiswechselmatrix</a>) ist dabei gerade die vorhergehende Drehung. </p><p>Seien <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> zwei Drehmatrizen um die beiden globalen Achsen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. Zur Berechnung der Drehmatrix zu der Reihenfolge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,H^{\prime })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,H^{\prime })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d2b7fceafc853253d6a459d7779c97f35fe952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.458ex; height:3.009ex;" alt="{\displaystyle (G,H^{\prime })}"></span> beobachtet man, dass die Drehmatrix für die zweite Drehung um <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> der basistransformierten Matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {B}}=ABA^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">~<!-- ~ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>A</mi> <mi>B</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {B}}=ABA^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbc0be8cc054493743fb953f1da7ea97753618a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.445ex; height:2.676ex;" alt="{\displaystyle {\tilde {B}}=ABA^{-1}}"></span> entsprechen muss. Dadurch erhält man für die resultierende Gesamtdrehmatrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\tilde {B}}A=ABA^{-1}A=AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">~<!-- ~ --></mo> </mover> </mrow> </mrow> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\tilde {B}}A=ABA^{-1}A=AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85c1fa5c0236f903bafa3cd48bdcb92d80e65240" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.402ex; height:2.676ex;" alt="{\displaystyle C={\tilde {B}}A=ABA^{-1}A=AB}"></span>. Für eine größere Anzahl von Drehungen erfolgt der Nachweis analog. </p><p>Bei drei aktiven Drehungen (A wird zuerst ausgeführt, dann B, dann C) ergibt sich die Gesamtdrehmatrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=((AB)\cdot C\cdot {(AB)}^{T})\cdot (A\cdot B\cdot A^{T})\cdot A=ABC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>C</mi> <mo>⋅<!-- ⋅ --></mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅<!-- ⋅ --></mo> <mi>B</mi> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅<!-- ⋅ --></mo> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mi>B</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=((AB)\cdot C\cdot {(AB)}^{T})\cdot (A\cdot B\cdot A^{T})\cdot A=ABC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d8c79a33ed005137246150fea02d5ac4de22fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.259ex; height:3.343ex;" alt="{\displaystyle D=((AB)\cdot C\cdot {(AB)}^{T})\cdot (A\cdot B\cdot A^{T})\cdot A=ABC}"></span> unter Verwendung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{T}=A^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{T}=A^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c78ac72f041be20d3e6c682a4b843391c6666d68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.307ex; height:2.676ex;" alt="{\displaystyle A^{T}=A^{-1}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(A\cdot B)}^{T}=B^{T}\cdot A^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅<!-- ⋅ --></mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>⋅<!-- ⋅ --></mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(A\cdot B)}^{T}=B^{T}\cdot A^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce6d21425beeabec3ece0962505a1a77edb0177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.448ex; height:3.343ex;" alt="{\displaystyle {(A\cdot B)}^{T}=B^{T}\cdot A^{T}}"></span>. </p><p>Durch diese Darstellung ergibt sich, dass sich die Drehmatrix für eine beliebige Drehreihenfolge in nacheinander gedrehten Achsen durch die einfache Multiplikation von Drehmatrizen um globale Koordinatenachsen ergibt – allerdings in umgekehrter Reihenfolge. </p> <div class="mw-heading mw-heading2"><h2 id="Ergebnis,_Interpretation"><span id="Ergebnis.2C_Interpretation"></span>Ergebnis, Interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=16" title="Abschnitt bearbeiten: Ergebnis, Interpretation" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=16" title="Quellcode des Abschnitts bearbeiten: Ergebnis, Interpretation"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Das erhaltene Koordinatensystem mit den Achsen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b827334f818c61a8b86e5d1ea9a3203c8e074b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.134ex; height:2.509ex;" alt="{\displaystyle X''}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Y</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3d2e9a4ff5f48169a01db9117bb418865177a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:3.037ex; height:2.343ex;" alt="{\displaystyle Y''}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e8aa693bfc0f7c74f19f5c1934808f88ec97e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle Z''}"></span> ist das sogenannte körperfeste System. Die Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ<!-- ϕ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> geben dabei die Lage der <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Z</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07e8aa693bfc0f7c74f19f5c1934808f88ec97e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.846ex; height:2.509ex;" alt="{\displaystyle Z''}"></span>-Achse gegenüber dem körperfesten System an („Drehung“ und „Kippung“); der Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ<!-- ψ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> beschreibt die Eigendrehung des Körpers um sie. Dem entsprechen folgende Namenskonventionen: </p> <ul><li><a href="/wiki/Flugzeug#Flugsteuerung" title="Flugzeug">Flugsteuerung</a> (<a href="/wiki/Roll-Pitch-Yaw-Winkel" class="mw-redirect" title="Roll-Pitch-Yaw-Winkel">Rollwinkel, Nickwinkel, Gierwinkel</a>)</li> <li><a href="/wiki/Kreisel" title="Kreisel">Kreiseltheorie</a>: <i><a href="/wiki/Pr%C3%A4zession" title="Präzession">Präzession</a></i>, <i><a href="/wiki/Nutation_(Physik)" title="Nutation (Physik)">Nutation</a></i> und <i>Spin</i> oder <i>Eigenrotation</i></li> <li><i><a href="/wiki/Azimut" title="Azimut">Azimut</a></i>, <i><a href="/wiki/H%C3%B6henwinkel" class="mw-redirect" title="Höhenwinkel">Höhenwinkel</a></i> oder <i>Elevation</i> und <i>Rotation</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="Mathematische_Eigenschaften">Mathematische Eigenschaften</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=17" title="Abschnitt bearbeiten: Mathematische Eigenschaften" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=17" title="Quellcode des Abschnitts bearbeiten: Mathematische Eigenschaften"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die Abbildung, die den Euler-Winkeln die zugehörige Drehmatrix zuordnet, besitzt <a href="/wiki/Kritischer_Punkt_(Mathematik)" title="Kritischer Punkt (Mathematik)">kritische Punkte</a>, in denen diese Zuordnung nicht lokal umkehrbar ist und man von einem <a href="/wiki/Gimbal_Lock" title="Gimbal Lock">Gimbal Lock</a> spricht. Im Fall der og. x- oder y-Konvention tritt dieser stets dann auf, wenn der zweite Rotationswinkel gleich null wird und der <a href="/wiki/Axialvektor" class="mw-redirect" title="Axialvektor">Drehvektor</a> der ersten Drehung damit derselbe ist wie der Drehvektor der zweiten Drehung. Das aber bedeutet, dass es für eine Rotation um die <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>-Achse beliebig viele Euler-Winkel mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =Z+Z'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mi>Z</mi> <mo>+</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =Z+Z'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717d794bf2808130b1d3eb2e3306b8a6df574c3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.5ex; height:2.676ex;" alt="{\displaystyle \alpha =Z+Z'}"></span> gibt. </p><p>Bei der Definition der Lagewinkel nach der Luftfahrtnorm liegen die kritischen Punkte bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \theta =\pm {\frac {\pi }{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>θ<!-- θ --></mi> <mo>=</mo> <mo>±<!-- ± --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π<!-- π --></mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \theta =\pm {\frac {\pi }{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7bd6a11fbbdd5afeb2f95e96cdf0b42ed60cc01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.775ex; height:3.176ex;" alt="{\displaystyle \textstyle \theta =\pm {\frac {\pi }{2}}}"></span>. </p><p>Nach <a href="/wiki/Kurt_Magnus_(Ingenieur)" title="Kurt Magnus (Ingenieur)">Kurt Magnus</a><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> ist bei Kreisel-Problemen, bei denen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ<!-- θ --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7bc6e34b53e0e8a8815159c356b1acccf7ea24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta =0}"></span> möglich ist, die Beschreibung mit Eulerwinkeln (x-Konvention) nicht möglich und man verwendet stattdessen Kardanwinkel. </p> <div class="mw-heading mw-heading2"><h2 id="Nachteile,_Alternativen"><span id="Nachteile.2C_Alternativen"></span>Nachteile, Alternativen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=18" title="Abschnitt bearbeiten: Nachteile, Alternativen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=18" title="Quellcode des Abschnitts bearbeiten: Nachteile, Alternativen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zur Darstellung von Drehungen haben Euler-Winkel mehrere Nachteile: </p> <ul><li>Die oben erwähnte Singularität führt dazu, dass eine einzige Drehung durch unterschiedliche Euler-Drehungen ausgedrückt werden kann. Dies führt zu einem Phänomen, das als <a href="/wiki/Gimbal_Lock" title="Gimbal Lock">Gimbal Lock</a> bekannt ist.</li> <li>Die korrekte Kombination von Drehungen im Euler-System ist nicht intuitiv anzugeben, da sich die Drehachsen verändern.</li></ul> <p>Anstatt mit den Eulerwinkeln kann man jede Drehung auch durch einen Vektor angeben, der durch seine Orientierung die Lage der Achse und den Drehsinn angibt, und durch seinen Betrag den Drehwinkel (siehe z. B.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> oder <a href="/wiki/Orthogonaler_Tensor" title="Orthogonaler Tensor">Orthogonaler Tensor</a>). Eine andere Möglichkeit, die Orientierung zu beschreiben und teils diese Nachteile zu umgehen, sind <a href="/wiki/Quaternionen" class="mw-redirect" title="Quaternionen">Quaternionen</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Anwendungen">Anwendungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=19" title="Abschnitt bearbeiten: Anwendungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=19" title="Quellcode des Abschnitts bearbeiten: Anwendungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Datei:MAUD-MTEX-TiAl-hasylab-2003-Liss.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/MAUD-MTEX-TiAl-hasylab-2003-Liss.png/220px-MAUD-MTEX-TiAl-hasylab-2003-Liss.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/MAUD-MTEX-TiAl-hasylab-2003-Liss.png/330px-MAUD-MTEX-TiAl-hasylab-2003-Liss.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/MAUD-MTEX-TiAl-hasylab-2003-Liss.png/440px-MAUD-MTEX-TiAl-hasylab-2003-Liss.png 2x" data-file-width="585" data-file-height="388" /></a><figcaption>Textur Polfiguren von gamma-TiAl in einer alpha2-gamma Zweiphasenlegierung.<sup id="cite_ref-Liss_10-0" class="reference"><a href="#cite_note-Liss-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <table class="wikitable float-right infobox toptextcells" style="font-size:90%; vertical-align:top; text-align:left; empty-cells:collapse; min-width:200px; max-width:350px;"> <tbody><tr valign="top"> <th colspan="2" align="center"><a href="/wiki/Roll-Nick-Gier-Winkel" title="Roll-Nick-Gier-Winkel">Roll-Nick-Gier-Winkel</a> (<a class="mw-selflink selflink">Eulerwinkel</a>) </th></tr> <tr> <td colspan="2" align="center" style="border-bottom: 0px"><figure class="mw-default-size mw-halign-center skin-invert-image" typeof="mw:File/Frameless"><a href="/wiki/Datei:Roll_pitch_yaw_gravitation_center_de.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Roll_pitch_yaw_gravitation_center_de.png/150px-Roll_pitch_yaw_gravitation_center_de.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Roll_pitch_yaw_gravitation_center_de.png/225px-Roll_pitch_yaw_gravitation_center_de.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Roll_pitch_yaw_gravitation_center_de.png/300px-Roll_pitch_yaw_gravitation_center_de.png 2x" data-file-width="1080" data-file-height="1080" /></a><figcaption></figcaption></figure> </td></tr> <tr valign="top"> <td style="border-top: 0px; border-right: 0px"><span style="visibility:hidden;">0</span> <a href="/wiki/Rotationsachse" title="Rotationsachse">Rotationsachsen</a>: </td> <td style="border-top: 0px; border-left: 0px">Bewegung: </td></tr> <tr> <td>↙ <a href="/wiki/L%C3%A4ngsachse" title="Längsachse">Längsachse <small>(Roll-/Wankachse)</small></a>: </td> <td><a href="/wiki/Rollen_(L%C3%A4ngsachse)" title="Rollen (Längsachse)">Rollen</a>, <a href="/wiki/Wanken" title="Wanken">Wanken</a> </td></tr> <tr> <td>↖ <a href="/wiki/Querachse" title="Querachse">Querachse <small>(Nickachse)</small></a>: </td> <td><a href="/wiki/Querachse" title="Querachse">Nicken, Stampfen</a> </td></tr> <tr> <td>↓ <a href="/wiki/Gierachse" title="Gierachse">Vertikalachse <small>(Gierachse)</small></a>: </td> <td><a href="/wiki/Gierachse" title="Gierachse">Gieren <small>(Schlingern)</small></a> </td></tr></tbody></table> <p>In der <a href="/wiki/Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">Theoretischen Physik</a> werden die eulerschen Winkel zur Beschreibung des <a href="/wiki/Starrer_K%C3%B6rper" title="Starrer Körper">Starren Körpers</a> benutzt. Eine praktische Anwendung ergibt die bekannte <a href="/wiki/Kardanische_Aufh%C3%A4ngung" title="Kardanische Aufhängung">kardanische Aufhängung</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> der technischen Mechanik. </p><p>Bei bewegten Objekten (z. B. Fahrzeugen oder Flugzeugen) bezeichnet man die Euler-Winkel der Hauptlagen als <a href="/wiki/Roll-Nick-Gier-Winkel" title="Roll-Nick-Gier-Winkel">Roll-Nick-Gier-Winkel</a>. </p><p>In der <a href="/wiki/Kristallographie" title="Kristallographie">Kristallographie</a> werden die eulerschen Winkel zur Beschreibung der Kreise des <a href="/wiki/Einkristalldiffraktometer" title="Einkristalldiffraktometer">Einkristalldiffraktometers</a> (mit einer kardanischen Aufhängung aus zwei senkrecht aufeinander stehenden Drehkreisen, die den Euler-Winkeln entspricht und Euler-Wiege genannt wird)<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> und zur Beschreibung der <a href="/wiki/Orientierungsdichteverteilungsfunktion" title="Orientierungsdichteverteilungsfunktion">Orientierungsdichteverteilungsfunktion</a> von <a href="/wiki/Textur_(Kristallographie)" title="Textur (Kristallographie)">Texturen</a> verwendet. </p><p>In der <a href="/wiki/Astronomie" title="Astronomie">Astronomie</a> sind die eulerschen Winkel unter anderen Bezeichnungen als <i><a href="/wiki/Bahnelement" title="Bahnelement">Bahnelement</a></i> eines <a href="/wiki/Astronomisches_Objekt" title="Astronomisches Objekt">Objekts</a> geläufig. </p><p>In der <a href="/wiki/Computergrafik" title="Computergrafik">Computergrafik</a> werden die eulerschen Winkel zur Beschreibung der Orientierung eines Objektes verwendet. </p><p>In der Festkörper-<a href="/wiki/NMR-Spektroskopie" class="mw-redirect" title="NMR-Spektroskopie">NMR</a> werden die eulerschen Winkel zur theoretischen Beschreibung und zur Simulation von Spektren benutzt. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=20" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=20" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Herbert Goldstein</a>: <i>Klassische Mechanik</i>, Wiley-VCH 2006</li> <li>Georg Rill, Thomas Schaeffer: <i>Grundlagen und Methodik der Mehrkörpersimulation.</i> Vieweg + Teubner, Wiesbaden 2010, <a href="/wiki/Spezial:ISBN-Suche/9783658160098" class="internal mw-magiclink-isbn">ISBN 978-3-658-16009-8</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=21" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=21" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noresize noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Commons"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div><b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Euler_angles?uselang=de"><span lang="en">Commons</span>: Euler angles</a></span></b> – Album mit Bildern, Videos und Audiodateien</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><div class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wikibooks"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span></div><b><a href="https://de.wikibooks.org/wiki/Mechanik_starrer_K%C3%B6rper#Die_eulerschen_Winkel" class="extiw" title="b:Mechanik starrer Körper">Wikibooks: <i>Die Mechanik starrer Körper</i>, Die eulerschen Winkel</a></b> – Lern- und Lehrmaterialien</div> <ul><li>Eric Weisstein: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/EulerAngles.html"><i>Euler Angles</i>.</a> In: <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit&section=22" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Eulersche_Winkel&action=edit&section=22" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-goldstein-1"><span class="mw-cite-backlink"><a href="#cite_ref-goldstein_1-0">↑</a></span> <span class="reference-text"> Herbert Goldstein, Charles P. Poole Jr., John L. Safko: <cite style="font-style:italic">Klassische Mechanik</cite>. 3., vollständig und erweiterte Auflage. WILEY-VCH, 2006, <a href="/wiki/Spezial:ISBN-Suche/3527405895" class="internal mw-magiclink-isbn">ISBN 3-527-40589-5</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>161<span style="display:inline-block;width:.2em"> </span>ff</span>. (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=bbsCZN1U17cC&pg=PA161#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.au=Herbert+Goldstein%2C+Charles+P.+Poole+Jr.%2C+John+L.+Safko&rft.btitle=Klassische+Mechanik&rft.date=2006&rft.edition=3.%2C+vollst%C3%A4ndig+und+erweiterte&rft.genre=book&rft.isbn=3527405895&rft.pages=161+ff&rft.pub=WILEY-VCH" style="display:none"> </span>: „Wir können die Transformation von einem kartesischen Koordinatensystem auf ein anderes mittels dreier aufeinander folgender Drehungen durchführen, die in einer bestimmten Reihenfolge erfolgen müssen.“</span> </li> <li id="cite_note-ChengGupta89-2"><span class="mw-cite-backlink"><a href="#cite_ref-ChengGupta89_2-0">↑</a></span> <span class="reference-text">Hui Cheng, K. C. Gupta: <cite style="font-style:italic">A Historical Note on Final Rotations</cite>. In: <cite style="font-style:italic">Journal of Applied Mechanics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>56</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, März 1989, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>139–145</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1115/1.3176034">10.1115/1.3176034</a></span> (<a rel="nofollow" class="external text" href="https://pdfs.semanticscholar.org/2d4f/d2652bb1a20cbe19229cea6acd43f6caee46.pdf">Volltext</a> [PDF; abgerufen am 13. Dezember 2018]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.atitle=A+Historical+Note+on+Final+Rotations&rft.au=Hui+Cheng%2C+K.+C.+Gupta&rft.date=1989-03&rft.doi=10.1115%2F1.3176034&rft.genre=journal&rft.issue=1&rft.jtitle=Journal+of+Applied+Mechanics&rft.pages=139-145&rft.volume=56" style="display:none"> </span></span> </li> <li id="cite_note-E478-3"><span class="mw-cite-backlink"><a href="#cite_ref-E478_3-0">↑</a></span> <span class="reference-text">L. Eulerus: <cite style="font-style:italic">Formulae generales pro translatione quacunque corporum rigidorum</cite>. In: <cite style="font-style:italic">Novi Commentarii academiae scientiarum Petropolitanae</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>20 (1775)</span>, 1776, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>189–207</span> (<a rel="nofollow" class="external text" href="http://www.17centurymaths.com/contents//euler/e478tr.pdf">Online</a> mit englischer Übersetzung [PDF; <span style="white-space:nowrap">188<span style="display:inline-block;width:.2em"> </span>kB</span>; abgerufen am 13. Dezember 2018] <i>translatio</i> ist hier eine beliebige <a href="/wiki/Bewegung_(Mathematik)" title="Bewegung (Mathematik)">Bewegung</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.atitle=Formulae+generales+pro+translatione+quacunque+corporum+rigidorum&rft.au=L.+Eulerus&rft.btitle=Novi+Commentarii+academiae+scientiarum+Petropolitanae&rft.date=1776&rft.genre=book&rft.pages=189-207&rft.volume=20+%281775%29" style="display:none"> </span></span> </li> <li id="cite_note-E479-4"><span class="mw-cite-backlink"><a href="#cite_ref-E479_4-0">↑</a></span> <span class="reference-text">L. Eulerus: <cite style="font-style:italic">Nova methodus motum corporum rigidorum determinandi</cite>. In: <cite style="font-style:italic">Novi Commentarii academiae scientiarum Petropolitanae</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>20 (1775)</span>, 1776, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>208–238</span> (<a rel="nofollow" class="external text" href="http://math.dartmouth.edu/~euler/docs/originals/E479.pdf">Online</a> [PDF; <span style="white-space:nowrap">1,6<span style="display:inline-block;width:.2em"> </span>MB</span>; abgerufen am 13. Dezember 2018] die Abbildungsgleichungen stehen in §13).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.atitle=Nova+methodus+motum+corporum+rigidorum+determinandi&rft.au=L.+Eulerus&rft.btitle=Novi+Commentarii+academiae+scientiarum+Petropolitanae&rft.date=1776&rft.genre=book&rft.pages=208-238&rft.volume=20+%281775%29" style="display:none"> </span></span> </li> <li id="cite_note-E825-5"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-E825_5-0">a</a></sup> <sup><a href="#cite_ref-E825_5-1">b</a></sup> <sup><a href="#cite_ref-E825_5-2">c</a></sup></span> <span class="reference-text">Leonardus Eulerus: <cite style="font-style:italic">De motu corporum circa punctum fixum mobilium</cite>. In: <cite style="font-style:italic">Leonardi Euleri opera postuma mathematica et physica : anno MDCCCXLIV detecta</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>2</span>, 1862, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>43–62</span> (<a rel="nofollow" class="external text" href="http://math.dartmouth.edu/~euler/docs/originals/E825.pdf">Online</a> [PDF; <span style="white-space:nowrap">1,4<span style="display:inline-block;width:.2em"> </span>MB</span>; abgerufen am 13. Dezember 2018]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.atitle=De+motu+corporum+circa+punctum+fixum+mobilium&rft.au=Leonardus+Eulerus&rft.btitle=Leonardi+Euleri+opera+postuma+mathematica+et+physica+%3A+anno+MDCCCXLIV+detecta&rft.date=1862&rft.genre=book&rft.pages=43-62&rft.volume=2" style="display:none"> </span> – <a rel="nofollow" class="external text" href="http://math.dartmouth.edu/~euler/docs/originals/E825.figure.pdf">zugehörige Abbildungen</a></span> </li> <li id="cite_note-Lagrange-6"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Lagrange_6-0">a</a></sup> <sup><a href="#cite_ref-Lagrange_6-1">b</a></sup> <sup><a href="#cite_ref-Lagrange_6-2">c</a></sup></span> <span class="reference-text">La Grange: <cite style="font-style:italic">Analytische Mechanik</cite>. Göttingen 1797 (<a rel="nofollow" class="external text" href="https://archive.org/download/analytischemech00murhgoog/analytischemech00murhgoog.pdf">Scan</a> [PDF; abgerufen am 13. Dezember 2018] Originaltitel: <cite style="font-style:italic">Mécanique Analytique</cite>. 1788. Übersetzt von Friedrich Wilhelm August Murhard).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.au=La+Grange&rft.btitle=Analytische+Mechanik&rft.date=1797&rft.genre=book&rft.place=G%C3%B6ttingen" style="display:none"> </span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text">Ein ausführlicher Beweis findet sich in: G. Fischer: <i>Lernbuch Lineare Algebra und Analytische Geometrie.</i> 2. Auflage 2012, in 5.3.6</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><a href="#cite_ref-8">↑</a></span> <span class="reference-text">Magnus, Kreisel, Springer, 1971, S. 32</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><a href="#cite_ref-9">↑</a></span> <span class="reference-text">Herbert Goldstein, Charles P. Poole, Jr., John L. Safko, Sr.: <cite style="font-style:italic">Klassische Mechanik</cite>. 3. Auflage. John Wiley & Sons, 2012, <a href="/wiki/Spezial:ISBN-Suche/9783527405893" class="internal mw-magiclink-isbn">ISBN 978-3-527-40589-3</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.au=Herbert+Goldstein%2C+Charles+P.+Poole%2C+Jr.%2C+...&rft.btitle=Klassische+Mechanik&rft.date=2012&rft.edition=3&rft.genre=book&rft.isbn=9783527405893&rft.pub=John+Wiley+%26+Sons" style="display:none"> </span></span> </li> <li id="cite_note-Liss-10"><span class="mw-cite-backlink"><a href="#cite_ref-Liss_10-0">↑</a></span> <span class="reference-text">Liss KD, Bartels A, Schreyer A, Clemens H: <cite style="font-style:italic">High energy X-rays: A tool for advanced bulk investigations in materials science and physics</cite>. In: <cite style="font-style:italic">Textures Microstruct.</cite> 35. Jahrgang, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>3/4</span>, 2003, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>219–52</span>, <a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1080/07303300310001634952">10.1080/07303300310001634952</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Eulersche+Winkel&rft.atitle=High+energy+X-rays%3A+A+tool+for+advanced+bulk+investigations+in+materials+science+and+physics&rft.au=Liss+KD%2C+Bartels+A%2C+Schreyer+A%2C+...&rft.date=2003&rft.doi=10.1080%2F07303300310001634952&rft.genre=journal&rft.issue=3%2F4&rft.jtitle=Textures+Microstruct.&rft.pages=219-52&rft.volume=35.+Jahrgang" style="display:none"> </span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><a href="#cite_ref-11">↑</a></span> <span class="reference-text">Der Zusammenhang zwischen den eulerschen Winkeln und der kardanischen Aufhängung ist u. a. in Kapitel 11.7 des folgenden Buches dargestellt: U. Krey, A. Owen: <i>Basic Theoretical Physics – A Concise Overview.</i> Springer-Verlag, Berlin 2007.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><a href="#cite_ref-12">↑</a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.spektrum.de/lexikon/geowissenschaften/euler-wiege/4403">Euler-Wiege, Lexikon der Geowissenschaften, Spektrum</a></span> </li> </ol> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="normdaten"> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Normdaten (Sachbegriff): <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="plainlinks-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4489302-4">4489302-4</a></span> <span class="noprint">(<a rel="nofollow" class="external text" href="https://lobid.org/gnd/4489302-4">lobid</a>, <a rel="nofollow" class="external text" href="https://swb.bsz-bw.de/DB=2.104/SET=1/TTL=1/CMD?retrace=0&trm_old=&ACT=SRCHA&IKT=2999&SRT=RLV&TRM=4489302-4">OGND</a><span class="metadata">, <a rel="nofollow" class="external text" href="https://prometheus.lmu.de/gnd/4489302-4">AKS</a></span>)</span> | <a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <span class="plainlinks-print"><a rel="nofollow" class="external text" href="https://lccn.loc.gov/sh85045551">sh85045551</a></span> <span class="metadata"></span></div> </div></div></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Eulersche_Winkel&oldid=243597841">https://de.wikipedia.org/w/index.php?title=Eulersche_Winkel&oldid=243597841</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Analytische_Geometrie" title="Kategorie:Analytische Geometrie">Analytische Geometrie</a></li><li><a href="/wiki/Kategorie:Kristallographie" title="Kategorie:Kristallographie">Kristallographie</a></li><li><a href="/wiki/Kategorie:Klassische_Mechanik" title="Kategorie:Klassische Mechanik">Klassische Mechanik</a></li><li><a href="/wiki/Kategorie:Winkel" title="Kategorie:Winkel">Winkel</a></li><li><a href="/wiki/Kategorie:Leonhard_Euler_als_Namensgeber" title="Kategorie:Leonhard Euler als Namensgeber">Leonhard Euler als Namensgeber</a></li><li><a href="/wiki/Kategorie:Koordinatensystem" title="Kategorie:Koordinatensystem">Koordinatensystem</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Versteckte Kategorie: <ul><li><a href="/wiki/Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Cite_journal/tempor%C3%A4r" title="Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Cite journal/temporär">Wikipedia:Vorlagenfehler/Vorlage:Cite journal/temporär</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Eulersche+Winkel" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. Es ist jedoch nicht zwingend erforderlich."><span>Benutzerkonto erstellen</span></a></li><li id="pt-login" class="mw-list-item"><a href="/w/index.php?title=Spezial:Anmelden&returnto=Eulersche+Winkel" title="Anmelden ist zwar keine Pflicht, wird aber gerne gesehen. [o]" accesskey="o"><span>Anmelden</span></a></li> </ul> </div> </nav> <div id="left-navigation"> <nav id="p-namespaces" class="mw-portlet mw-portlet-namespaces vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-namespaces-label" > <h3 id="p-namespaces-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Namensräume</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected mw-list-item"><a href="/wiki/Eulersche_Winkel" title="Seiteninhalt anzeigen [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="mw-list-item"><a href="/wiki/Diskussion:Eulersche_Winkel" rel="discussion" title="Diskussion zum Seiteninhalt [t]" accesskey="t"><span>Diskussion</span></a></li> </ul> </div> </nav> <nav id="p-variants" class="mw-portlet mw-portlet-variants emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-variants-label" > <input type="checkbox" id="p-variants-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-variants" class="vector-menu-checkbox" aria-labelledby="p-variants-label" > <label id="p-variants-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Deutsch</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> </div> <div id="right-navigation"> <nav id="p-views" class="mw-portlet mw-portlet-views vector-menu-tabs vector-menu-tabs-legacy vector-menu" aria-labelledby="p-views-label" > <h3 id="p-views-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Ansichten</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected mw-list-item"><a href="/wiki/Eulersche_Winkel"><span>Lesen</span></a></li><li id="ca-ve-edit" class="mw-list-item"><a href="/w/index.php?title=Eulersche_Winkel&veaction=edit" title="Diese Seite mit dem VisualEditor bearbeiten [v]" accesskey="v"><span>Bearbeiten</span></a></li><li id="ca-edit" class="collapsible mw-list-item"><a href="/w/index.php?title=Eulersche_Winkel&action=edit" title="Den Quelltext dieser Seite bearbeiten [e]" accesskey="e"><span>Quelltext bearbeiten</span></a></li><li id="ca-history" class="mw-list-item"><a href="/w/index.php?title=Eulersche_Winkel&action=history" title="Frühere Versionen dieser Seite [h]" accesskey="h"><span>Versionsgeschichte</span></a></li> </ul> </div> </nav> <nav id="p-cactions" class="mw-portlet mw-portlet-cactions emptyPortlet vector-menu-dropdown vector-menu" aria-labelledby="p-cactions-label" title="Weitere Optionen" > <input type="checkbox" id="p-cactions-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-cactions" class="vector-menu-checkbox" aria-labelledby="p-cactions-label" > <label id="p-cactions-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Weitere</span> </label> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </nav> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <h3 >Suche</h3> <form action="/w/index.php" id="searchform" class="vector-search-box-form"> <div id="simpleSearch" class="vector-search-box-inner" data-search-loc="header-navigation"> <input class="vector-search-box-input" type="search" name="search" placeholder="Wikipedia durchsuchen" aria-label="Wikipedia durchsuchen" autocapitalize="sentences" title="Durchsuche die Wikipedia [f]" accesskey="f" id="searchInput" > <input type="hidden" name="title" value="Spezial:Suche"> <input id="mw-searchButton" class="searchButton mw-fallbackSearchButton" type="submit" name="fulltext" title="Suche nach Seiten, die diesen Text enthalten" value="Suchen"> <input id="searchButton" class="searchButton" type="submit" name="go" title="Gehe direkt zu der Seite mit genau diesem Namen, falls sie vorhanden ist." value="Artikel"> </div> </form> </div> </div> </div> <div id="mw-panel" class="vector-legacy-sidebar"> <div id="p-logo" role="banner"> <a class="mw-wiki-logo" href="/wiki/Wikipedia:Hauptseite" title="Hauptseite"></a> </div> <nav id="p-navigation" class="mw-portlet mw-portlet-navigation vector-menu-portal portal vector-menu" aria-labelledby="p-navigation-label" > <h3 id="p-navigation-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Navigation</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Hauptseite" title="Hauptseite besuchen [z]" accesskey="z"><span>Hauptseite</span></a></li><li id="n-topics" class="mw-list-item"><a href="/wiki/Portal:Wikipedia_nach_Themen"><span>Themenportale</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Spezial:Zuf%C3%A4llige_Seite" title="Zufällige Seite aufrufen [x]" accesskey="x"><span>Zufälliger Artikel</span></a></li> </ul> </div> </nav> <nav id="p-Mitmachen" class="mw-portlet mw-portlet-Mitmachen vector-menu-portal portal vector-menu" aria-labelledby="p-Mitmachen-label" > <h3 id="p-Mitmachen-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Mitmachen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-Artikel-verbessern" class="mw-list-item"><a href="/wiki/Wikipedia:Beteiligen"><span>Artikel verbessern</span></a></li><li id="n-Neuerartikel" class="mw-list-item"><a href="/wiki/Hilfe:Neuen_Artikel_anlegen"><span>Neuen Artikel anlegen</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Autorenportal" title="Info-Zentrum über Beteiligungsmöglichkeiten"><span>Autorenportal</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Hilfe:%C3%9Cbersicht" title="Übersicht über Hilfeseiten"><span>Hilfe</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Spezial:Letzte_%C3%84nderungen" title="Liste der letzten Änderungen in Wikipedia [r]" accesskey="r"><span>Letzte Änderungen</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:Kontakt" title="Kontaktmöglichkeiten"><span>Kontakt</span></a></li><li id="n-sitesupport" class="mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_de.wikipedia.org&uselang=de" title="Unterstütze uns"><span>Spenden</span></a></li> </ul> </div> </nav> <nav id="p-tb" class="mw-portlet mw-portlet-tb vector-menu-portal portal vector-menu" aria-labelledby="p-tb-label" > <h3 id="p-tb-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Spezial:Linkliste/Eulersche_Winkel" title="Liste aller Seiten, die hierher verlinken [j]" accesskey="j"><span>Links auf diese Seite</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Spezial:%C3%84nderungen_an_verlinkten_Seiten/Eulersche_Winkel" rel="nofollow" title="Letzte Änderungen an Seiten, die von hier verlinkt sind [k]" accesskey="k"><span>Änderungen an verlinkten Seiten</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Spezial:Spezialseiten" title="Liste aller Spezialseiten [q]" accesskey="q"><span>Spezialseiten</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Eulersche_Winkel&oldid=243597841" title="Dauerhafter Link zu dieser Seitenversion"><span>Permanenter Link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Eulersche_Winkel&action=info" title="Weitere Informationen über diese Seite"><span>Seiteninformationen</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Spezial:Zitierhilfe&page=Eulersche_Winkel&id=243597841&wpFormIdentifier=titleform" title="Hinweise, wie diese Seite zitiert werden kann"><span>Artikel zitieren</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Spezial:URL-K%C3%BCrzung&url=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FEulersche_Winkel"><span>Kurzlink</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Spezial:QrCode&url=https%3A%2F%2Fde.wikipedia.org%2Fwiki%2FEulersche_Winkel"><span>QR-Code herunterladen</span></a></li> </ul> </div> </nav> <nav id="p-coll-print_export" class="mw-portlet mw-portlet-coll-print_export vector-menu-portal portal vector-menu" aria-labelledby="p-coll-print_export-label" > <h3 id="p-coll-print_export-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Drucken/exportieren</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Spezial:DownloadAsPdf&page=Eulersche_Winkel&action=show-download-screen"><span>Als PDF herunterladen</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Eulersche_Winkel&printable=yes" title="Druckansicht dieser Seite [p]" accesskey="p"><span>Druckversion</span></a></li> </ul> </div> </nav> <nav id="p-wikibase-otherprojects" class="mw-portlet mw-portlet-wikibase-otherprojects vector-menu-portal portal vector-menu" aria-labelledby="p-wikibase-otherprojects-label" > <h3 id="p-wikibase-otherprojects-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In anderen Projekten</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Euler_angles" hreflang="en"><span>Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q751290" title="Link zum verbundenen Objekt im Datenrepositorium [g]" accesskey="g"><span>Wikidata-Datenobjekt</span></a></li> </ul> </div> </nav> <nav id="p-lang" class="mw-portlet mw-portlet-lang vector-menu-portal portal vector-menu" aria-labelledby="p-lang-label" > <h3 id="p-lang-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">In anderen Sprachen</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%88%D8%A7%D9%8A%D8%A7_%D8%A3%D9%88%D9%8A%D9%84%D8%B1" title="زوايا أويلر – Arabisch" lang="ar" hreflang="ar" data-title="زوايا أويلر" data-language-autonym="العربية" data-language-local-name="Arabisch" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Angles_d%27Euler" title="Angles d'Euler – Katalanisch" lang="ca" hreflang="ca" data-title="Angles d'Euler" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CF%89%CE%BD%CE%AF%CE%B5%CF%82_%CE%8C%CE%B9%CE%BB%CE%B5%CF%81" title="Γωνίες Όιλερ – Griechisch" lang="el" hreflang="el" data-title="Γωνίες Όιλερ" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euler_angles" title="Euler angles – Englisch" lang="en" hreflang="en" data-title="Euler angles" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81ngulos_de_Euler" title="Ángulos de Euler – Spanisch" lang="es" hreflang="es" data-title="Ángulos de Euler" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eulerren_angeluak" title="Eulerren angeluak – Baskisch" lang="eu" hreflang="eu" data-title="Eulerren angeluak" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D8%A7%D9%88%DB%8C%D9%87%E2%80%8C%D9%87%D8%A7%DB%8C_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="زاویههای اویلر – Persisch" lang="fa" hreflang="fa" data-title="زاویههای اویلر" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Angles_d%27Euler" title="Angles d'Euler – Französisch" lang="fr" hreflang="fr" data-title="Angles d'Euler" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Na_huillinneacha_Euler" title="Na huillinneacha Euler – Irisch" lang="ga" hreflang="ga" data-title="Na huillinneacha Euler" data-language-autonym="Gaeilge" data-language-local-name="Irisch" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%96%D7%95%D7%95%D7%99%D7%95%D7%AA_%D7%90%D7%95%D7%99%D7%9C%D7%A8" title="זוויות אוילר – Hebräisch" lang="he" hreflang="he" data-title="זוויות אוילר" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Euler-sz%C3%B6gek" title="Euler-szögek – Ungarisch" lang="hu" hreflang="hu" data-title="Euler-szögek" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sudut_Euler" title="Sudut Euler – Indonesisch" lang="id" hreflang="id" data-title="Sudut Euler" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Angoli_di_Eulero" title="Angoli di Eulero – Italienisch" lang="it" hreflang="it" data-title="Angoli di Eulero" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%AA%E3%82%A4%E3%83%A9%E3%83%BC%E8%A7%92" title="オイラー角 – Japanisch" lang="ja" hreflang="ja" data-title="オイラー角" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%B9%D0%BB%D0%B5%D1%80_%D0%B1%D2%B1%D1%80%D1%8B%D1%88%D1%82%D0%B0%D1%80%D1%8B" title="Эйлер бұрыштары – Kasachisch" lang="kk" hreflang="kk" data-title="Эйлер бұрыштары" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%EA%B0%81" title="오일러 각 – Koreanisch" lang="ko" hreflang="ko" data-title="오일러 각" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hoeken_van_Euler" title="Hoeken van Euler – Niederländisch" lang="nl" hreflang="nl" data-title="Hoeken van Euler" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Eulervinkler" title="Eulervinkler – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Eulervinkler" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/K%C4%85ty_Eulera" title="Kąty Eulera – Polnisch" lang="pl" hreflang="pl" data-title="Kąty Eulera" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%82ngulos_de_Euler" title="Ângulos de Euler – Portugiesisch" lang="pt" hreflang="pt" data-title="Ângulos de Euler" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Unghiurile_lui_Euler" title="Unghiurile lui Euler – Rumänisch" lang="ro" hreflang="ro" data-title="Unghiurile lui Euler" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%B3%D0%BB%D1%8B_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Углы Эйлера – Russisch" lang="ru" hreflang="ru" data-title="Углы Эйлера" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Eulerjevi_koti" title="Eulerjevi koti – Slowenisch" lang="sl" hreflang="sl" data-title="Eulerjevi koti" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9E%D1%98%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D0%B8_%D0%B8_%D0%A2%D0%B5%D1%98%D1%82-%D0%91%D1%80%D0%B0%D1%98%D0%B0%D0%BD%D0%BE%D0%B2%D0%B8_%D1%83%D0%B3%D0%BB%D0%BE%D0%B2%D0%B8" title="Ојлерови и Тејт-Брајанови углови – Serbisch" lang="sr" hreflang="sr" data-title="Ојлерови и Тејт-Брајанови углови" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Eulervinklar" title="Eulervinklar – Schwedisch" lang="sv" hreflang="sv" data-title="Eulervinklar" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%BE%D0%B2%D1%96_%D0%BA%D1%83%D1%82%D0%B8" title="Ейлерові кути – Ukrainisch" lang="uk" hreflang="uk" data-title="Ейлерові кути" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/G%C3%B3c_Euler" title="Góc Euler – Vietnamesisch" lang="vi" hreflang="vi" data-title="Góc Euler" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AC%A7%E6%8B%89%E8%A7%92" title="欧拉角 – Chinesisch" lang="zh" hreflang="zh" data-title="欧拉角" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%AD%90%E6%8B%89%E8%A7%92" title="歐拉角 – Kantonesisch" lang="yue" hreflang="yue" data-title="歐拉角" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q751290#sitelinks-wikipedia" title="Links auf Artikel in anderen Sprachen bearbeiten" class="wbc-editpage">Links bearbeiten</a></span></div> </div> </nav> </div> </div> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Diese Seite wurde zuletzt am 31. März 2024 um 01:42 Uhr bearbeitet.</li> <li id="footer-info-copyright"><div id="footer-info-copyright-stats" class="noprint"><a rel="nofollow" class="external text" href="https://pageviews.wmcloud.org/?pages=Eulersche_Winkel&project=de.wikipedia.org">Abrufstatistik</a> · <a rel="nofollow" class="external text" href="https://xtools.wmcloud.org/authorship/de.wikipedia.org/Eulersche_Winkel?uselang=de">Autoren</a> </div><div id="footer-info-copyright-separator"><br /></div><div id="footer-info-copyright-info"> <p>Der Text ist unter der Lizenz <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">„Creative-Commons Namensnennung – Weitergabe unter gleichen Bedingungen“</a> verfügbar; Informationen zu den Urhebern und zum Lizenzstatus eingebundener Mediendateien (etwa Bilder oder Videos) können im Regelfall durch Anklicken dieser abgerufen werden. Möglicherweise unterliegen die Inhalte jeweils zusätzlichen Bedingungen. Durch die Nutzung dieser Website erklären Sie sich mit den <span class="plainlinks"><a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/de">Nutzungsbedingungen</a> und der <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Privacy_policy/de">Datenschutzrichtlinie</a></span> einverstanden.<br /> </p> Wikipedia® ist eine eingetragene Marke der Wikimedia Foundation Inc.</div></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy/de">Datenschutz</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%C3%9Cber_Wikipedia">Über Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Impressum">Impressum</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Verhaltenskodex</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Entwickler</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/de.wikipedia.org">Statistiken</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Stellungnahme zu Cookies</a></li> <li id="footer-places-mobileview"><a href="//de.m.wikipedia.org/w/index.php?title=Eulersche_Winkel&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile Ansicht</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> <script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("This page is using the deprecated ResourceLoader module \"codex-search-styles\".\n[1.43] Use a CodexModule with codexComponents to set your specific components used: https://www.mediawiki.org/wiki/Codex#Using_a_limited_subset_of_components");mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-mqqtn","wgBackendResponseTime":153,"wgPageParseReport":{"limitreport":{"cputime":"0.404","walltime":"0.896","ppvisitednodes":{"value":3532,"limit":1000000},"postexpandincludesize":{"value":24648,"limit":2097152},"templateargumentsize":{"value":3295,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":25280,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 224.541 1 -total"," 45.74% 102.711 8 Vorlage:Literatur"," 19.67% 44.170 1 Vorlage:Cite_journal"," 9.20% 20.660 1 Vorlage:Commons"," 8.45% 18.967 1 Vorlage:Normdaten"," 7.55% 16.942 1 Vorlage:MathWorld"," 6.40% 14.378 1 Vorlage:Google_Buch"," 5.02% 11.262 1 Vorlage:Wikidata-Registrierung"," 4.39% 9.850 10 Vorlage:Str_find"," 4.16% 9.343 1 Vorlage:Cite_book/Meldung2"]},"scribunto":{"limitreport-timeusage":{"value":"0.060","limit":"10.000"},"limitreport-memusage":{"value":4152401,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-8cb5f4d85-c4rwm","timestamp":"20241126194400","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Eulersche Winkel","url":"https:\/\/de.wikipedia.org\/wiki\/Eulersche_Winkel","sameAs":"http:\/\/www.wikidata.org\/entity\/Q751290","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q751290","author":{"@type":"Organization","name":"Autoren der Wikimedia-Projekte"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-06-14T06:22:57Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/8\/85\/Euler2a.gif","headline":"rotatorische Transformation kartesischer Koordinaten"}</script> </body> </html>