Ellissoide di riferimento

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data-event-name="pinnable-header.vector-toc.unpin">nascondi</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inizio</div> </a> </li> <li id="toc-Proprietà_dell'ellissoide" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proprietà_dell'ellissoide"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Proprietà dell'ellissoide</span> </div> </a> <button aria-controls="toc-Proprietà_dell'ellissoide-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Attiva/disattiva la sottosezione Proprietà dell'ellissoide</span> </button> <ul id="toc-Proprietà_dell'ellissoide-sublist" class="vector-toc-list"> <li id="toc-Ellisse_dato_dalla_sezione_traversa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ellisse_dato_dalla_sezione_traversa"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Ellisse dato dalla sezione traversa</span> </div> </a> <ul id="toc-Ellisse_dato_dalla_sezione_traversa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ellissoide_triassiale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ellissoide_triassiale"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Ellissoide triassiale</span> </div> </a> <ul id="toc-Ellissoide_triassiale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Coordinate_geografiche_ellittiche" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coordinate_geografiche_ellittiche"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Coordinate geografiche ellittiche</span> </div> </a> <ul id="toc-Coordinate_geografiche_ellittiche-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ellissoidi_usati_per_la_definizione_dei_punti_sulla_Terra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ellissoidi_usati_per_la_definizione_dei_punti_sulla_Terra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Ellissoidi usati per la definizione dei punti sulla Terra</span> </div> </a> <ul id="toc-Ellissoidi_usati_per_la_definizione_dei_punti_sulla_Terra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ellissoidi_di_riferimento_per_altri_corpi_celesti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ellissoidi_di_riferimento_per_altri_corpi_celesti"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ellissoidi di riferimento per altri corpi celesti</span> </div> </a> <ul id="toc-Ellissoidi_di_riferimento_per_altri_corpi_celesti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voci_correlate" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voci_correlate"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Collegamenti_esterni" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Collegamenti_esterni"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Collegamenti esterni</span> </div> </a> <ul id="toc-Collegamenti_esterni-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Indice" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostra/Nascondi l'indice" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostra/Nascondi l'indice</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ellissoide di riferimento</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Vai a una voce in un'altra lingua. Disponibile in 27 lingue" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-27" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">27 lingue</span> </label> <div class="vector-dropdown-content"> <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%B3%D8%B7%D8%AD_%D9%86%D8%A7%D9%82%D8%B5%D9%8A_%D9%85%D8%B1%D8%AC%D8%B9%D9%8A" title="سطح ناقصي مرجعي - arabo" lang="ar" hreflang="ar" data-title="سطح ناقصي مرجعي" data-language-autonym="العربية" data-language-local-name="arabo" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Elipsoide_de_referencia" title="Elipsoide de referencia - asturiano" lang="ast" hreflang="ast" data-title="Elipsoide de referencia" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/El%C2%B7lipsoide_de_refer%C3%A8ncia" title="El·lipsoide de referència - catalano" lang="ca" hreflang="ca" data-title="El·lipsoide de referència" data-language-autonym="Català" data-language-local-name="catalano" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Referen%C4%8Dn%C3%AD_elipsoid" title="Referenční elipsoid - ceco" lang="cs" hreflang="cs" data-title="Referenční elipsoid" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Referenceellipsoide" title="Referenceellipsoide - danese" lang="da" hreflang="da" data-title="Referenceellipsoide" data-language-autonym="Dansk" data-language-local-name="danese" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Referenzellipsoid" title="Referenzellipsoid - tedesco" lang="de" hreflang="de" data-title="Referenzellipsoid" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BB%CE%BB%CE%B5%CE%B9%CF%88%CE%BF%CE%B5%CE%B9%CE%B4%CE%AD%CF%82_%CE%B1%CE%BD%CE%B1%CF%86%CE%BF%CF%81%CE%AC%CF%82" title="Ελλειψοειδές αναφοράς - greco" lang="el" hreflang="el" data-title="Ελλειψοειδές αναφοράς" data-language-autonym="Ελληνικά" data-language-local-name="greco" 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/Earth_ellipsoid" title="Earth ellipsoid - inglese" lang="en" hreflang="en" data-title="Earth ellipsoid" data-language-autonym="English" data-language-local-name="inglese" 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/Elipsoide_de_referencia" title="Elipsoide de referencia - spagnolo" lang="es" hreflang="es" data-title="Elipsoide de referencia" data-language-autonym="Español" data-language-local-name="spagnolo" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Referentsellipsoid" title="Referentsellipsoid - estone" lang="et" hreflang="et" data-title="Referentsellipsoid" data-language-autonym="Eesti" data-language-local-name="estone" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Erreferentziazko_elipsoide" title="Erreferentziazko elipsoide - basco" lang="eu" hreflang="eu" data-title="Erreferentziazko elipsoide" data-language-autonym="Euskara" data-language-local-name="basco" 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%A8%DB%8C%D8%B6%DB%8C%E2%80%8C%D9%88%D8%A7%D8%B1_%D9%85%D8%B1%D8%AC%D8%B9" title="بیضی‌وار مرجع - persiano" lang="fa" hreflang="fa" data-title="بیضی‌وار مرجع" data-language-autonym="فارسی" data-language-local-name="persiano" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vertausellipsoidi" title="Vertausellipsoidi - finlandese" lang="fi" hreflang="fi" data-title="Vertausellipsoidi" data-language-autonym="Suomi" data-language-local-name="finlandese" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_ellipso%C3%AFdal_de_la_Terre" title="Modèle ellipsoïdal de la Terre - francese" lang="fr" hreflang="fr" data-title="Modèle ellipsoïdal de la Terre" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Elipsoide_de_referencia" title="Elipsoide de referencia - galiziano" lang="gl" hreflang="gl" data-title="Elipsoide de referencia" data-language-autonym="Galego" data-language-local-name="galiziano" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B5%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86-%D1%8D%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Референц-эллипсоид - kazako" lang="kk" hreflang="kk" data-title="Референц-эллипсоид" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Referentie-ellipso%C3%AFde" title="Referentie-ellipsoïde - olandese" lang="nl" hreflang="nl" data-title="Referentie-ellipsoïde" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Elipsoide_de_refer%C3%AAncia" title="Elipsoide de referência - portoghese" lang="pt" hreflang="pt" data-title="Elipsoide de referência" data-language-autonym="Português" data-language-local-name="portoghese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B5%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86-%D1%8D%D0%BB%D0%BB%D0%B8%D0%BF%D1%81%D0%BE%D0%B8%D0%B4" title="Референц-эллипсоид - russo" lang="ru" hreflang="ru" data-title="Референц-эллипсоид" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Reference_ellipsoid" title="Reference ellipsoid - Simple English" lang="en-simple" hreflang="en-simple" data-title="Reference ellipsoid" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Referentni_elipsoid" title="Referentni elipsoid - serbo" lang="sr" hreflang="sr" data-title="Referentni elipsoid" data-language-autonym="Српски / srpski" data-language-local-name="serbo" 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/Referensellipsoid" title="Referensellipsoid - svedese" lang="sv" hreflang="sv" data-title="Referensellipsoid" data-language-autonym="Svenska" data-language-local-name="svedese" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Referans_elipsoid" title="Referans elipsoid - turco" lang="tr" hreflang="tr" data-title="Referans elipsoid" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%B5%D1%84%D0%B5%D1%80%D0%B5%D0%BD%D1%86-%D0%B5%D0%BB%D1%96%D0%BF%D1%81%D0%BE%D1%97%D0%B4" title="Референц-еліпсоїд - ucraino" lang="uk" hreflang="uk" data-title="Референц-еліпсоїд" data-language-autonym="Українська" data-language-local-name="ucraino" 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/Ellipsoid_quy_chi%E1%BA%BFu" title="Ellipsoid quy chiếu - vietnamita" lang="vi" hreflang="vi" data-title="Ellipsoid quy chiếu" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" 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/%E5%8F%82%E8%80%83%E6%A4%AD%E7%90%83" title="参考椭球 - cinese" lang="zh" hreflang="zh" data-title="参考椭球" data-language-autonym="中文" data-language-local-name="cinese" 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/%E5%8F%83%E8%80%83%E6%A9%A2%E7%90%83%E9%AB%94" title="參考橢球體 - cantonese" lang="yue" hreflang="yue" data-title="參考橢球體" data-language-autonym="粵語" data-language-local-name="cantonese" 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/Q1335878#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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data-event-name="pinnable-header.vector-appearance.unpin">nascondi</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Da Wikipedia, l'enciclopedia libera.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="it" dir="ltr"><p>In <a href="/wiki/Geodesia" title="Geodesia">geodesia</a>, un <b>ellissoide di riferimento</b> è una superficie definita matematicamente che approssima il <a href="/wiki/Geoide" title="Geoide">geoide</a> (con un errore accettabile), la vera <a href="/wiki/Figura_della_Terra" title="Figura della Terra">forma della Terra</a>, o di un altro corpo celeste. A causa della relativa semplicità, gli ellissoidi di riferimento sono usati comunemente come superficie di riferimento per definire una <a href="/wiki/Rete_geodetica" title="Rete geodetica">rete geodetica</a> e qualunque punto dello spazio di cui sia definita la <a href="/wiki/Latitudine" title="Latitudine">latitudine</a>, la <a href="/wiki/Longitudine" title="Longitudine">longitudine</a> e l'<a href="/wiki/Altimetria" title="Altimetria">elevazione</a> sull'ellissoide. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proprietà_dell'ellissoide"><span id="Propriet.C3.A0_dell.27ellissoide"></span>Proprietà dell'ellissoide</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=1" title="Modifica la sezione Proprietà dell'ellissoide" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=1" title="Edit section's source code: Proprietà dell'ellissoide"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dal punto di vista matematico, un ellissoide di riferimento è usualmente uno <a href="/wiki/Sferoide" title="Sferoide">sferoide</a> <a href="/w/index.php?title=Oblato_(geometria)&action=edit&redlink=1" class="new" title="Oblato (geometria) (la pagina non esiste)">oblato</a> (appiattito) i cui semiassi sono definiti: </p> <ul><li>raggio <a href="/wiki/Equatore" title="Equatore">equatoriale</a> (il <a href="/wiki/Semiasse_maggiore" title="Semiasse maggiore">semiasse maggiore</a> <span class="mwe-math-element"><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>);</li> <li>raggio <a href="/wiki/Polo_geografico" title="Polo geografico">polare</a> (il <a href="/wiki/Semiasse_minore" title="Semiasse minore">semiasse minore</a> <span class="mwe-math-element"><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>).</li></ul> <p>Nel lavoro con geometria ellittica, diversi parametri sono comunemente utilizzati, che sono tutte funzioni trigonometriche di un'ellisse di <a href="/w/index.php?title=Eccentricit%C3%A0_angolare&action=edit&redlink=1" class="new" title="Eccentricità angolare (la pagina non esiste)">eccentricità angolare</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle o\!\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o\!\varepsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da64449099b6dc2036ca31f9d6db3add2ea31434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.824ex; height:1.676ex;" alt="{\displaystyle o\!\varepsilon }"></span>: </p> <dl><dd><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 o\!\varepsilon =\arccos \left({\frac {b}{a}}\right)=2\arctan \left({\sqrt {\frac {a-b}{a+b}}}\right);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o\!\varepsilon =\arccos \left({\frac {b}{a}}\right)=2\arctan \left({\sqrt {\frac {a-b}{a+b}}}\right);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1738612e2c6703dd5ad6b16e0c06505cc14aa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.037ex; height:6.343ex;" alt="{\displaystyle o\!\varepsilon =\arccos \left({\frac {b}{a}}\right)=2\arctan \left({\sqrt {\frac {a-b}{a+b}}}\right);}"></span></dd></dl></dd></dl> <p>La rotazione della Terra causa un rigonfiamento all'equatore ed un appiattimento ai poli, cosicché il raggio equatoriale è maggiore del raggio polare: <span class="mwe-math-element"><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>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span>. Quest’appiattimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> determina quanto lo sferoide si avvicina alla forma sferica ed è definito da: </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 f={\mbox{ver}}(o\!\varepsilon )=2\sin \left({\frac {o\!\varepsilon }{2}}\right)^{2}=1-\cos(o\!\varepsilon )={\frac {a-b}{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>ver</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\mbox{ver}}(o\!\varepsilon )=2\sin \left({\frac {o\!\varepsilon }{2}}\right)^{2}=1-\cos(o\!\varepsilon )={\frac {a-b}{a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70af229475f099c883c545ff719ee7c622b90ff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.057ex; height:5.343ex;" alt="{\displaystyle f={\mbox{ver}}(o\!\varepsilon )=2\sin \left({\frac {o\!\varepsilon }{2}}\right)^{2}=1-\cos(o\!\varepsilon )={\frac {a-b}{a}}.}"></span></dd></dl> <p>Per la Terra, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> vale circa 1/300, che si traduce in una differenza di circa 20 km, e si sta lentamente riducendo su una scala di tempo geologica. </p><p>Anche il rigonfiamento equatoriale subisce lente variazioni. Nel 1998 un’inversione di tendenza ha portato il valore ad aumentare, forse a causa di una redistribuzione delle masse oceaniche dovuta alle correnti. <sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Facendo dei confronti, la Luna è meno ellittica della Terra, con un appiattimento di meno di 1/825, mentre <a href="/wiki/Giove_(astronomia)" title="Giove (astronomia)">Giove</a> è visibilmente appiattito con circa 1/15. </p><p>Tradizionalmente quando si definisce un ellissoide di riferimento si specifica il raggio equatoriale <span class="mwe-math-element"><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> (usualmente in <a href="/wiki/Metro" title="Metro">metri</a>) e l'inverso del rapporto di appiattimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}"></span>. Il raggio polare è quindi ricavato come: </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 b=a\cos(o\!\varepsilon )=a(1-f).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=a\cos(o\!\varepsilon )=a(1-f).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4353126dfa078be2f7efcddbb24d15c1262107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.523ex; height:2.843ex;" alt="{\displaystyle b=a\cos(o\!\varepsilon )=a(1-f).}"></span></dd></dl> <p>L'appiattimento teorico calcolato considerando la gravità e la forza centrifuga vale: </p> <dl><dd><dl><dd><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 q={\frac {a^{3}\omega ^{2}}{GM}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>G</mi> <mi>M</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a619a580a1225e416af2ede1f0d0bc40385e3099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.822ex; height:5.843ex;" alt="{\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}}\,,}"></span></dd></dl></dd></dl></dd></dl> <p>dove <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> è la <a href="/wiki/Velocit%C3%A0_angolare" title="Velocità angolare">velocità angolare</a>, <span class="mwe-math-element"><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> è la <a href="/wiki/Costante_gravitazionale" class="mw-redirect" title="Costante gravitazionale">costante gravitazionale</a>, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> è la massa del pianeta. </p><p><sup id="cite_ref-IAU_2-0" class="reference"><a href="#cite_note-IAU-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Per la Terra <span class="mwe-math-element"><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^{-1}\approx 289}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≈</mo> <mn>289</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{-1}\approx 289}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cffa3b747a40ae41812b2686bec9c3af1a821700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.998ex; height:3.009ex;" alt="{\displaystyle q^{-1}\approx 289}"></span>, vicino al valore misurato di <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\approx 298.257}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>≈</mo> <mn>298.257</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\approx 298.257}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f053892efccdbf21725b5af6b0cfb4e4ee7cb1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.373ex; height:3.009ex;" alt="{\displaystyle f^{-1}\approx 298.257}"></span>. La differenza è dovuta alla disomogeneità della densità della Terra, in particolare alla rigidezza del nucleo, che ha una densità notevolmente superiore al mantello. </p> <div class="mw-heading mw-heading3"><h3 id="Ellisse_dato_dalla_sezione_traversa">Ellisse dato dalla sezione traversa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=2" title="Modifica la sezione Ellisse dato dalla sezione traversa" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=2" title="Edit section's source code: Ellisse dato dalla sezione traversa"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uno sferoide è una <b>figura di rotazione</b> generata dalla rotazione di un'ellisse attorno all'asse minore. Coerentemente, l'asse minore coincide con l'asse di rotazione della Terra (distinto dall'asse magnetico e dall'asse orbitale). L'appiattimento dello sferoide è legato all'<a href="/wiki/Eccentricit%C3%A0_(matematica)" title="Eccentricità (matematica)">eccentricità</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>, dell'ellisse: </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 e^{2}=\sin(o\!\varepsilon )^{2}=f(2-f)={\frac {a^{2}-b^{2}}{a^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2}=\sin(o\!\varepsilon )^{2}=f(2-f)={\frac {a^{2}-b^{2}}{a^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bea0ff6d4fd437affbdddc03f222bbb14840b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.005ex; height:6.009ex;" alt="{\displaystyle e^{2}=\sin(o\!\varepsilon )^{2}=f(2-f)={\frac {a^{2}-b^{2}}{a^{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ellissoide_triassiale">Ellissoide triassiale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=3" title="Modifica la sezione Ellissoide triassiale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=3" title="Edit section's source code: Ellissoide triassiale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si definisce sferoide un ellissoide con due dei tre assi uguali. Raramente viene usato ai fini geoidici un <b>ellissoide scaleno</b>, in cui i tre assi sono diversi tra loro, detto anche <b>triassiale</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 a_{x},\,a_{y},\,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{x},\,a_{y},\,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f21995517e8ed26d3cdd3bdfce38c7af2f6ba27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.521ex; height:2.843ex;" alt="{\displaystyle a_{x},\,a_{y},\,b}"></span>. Viene usato per modellare corpi celesti minori, come piccole lune ed asteroidi. Ad esempio <a href="/wiki/Telesto_(astronomia)" title="Telesto (astronomia)">Telesto</a>, una luna triassiale di Saturno, ha appiattimenti di 1/3 e 1/2. </p> <div class="mw-heading mw-heading2"><h2 id="Coordinate_geografiche_ellittiche">Coordinate geografiche ellittiche</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=4" title="Modifica la sezione Coordinate geografiche ellittiche" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=4" title="Edit section's source code: Coordinate geografiche ellittiche"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sulla base degli ellissoidi di riferimento si definiscono i sistemi di <a href="/wiki/Coordinate_geografiche" title="Coordinate geografiche">coordinate geografiche</a>, che identificano i punti sulla superficie dei corpi celesti in termini di <a href="/wiki/Latitudine" title="Latitudine">latitudine</a> (nord-sud) e <a href="/wiki/Longitudine" title="Longitudine">longitudine</a> (est-ovest). </p><p>La longitudine è la misura dell'angolo di rotazione tra il meridiano zero e il punto da misurare. Per convenzione, nel caso di Terra, Sole e Luna, l'angolo viene espresso in gradi che spaziano tra i −180º e i +180º, per gli altri corpi celesti si usano invece i valori da 0° a 360°. </p><p>La latitudine è la distanza angolare di un punto dai poli o dall'equatore, misurata lungo un meridiano. Essa assume valori compresi fra −90º e +90º, con lo zero in corrispondenza dell'equatore. </p><p>La comune latitudine, ovvero la latitudine geografica, è l'angolo compreso fra il piano equatoriale e una linea che è normale all'ellissoide di riferimento. Siccome è dipendente dall'appiattimento, essa può essere leggermente differente dalla <i>latitudine geocentrica</i> che è l'angolo fra il piano equatoriale e una linea dal centro dell'ellissoide. Per corpi non terrestri si usano invece i termini <i>planetografica</i> e <i>planetocentrica</i>. </p><p>Questi sistemi prevedono inoltre la scelta di un <a href="/wiki/Meridiano_(geografia)" class="mw-redirect" title="Meridiano (geografia)">meridiano</a> di riferimento o "meridiano zero". Nel caso della Terra si assume di regola il <a href="/wiki/Meridiano_di_Greenwich" title="Meridiano di Greenwich">meridiano di Greenwich</a>; per gli altri corpi celesti si utilizza come punto di riferimento un oggetto superficiale ben riconoscibile. Ad esempio, nel caso di Marte, il meridiano di riferimento passa per il centro del <a href="/wiki/Cratere_Airy-0" title="Cratere Airy-0">cratere Airy-0</a>. </p><p>È possibile che molti sistemi di coordinate differenti siano definiti sullo stesso ellissoide di riferimento. </p><p>Le coordinate di un punto geodetico sono normalmente riportate come latitudine e longitudine geodetica: cioè la direzione nello spazio della normale geodetica contenente il punto e l'altezza del punto sull'ellissoide di riferimento. Utilizzando queste coordinate (Latitudine <span class="mwe-math-element"><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</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>, longitudine <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> e altezza <i>h</i>) è possibile calcolare le coordinate rettangolari geodetiche come segue: </p> <dl><dd><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 X_{t}=[N+h]\cos(\phi )\cos(\lambda );}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">]</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{t}=[N+h]\cos(\phi )\cos(\lambda );}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9944f9d894c0e89c1d82c22561075556117d9710" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.388ex; height:2.843ex;" alt="{\displaystyle X_{t}=[N+h]\cos(\phi )\cos(\lambda );}"></span></dd> <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 Y_{t}=[N+h]\cos(\phi )\sin(\lambda );}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">]</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{t}=[N+h]\cos(\phi )\sin(\lambda );}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3ed5d262dc5bfc686d1d98ea9ed5838db0d9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.559ex; height:2.843ex;" alt="{\displaystyle Y_{t}=[N+h]\cos(\phi )\sin(\lambda );}"></span></dd> <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 Z_{t}=[N\cos(o\!\varepsilon )^{2}+h]\sin(\phi );}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>h</mi> <mo stretchy="false">]</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{t}=[N\cos(o\!\varepsilon )^{2}+h]\sin(\phi );}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f8bca83c1cd3f02fdbe60283e68e4d357285e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.319ex; height:3.176ex;" alt="{\displaystyle Z_{t}=[N\cos(o\!\varepsilon )^{2}+h]\sin(\phi );}"></span></dd></dl></dd></dl> <p>dove </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 N=N(\phi )={\frac {a}{\sqrt {1-(\sin(\phi )\sin(o\!\varepsilon ))^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msqrt> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=N(\phi )={\frac {a}{\sqrt {1-(\sin(\phi )\sin(o\!\varepsilon ))^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8abdc21b58f8b1c9b78742802d20a7d894391085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.472ex; height:6.009ex;" alt="{\displaystyle N=N(\phi )={\frac {a}{\sqrt {1-(\sin(\phi )\sin(o\!\varepsilon ))^{2}}}}}"></span></dd></dl> <p>è il <a href="/wiki/Raggio_di_curvatura" class="mw-redirect" title="Raggio di curvatura">raggio di curvatura</a> nel <a href="/w/index.php?title=Primo_verticale&action=edit&redlink=1" class="new" title="Primo verticale (la pagina non esiste)">primo verticale</a>. </p><p>Al contrario dedurre <span class="mwe-math-element"><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</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>, <span class="mwe-math-element"><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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> e <i>h</i> dalle coordinate rettangolari richiede di procedere per <a href="/wiki/Iterazione" title="Iterazione">iterazione</a> </p><p>Se <span class="mwe-math-element"><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 _{c}=\arctan(\sec(o\!\varepsilon )^{2}\tan(\psi _{t}))\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{c}=\arctan(\sec(o\!\varepsilon )^{2}\tan(\psi _{t}))\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa07a6cc63522290b9e2fc78a7a70f2a171fc3f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.912ex; height:3.176ex;" alt="{\displaystyle \phi _{c}=\arctan(\sec(o\!\varepsilon )^{2}\tan(\psi _{t}))\;}"></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 \phi _{p}=\phi _{c}:\;\phi _{c}=\arctan \!\left({\frac {\qquad \;\;a^{2}Z_{t}\quad \,+{\frac {1}{4}}[N(\phi _{p})\sin(\phi _{p})]^{3}\sin(2o\!\varepsilon )^{2}}{\!\!\!\!\!a^{2}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}\,-[N(\phi _{p})\cos(\phi _{p})]^{3}\sin(o\!\varepsilon )^{2}}}\right);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>:</mo> <mspace width="thickmathspace" /> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="2em" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="1em" /> <mspace width="thinmathspace" /> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>−</mo> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{p}=\phi _{c}:\;\phi _{c}=\arctan \!\left({\frac {\qquad \;\;a^{2}Z_{t}\quad \,+{\frac {1}{4}}[N(\phi _{p})\sin(\phi _{p})]^{3}\sin(2o\!\varepsilon )^{2}}{\!\!\!\!\!a^{2}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}\,-[N(\phi _{p})\cos(\phi _{p})]^{3}\sin(o\!\varepsilon )^{2}}}\right);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7d3c19bd10fad2f49a23a726cb14099688e7d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:71.382ex; height:10.509ex;" alt="{\displaystyle \phi _{p}=\phi _{c}:\;\phi _{c}=\arctan \!\left({\frac {\qquad \;\;a^{2}Z_{t}\quad \,+{\frac {1}{4}}[N(\phi _{p})\sin(\phi _{p})]^{3}\sin(2o\!\varepsilon )^{2}}{\!\!\!\!\!a^{2}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}\,-[N(\phi _{p})\cos(\phi _{p})]^{3}\sin(o\!\varepsilon )^{2}}}\right);}"></span> </p><p>si ripete finché <span class="mwe-math-element"><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 _{c}=\phi _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{c}=\phi _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fece415172a647762bb9372f735a4dc18d56db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.873ex; height:2.843ex;" alt="{\displaystyle \phi _{c}=\phi _{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 \phi =\phi _{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\phi _{c}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/210222569c9398629335d9689b8ad77fa0961e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.46ex; height:2.509ex;" alt="{\displaystyle \phi =\phi _{c}.}"></span> </p><p>O, introducendo le latitudini geocentrica, <span class="mwe-math-element"><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</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>, o parametrica o ridotta, <span class="mwe-math-element"><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</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>, si ha: </p><p><span class="mwe-math-element"><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 _{t}=\arctan \left({\frac {Z_{t}}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}\right)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msqrt> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{t}=\arctan \left({\frac {Z_{t}}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}\right)\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28ad16eb2d86e756d2dfee1e7633e9206e35afd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:28.622ex; height:10.509ex;" alt="{\displaystyle \psi _{t}=\arctan \left({\frac {Z_{t}}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}\right)\;}"></span> and <span class="mwe-math-element"><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 _{c}=\arctan(\sec(o\!\varepsilon )\tan(\psi _{t}))\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{c}=\arctan(\sec(o\!\varepsilon )\tan(\psi _{t}))\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9e45ae9642f502a01a2fba8afbae139825ace3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.788ex; height:2.843ex;" alt="{\displaystyle \beta _{c}=\arctan(\sec(o\!\varepsilon )\tan(\psi _{t}))\;}"></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 \phi _{p}=\phi _{c}:\;\phi _{c}=\arctan \!\left({\frac {\qquad \,Z_{t}\qquad +b\sin(\beta _{c})^{3}\tan(o\!\varepsilon )^{2}}{{\sqrt {X_{t}^{2}+Y_{t}^{2}}}\;-a\cos(\beta _{c})^{3}\sin(o\!\varepsilon )^{2}}}\right);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>:</mo> <mspace width="thickmathspace" /> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="2em" /> <mspace width="thinmathspace" /> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mspace width="2em" /> <mo>+</mo> <mi>b</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> <mspace width="thickmathspace" /> <mo>−</mo> <mi>a</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{p}=\phi _{c}:\;\phi _{c}=\arctan \!\left({\frac {\qquad \,Z_{t}\qquad +b\sin(\beta _{c})^{3}\tan(o\!\varepsilon )^{2}}{{\sqrt {X_{t}^{2}+Y_{t}^{2}}}\;-a\cos(\beta _{c})^{3}\sin(o\!\varepsilon )^{2}}}\right);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/994ff06fdb1b6dd00356638a57bab5518c4618ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:61.015ex; height:10.509ex;" alt="{\displaystyle \phi _{p}=\phi _{c}:\;\phi _{c}=\arctan \!\left({\frac {\qquad \,Z_{t}\qquad +b\sin(\beta _{c})^{3}\tan(o\!\varepsilon )^{2}}{{\sqrt {X_{t}^{2}+Y_{t}^{2}}}\;-a\cos(\beta _{c})^{3}\sin(o\!\varepsilon )^{2}}}\right);}"></span> </p><p><span class="mwe-math-element"><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}=\beta _{c}:\;\beta _{c}=\arctan \left(\cos(o\!\varepsilon )\tan(\phi _{c})\right);\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>:</mo> <mspace width="thickmathspace" /> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>;</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{p}=\beta _{c}:\;\beta _{c}=\arctan \left(\cos(o\!\varepsilon )\tan(\phi _{c})\right);\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340f699bf34e04b6514366580f478edf54276065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.872ex; height:3.009ex;" alt="{\displaystyle \beta _{p}=\beta _{c}:\;\beta _{c}=\arctan \left(\cos(o\!\varepsilon )\tan(\phi _{c})\right);\;}"></span> </p><p>Si ripete finché<span class="mwe-math-element"><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 _{c}=\phi _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{c}=\phi _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fece415172a647762bb9372f735a4dc18d56db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.873ex; height:2.843ex;" alt="{\displaystyle \phi _{c}=\phi _{p}}"></span> and <span class="mwe-math-element"><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 _{c}=\beta _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta _{c}=\beta _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b165f7c2136491a8dbe4d3875868e3705065276b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.733ex; height:2.843ex;" alt="{\displaystyle \beta _{c}=\beta _{p}}"></span>: </p><p><span class="mwe-math-element"><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 =\phi _{c};\quad \beta =\beta _{c};\quad \psi =\arctan(\cos(o\!\varepsilon )\tan(\beta )).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>;</mo> <mspace width="1em" /> <mi>β</mi> <mo>=</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>;</mo> <mspace width="1em" /> <mi>ψ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <mo stretchy="false">)</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =\phi _{c};\quad \beta =\beta _{c};\quad \psi =\arctan(\cos(o\!\varepsilon )\tan(\beta )).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3237901782cff8b5af0468c7eded74d3329a4e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.383ex; height:2.843ex;" alt="{\displaystyle \phi =\phi _{c};\quad \beta =\beta _{c};\quad \psi =\arctan(\cos(o\!\varepsilon )\tan(\beta )).}"></span> </p><p>Quando si trova <span class="mwe-math-element"><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</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>allora si può isolare h: </p><p><span class="mwe-math-element"><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=\sec(\phi ){\color {white}{\dot {\color {black}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}}}-N\;=\;\csc(\phi )Z_{t}-\cos(o\!\varepsilon )^{2}N,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>sec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mo>−</mo> <mi>N</mi> <mspace width="thickmathspace" /> <mo>=</mo> <mspace width="thickmathspace" /> <mi>csc</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=\sec(\phi ){\color {white}{\dot {\color {black}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}}}-N\;=\;\csc(\phi )Z_{t}-\cos(o\!\varepsilon )^{2}N,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9013217e029b65dea2f5f3bdbc1f5b65d3a4702b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:53.014ex; height:5.343ex;" alt="{\displaystyle h=\sec(\phi ){\color {white}{\dot {\color {black}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}}}-N\;=\;\csc(\phi )Z_{t}-\cos(o\!\varepsilon )^{2}N,}"></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 {}_{\color {white}8.}=\cos(\phi ){\color {white}{\dot {\color {black}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}}}\,+\,\sin(\phi )\left[Z_{t}+\sin(o\!\varepsilon )^{2}N\sin(\phi )\right]-N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mn>8.</mn> </mstyle> </mrow> </msub> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="white"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <mspace width="thinmathspace" /> <mo>+</mo> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>o</mi> <mspace width="negativethinmathspace" /> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}_{\color {white}8.}=\cos(\phi ){\color {white}{\dot {\color {black}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}}}\,+\,\sin(\phi )\left[Z_{t}+\sin(o\!\varepsilon )^{2}N\sin(\phi )\right]-N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68993a66c6bebc4224b497720e909a1e660afcd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:60.926ex; height:5.343ex;" alt="{\displaystyle {}_{\color {white}8.}=\cos(\phi ){\color {white}{\dot {\color {black}{\sqrt {X_{t}^{2}+Y_{t}^{2}}}}}}\,+\,\sin(\phi )\left[Z_{t}+\sin(o\!\varepsilon )^{2}N\sin(\phi )\right]-N.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Ellissoidi_usati_per_la_definizione_dei_punti_sulla_Terra">Ellissoidi usati per la definizione dei punti sulla Terra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=5" title="Modifica la sezione Ellissoidi usati per la definizione dei punti sulla Terra" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=5" title="Edit section's source code: Ellissoidi usati per la definizione dei punti sulla Terra"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Il riferimento attualmente più usato, grazie all'impiego nel contesto <a href="/wiki/GPS" title="GPS">GPS</a>, è il <a href="/wiki/WGS84" title="WGS84">WGS84</a>. </p><p>La cartografia italiana è realizzata impiegando l'ellissoide internazionale di <a href="/wiki/John_Fillmore_Hayford" title="John Fillmore Hayford">Hayford</a>, tranne il sistema catastale che adopera il sistema anteguerra basato sull'ellissoide di <a href="/wiki/Friedrich_Wilhelm_Bessel" title="Friedrich Wilhelm Bessel">Bessel</a>. </p><p>I parametri di seguito elencati definiscono la forma degli ellissoidi storicamente impiegati. </p> <table class="wikitable"> <tbody><tr> <th>Nome</th> <th>Semiasse Magg.(m)</th> <th>Semiasse Min.(m)</th> <th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}"></span></th> <th>Area applicazione </th></tr> <tr> <td>Sfera (6371 km)</td> <td>6 371 000</td> <td>6 371 000</td> <td>0</td> <td> </td></tr> <tr> <td>Timbalai</td> <td>6 377 298,56</td> <td>6 356 097,55</td> <td>300,801639166</td> <td> </td></tr> <tr> <td>Sferoide di Everest</td> <td>6 377 301,243</td> <td>6 356 100,228</td> <td>300,801694993</td> <td> </td></tr> <tr> <td>Everest modificato (Malaya) Revised Kertau</td> <td>6 377 304,063</td> <td>6 356 103,038993</td> <td>300,801699969</td> <td> </td></tr> <tr> <td><a href="/wiki/Pierre_Louis_Maupertuis" class="mw-redirect" title="Pierre Louis Maupertuis">Maupertuis</a> (1738)</td> <td>6 397 300</td> <td>6 363 806,283</td> <td>191</td> <td>Francia </td></tr> <tr> <td><a href="/wiki/George_Everest" title="George Everest">Everest</a> (1830)</td> <td>6 377 276,345</td> <td>6 356 075,413</td> <td>300,801697979</td> <td>India </td></tr> <tr> <td><a href="/wiki/George_Biddell_Airy" title="George Biddell Airy">Airy</a> (1830)</td> <td>6 377 563,396</td> <td>6 356 256,909</td> <td>299,3249646</td> <td>Gran Bretagna </td></tr> <tr> <td><i><b><a href="/wiki/Friedrich_Wilhelm_Bessel" title="Friedrich Wilhelm Bessel">Bessel</a> (1841)</b></i></td> <td>6 377 397,155</td> <td>6 356 078,963</td> <td>299,1528128</td> <td>Europa, Giappone. Sistema catastale italiano </td></tr> <tr> <td><a href="/wiki/Alexander_Ross_Clarke" title="Alexander Ross Clarke">Clarke</a> (1866)</td> <td>6 378 206,4</td> <td>6 356 583,8</td> <td>294,9786982</td> <td>Nord America </td></tr> <tr> <td><a href="/wiki/Alexander_Ross_Clarke" title="Alexander Ross Clarke">Clarke</a> (1880)</td> <td>6 378 249,145</td> <td>6 356 514,870</td> <td>293,465</td> <td>Francia, Africa </td></tr> <tr> <td><a href="/wiki/Friedrich_Robert_Helmert" title="Friedrich Robert Helmert">Helmert</a> (1906)</td> <td>6 378 200</td> <td>6 356 818,17</td> <td>298,3</td> <td> </td></tr> <tr> <td><a href="/wiki/John_Fillmore_Hayford" title="John Fillmore Hayford">Hayford</a> (1910)</td> <td>6 378 388</td> <td>6 356 911,946</td> <td>297</td> <td>USA, Italia </td></tr> <tr> <td><i><b>International (1924)</b></i></td> <td>6 378 388</td> <td>6 356 911,946</td> <td>297</td> <td>Europa. Italia: <a href="/wiki/Roma_40" title="Roma 40">Roma 40</a>, <a href="/wiki/ED50" title="ED50">ED50</a> </td></tr> <tr> <td>NAD 27</td> <td>6 378 206,4</td> <td>6 356 583,800</td> <td>294,978698208</td> <td>Nord America </td></tr> <tr> <td>Krasovskii (1940)</td> <td>6 378 245</td> <td>6 356 863,019</td> <td>298,3</td> <td>Russia </td></tr> <tr> <td><a href="/w/index.php?title=WGS-66&action=edit&redlink=1" class="new" title="WGS-66 (la pagina non esiste)">WGS-66</a> (1966)</td> <td>6 378 145</td> <td>6 356 759,769</td> <td>298,25</td> <td>USA / DoD (Dipartimento della difesa) </td></tr> <tr> <td>Australian National (1966)</td> <td>6 378 160</td> <td>6 356 774,719</td> <td>298,25</td> <td>Australia </td></tr> <tr> <td>New International (1967)</td> <td>6 378 157,5</td> <td>6 356 772,2</td> <td>298,24961539</td> <td> </td></tr> <tr> <td>GRS-67 (1967)</td> <td>6 378 160</td> <td>6 356 774,516</td> <td>298,247167427</td> <td> </td></tr> <tr> <td>South American (1969)</td> <td>6 378 160</td> <td>6 356 774,719</td> <td>298,25</td> <td>Sud America </td></tr> <tr> <td><a href="/w/index.php?title=WGS-72&action=edit&redlink=1" class="new" title="WGS-72 (la pagina non esiste)">WGS-72</a> (1972)</td> <td>6 378 135</td> <td>6 356 750,52</td> <td>298,26</td> <td>USA / DoD (Dipartimento della difesa) </td></tr> <tr> <td><a href="/w/index.php?title=GRS_80&action=edit&redlink=1" class="new" title="GRS 80 (la pagina non esiste)">GRS-80</a> (1979)</td> <td>6 378 137</td> <td>6 356 752,3141</td> <td>298,257222101</td> <td> </td></tr> <tr> <td><a href="/w/index.php?title=NAD83&action=edit&redlink=1" class="new" title="NAD83 (la pagina non esiste)">NAD 83</a></td> <td>6 378 137</td> <td>6 356 752,3</td> <td>298,257024899</td> <td>Nord America </td></tr> <tr> <td><i><b><a href="/wiki/World_Geodetic_System#A_new_World_Geodetic_System:_WGS84" class="mw-redirect" title="World Geodetic System">WGS-84</a></b></i> (1984)</td> <td>6 378 137</td> <td>6 356 752,3142</td> <td>298,257223563</td> <td><b>cartografia GPS</b> </td></tr> <tr> <td><a href="/wiki/IERS" class="mw-redirect" title="IERS">IERS</a> (1989)</td> <td>6 378 136</td> <td>6 356 751,302</td> <td>298,257</td> <td>Output degli attuali GPS </td></tr> <tr> <td>Per scopi generali</td> <td>6 378 135</td> <td>6 356 750</td> <td>298,25274725275</td> <td>L'intero globo </td></tr></tbody></table> <p>Per poter costituire un sistema di riferimento un ellissoide deve essere posizionato ed orientato. Tradizionalmente gli ellissoidi di riferimento (e la loro realizzazione o datum) sono definiti localmente per meglio approssimare il geoide locale: di conseguenza non sono geocentrici. I moderni datum geodetici sono stabiliti con l'uso di tecnologia GPS, e sono quindi geocentrici. Il motivo principale è che il moto orbitale dei satelliti è relativo al centro di massa della terra. Una conseguenza positiva è che l'ellissoide così definito mantiene la sua validità a livello globale (es. WGS 84). </p> <div class="mw-heading mw-heading2"><h2 id="Ellissoidi_di_riferimento_per_altri_corpi_celesti">Ellissoidi di riferimento per altri corpi celesti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=6" title="Modifica la sezione Ellissoidi di riferimento per altri corpi celesti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=6" title="Edit section's source code: Ellissoidi di riferimento per altri corpi celesti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Gli ellissoidi di riferimento sono utili anche per mappare altri corpi celesti, come i pianeti, i loro satelliti, gli asteroidi e i nuclei delle comete. Alcuni corpi, già accuratamente osservati, hanno alcuni ellissoidi di riferimento propri abbastanza precisi, come la <a href="/wiki/Luna" title="Luna">Luna</a> e <a href="/wiki/Marte_(astronomia)" title="Marte (astronomia)">Marte</a>; anche riguardo il pianeta <a href="/wiki/Venere_(astronomia)" title="Venere (astronomia)">Venere</a> esiste un sistema di riferimento, stabilito convenzionalmente sulla superficie ideale di una sfera, centrata con il pianeta, del raggio di 6051 km<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=7" title="Modifica la sezione Note" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=7" title="Edit section's source code: Note"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1"><b>^</b></a> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.gsfc.nasa.gov/topstory/20020801gravityfield.html">Satellites Reveal A Mystery Of Large Change In Earth's Gravity Field</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100428205036/http://www.gsfc.nasa.gov/topstory/20020801gravityfield.html">Archiviato</a> il 28 aprile 2010 in <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>., Aug. 1, 2002, <a href="/wiki/Goddard_Space_Flight_Center" title="Goddard Space Flight Center">Goddard Space Flight Center</a>.</span> </li> <li id="cite_note-IAU-2"><a href="#cite_ref-IAU_2-0"><b>^</b></a> <span class="reference-text">La ridefinizione di "pianeta" data nel 2006 dall'<a href="/wiki/Unione_Astronomica_Internazionale" title="Unione Astronomica Internazionale">Unione Astronomica Internazionale</a> ha fornito la regola (2): un pianeta assuma la forma dovuta all'<a href="/wiki/Equilibrio_idrostatico" title="Equilibrio idrostatico">equilibrio idrostatico</a> dove la gravità e la forza centrifuga si bilanciano. <a rel="nofollow" class="external text" href="http://www.iau2006.org/mirror/www.iau.org/iau0603/index.html">IAU 2006 General Assembly: Result of the IAU Resolution votes</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061107022302/http://www.iau2006.org/mirror/www.iau.org/iau0603/index.html">Archiviato</a> il 7 novembre 2006 in <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3"><b>^</b></a> <span class="reference-text"><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.eso.org/public/outreach/eduoff/vt-2004/Background/Infol2/EIS-D6.html"><span style="font-style:italic;">ESO</span></a>, su <span style="font-style:italic;">www.eso.org</span>. <small>URL consultato il 7 luglio 2024</small>.</cite></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=8" title="Modifica la sezione Bibliografia" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=8" title="Edit section's source code: Bibliografia"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>P. K. Seidelmann (Chair), et al. (2005), “Report Of The IAU/IAG Working Group On Cartographic Coordinates And Rotational Elements: 2003,” <i>Celestial Mechanics and Dynamical Astronomy</i>, 91, pp. 203 – 215. <ul><li>Web address: <a rel="nofollow" class="external free" href="https://web.archive.org/web/20111102230508/http://astrogeology.usgs.gov/Projects/WGCCRE/">https://web.archive.org/web/20111102230508/http://astrogeology.usgs.gov/Projects/WGCCRE/</a></li></ul></li> <li><i>OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 1: Common architecture</i>, Annex B.4. 2005-11-30 <ul><li>Web address: <a rel="nofollow" class="external free" href="http://www.opengeospatial.org">http://www.opengeospatial.org</a></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Voci_correlate">Voci correlate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=9" title="Modifica la sezione Voci correlate" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=9" title="Edit section's source code: Voci correlate"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Raggio_della_Terra" class="mw-redirect" title="Raggio della Terra">Raggio della Terra</a></li> <li><a href="/wiki/Ellitticit%C3%A0" title="Ellitticità">Ellitticità</a></li> <li><a href="/wiki/Forma_della_Terra" class="mw-redirect" title="Forma della Terra">Forma della Terra</a></li> <li><a href="/wiki/Geoide" title="Geoide">Geoide</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Collegamenti_esterni">Collegamenti esterni</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ellissoide_di_riferimento&veaction=edit&section=10" title="Modifica la sezione Collegamenti esterni" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ellissoide_di_riferimento&action=edit&section=10" title="Edit section's source code: Collegamenti esterni"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100110061804/http://posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs.html"><span style="font-style:italic;">Coordinate System Index</span></a>, su <span style="font-style:italic;">posc.org</span>. <small>URL consultato il 13 aprile 2007</small> <small>(archiviato dall'<abbr title="http://www.posc.org/Epicentre.2_2/DataModel/ExamplesofUsage/eu_cs.html">url originale</abbr> il 10 gennaio 2010)</small>.</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://publib.boulder.ibm.com/infocenter/db2luw/v8/topic/com.ibm.db2.udb.doc/opt/csb3022a.htm"><span style="font-style:italic;">Geographic coordinate system</span></a>, su <span style="font-style:italic;">publib.boulder.ibm.com</span>.</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070402113758/http://www.spenvis.oma.be/spenvis/help/background/coortran/coortran.html"><span style="font-style:italic;">Coordinate systems and transformations</span></a>, su <span style="font-style:italic;">spenvis.oma.be</span>. <small>URL consultato il 13 aprile 2007</small> <small>(archiviato dall'<abbr title="http://www.spenvis.oma.be/spenvis/help/background/coortran/coortran.html">url originale</abbr> il 2 aprile 2007)</small>.</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061005140011/http://www.agnld.uni-potsdam.de/~shw/3_References/0_GPS/GPSHelmert1.html"><span style="font-style:italic;">Coordinate Systems, Frames and Datums</span></a>, su <span style="font-style:italic;">agnld.uni-potsdam.de</span>. <small>URL consultato il 13 aprile 2007</small> <small>(archiviato dall'<abbr title="http://www.agnld.uni-potsdam.de/~shw/3_References/0_GPS/GPSHelmert1.html">url originale</abbr> il 5 ottobre 2006)</small>.</cite></li> <li><cite class="citation web" style="font-style:normal"> <a rel="nofollow" class="external text" href="https://www.mathworks.com/access/helpdesk/help/toolbox/aeroblks/ecefpositiontolla.html"><span style="font-style:italic;">Aerospace Blockset: ECEF Position to LLA</span></a>, su <span style="font-style:italic;">mathworks.com</span>.</cite></li></ul> <div class="noprint" style="width:100%; 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Ellissoide di riferimento - Wikipedia