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Transformare geometrică - Wikipedia

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Disponibil în 29 limbi" > <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-29" 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">29 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar badge-Q70893996 mw-list-item" title=""><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_(%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A9)" title="تحويل (هندسة رياضية) – arabă" lang="ar" hreflang="ar" data-title="تحويل (هندسة رياضية)" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/H%C9%99nd%C9%99si_%C3%A7evrilm%C9%99l%C9%99r" title="Həndəsi çevrilmələr – azeră" lang="az" hreflang="az" data-title="Həndəsi çevrilmələr" data-language-autonym="Azərbaycanca" data-language-local-name="azeră" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Geometrijske_transformacije" title="Geometrijske transformacije – bosniacă" lang="bs" hreflang="bs" data-title="Geometrijske transformacije" data-language-autonym="Bosanski" data-language-local-name="bosniacă" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Transformaci%C3%B3_geom%C3%A8trica" title="Transformació geomètrica – catalană" lang="ca" hreflang="ca" data-title="Transformació geomètrica" data-language-autonym="Català" data-language-local-name="catalană" 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/Geometrick%C3%A9_zobrazen%C3%AD" title="Geometrické zobrazení – cehă" lang="cs" hreflang="cs" data-title="Geometrické zobrazení" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%81%D0%B5%D0%BD_%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%B9%C4%95" title="Координатсен трансформацийĕ – ciuvașă" lang="cv" hreflang="cv" data-title="Координатсен трансформацийĕ" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Trawsffurfiad_geometrig" title="Trawsffurfiad geometrig – galeză" lang="cy" hreflang="cy" data-title="Trawsffurfiad geometrig" data-language-autonym="Cymraeg" data-language-local-name="galeză" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Koordinatentransformation" title="Koordinatentransformation – germană" lang="de" hreflang="de" data-title="Koordinatentransformation" data-language-autonym="Deutsch" data-language-local-name="germană" 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%93%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CF%8C%CF%82_%CE%BC%CE%B5%CF%84%CE%B1%CF%83%CF%87%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CF%83%CE%BC%CF%8C%CF%82" title="Γεωμετρικός μετασχηματισμός – greacă" lang="el" hreflang="el" data-title="Γεωμετρικός μετασχηματισμός" data-language-autonym="Ελληνικά" data-language-local-name="greacă" 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/Geometric_transformation" title="Geometric transformation – engleză" lang="en" hreflang="en" data-title="Geometric transformation" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Geometria_transformado" title="Geometria transformado – esperanto" lang="eo" hreflang="eo" data-title="Geometria transformado" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Transformaci%C3%B3n_geom%C3%A9trica" title="Transformación geométrica – spaniolă" lang="es" hreflang="es" data-title="Transformación geométrica" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Transformation_g%C3%A9om%C3%A9trique" title="Transformation géométrique – franceză" lang="fr" hreflang="fr" data-title="Transformation géométrique" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="העתקה גאומטרית – ebraică" lang="he" hreflang="he" data-title="העתקה גאומטרית" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Transzform%C3%A1ci%C3%B3_(matematika)" title="Transzformáció (matematika) – maghiară" lang="hu" hreflang="hu" data-title="Transzformáció (matematika)" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Transformasi_geometri" title="Transformasi geometri – indoneziană" lang="id" hreflang="id" data-title="Transformasi geometri" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E5%AD%A6%E7%9A%84%E5%A4%89%E6%8F%9B" title="幾何学的変換 – japoneză" lang="ja" hreflang="ja" data-title="幾何学的変換" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%A2eometrisk%C4%81_transform%C4%81cija" title="Ģeometriskā transformācija – letonă" lang="lv" hreflang="lv" data-title="Ģeometriskā transformācija" data-language-autonym="Latviešu" data-language-local-name="letonă" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Co%C3%B6rdinatentransformatie" title="Coördinatentransformatie – neerlandeză" lang="nl" hreflang="nl" data-title="Coördinatentransformatie" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przekszta%C5%82cenie_geometryczne" title="Przekształcenie geometryczne – poloneză" lang="pl" hreflang="pl" data-title="Przekształcenie geometryczne" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Transforma%C3%A7%C3%A3o_geom%C3%A9trica" title="Transformação geométrica – portugheză" lang="pt" hreflang="pt" data-title="Transformação geométrica" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Преобразование координат – rusă" lang="ru" hreflang="ru" data-title="Преобразование координат" data-language-autonym="Русский" data-language-local-name="rusă" 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/Transformation" title="Transformation – Simple English" lang="en-simple" hreflang="en-simple" data-title="Transformation" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Koordinattransformation" title="Koordinattransformation – suedeză" lang="sv" hreflang="sv" data-title="Koordinattransformation" data-language-autonym="Svenska" data-language-local-name="suedeză" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%89%E0%AE%B0%E0%AF%81%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="வடிவவியல் உருமாற்றம் – tamilă" lang="ta" hreflang="ta" data-title="வடிவவியல் உருமாற்றம்" data-language-autonym="தமிழ்" data-language-local-name="tamilă" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9B%E0%B8%A5%E0%B8%87%E0%B8%97%E0%B8%B2%E0%B8%87%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="การแปลงทางเรขาคณิต – thailandeză" lang="th" hreflang="th" data-title="การแปลงทางเรขาคณิต" data-language-autonym="ไทย" data-language-local-name="thailandeză" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B5_%D0%BF%D0%B5%D1%80%D0%B5%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Геометричне перетворення – ucraineană" lang="uk" hreflang="uk" data-title="Геометричне перетворення" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%B9%BE%E4%BD%95%E8%AE%8A%E6%8F%9B" title="幾何變換 – chineză" lang="zh" hreflang="zh" data-title="幾何變換" data-language-autonym="中文" data-language-local-name="chineză" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link 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aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspect"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspect</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ascunde</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">De la Wikipedia, enciclopedia liberă</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="ro" dir="ltr"><p>În <a href="/wiki/Matematic%C4%83" title="Matematică">matematică</a>, o <b>transformare geometrică</b> este o <a href="/wiki/No%C8%9Biune" title="Noțiune">noțiune</a> similară cu <a href="/wiki/Opera%C8%9Bie_algebric%C4%83" title="Operație algebrică">operație algebrică</a>, în care <a href="/wiki/Operand" title="Operand">operanzii</a> sunt elemente geometrice. Este o <a href="/wiki/Bijec%C8%9Bie" class="mw-redirect" title="Bijecție">bijecție</a> a unei <a href="/wiki/Mul%C8%9Bime" title="Mulțime">mulțimi</a> pe sine (sau pe o altă astfel de mulțime) cu caracteristici geometrice importante. Mai precis, este o <a href="/wiki/Func%C8%9Bie" title="Funcție">funcție</a> al cărei <a href="/wiki/Domeniu_(matematic%C4%83)" class="mw-redirect" title="Domeniu (matematică)">domeniu</a> și interval sunt mulțimi de <a href="/wiki/Punct_(geometrie)" title="Punct (geometrie)">puncte</a> — cel mai adesea ambele din <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> sau ambele din <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> — astfel încât funcția să fie <a href="/wiki/Func%C8%9Bie_injectiv%C4%83" title="Funcție injectivă">injectivă</a> și <a href="/wiki/Func%C8%9Bie_invers%C4%83" title="Funcție inversă">funcția inversă</a> să existe.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> Studiul <a href="/wiki/Geometrie" title="Geometrie">geometriei</a> poate fi abordat prin studiul acestor transformări<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup> care sunt utilizate ca <a href="/wiki/Opera%C8%9Bie_(matematic%C4%83)" title="Operație (matematică)">operații matematice</a> în construirea diferitelor forme geometrice. Inițierea acestei abordări se datorează matematicianului sovietic <a href="/wiki/Andrei_Kolmogorov" class="mw-redirect" title="Andrei Kolmogorov">Andrei Kolmogorov</a><sup id="cite_ref-russianedu_3-0" class="reference"><a href="#cite_note-russianedu-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup>. </p><p>Exemple: <a href="/wiki/Rota%C8%9Bie_(matematic%C4%83)" title="Rotație (matematică)">rotație</a>, <a href="/wiki/Transla%C8%9Bie" title="Translație">translație</a> etc. </p><p>Geometria modernă utilizează masiv această noțiune. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Clasificări"><span id="Clasific.C4.83ri"></span>Clasificări</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transformare_geometric%C4%83&amp;veaction=edit&amp;section=1" title="Modifică secțiunea: Clasificări" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Transformare_geometric%C4%83&amp;action=edit&amp;section=1" title="Edit section&#039;s source code: Clasificări"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Transformările geometrice pot fi clasificate în funcție de numărul <a href="/wiki/Operand" title="Operand">operanzilor</a> (deosebind astfel, de exemplu, transformările din <a href="/wiki/Plan_(geometrie)" title="Plan (geometrie)">plan</a> și transformările din spațiul tridimensional). De asemenea, ele pot fi clasificate în funcție de proprietățile pe care le conservă: </p> <ul><li><a href="/wiki/Deplasare_(geometrie)" title="Deplasare (geometrie)">Deplasările</a><sup id="cite_ref-LȚ_4-0" class="reference"><a href="#cite_note-LȚ-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup> conservă <a href="/wiki/Metric%C4%83" title="Metrică">distanțele</a> și <a href="/wiki/Unghi" title="Unghi">unghiurile orientate</a> (de exemplu <a href="/wiki/Transla%C8%9Bie_(geometrie)" title="Translație (geometrie)">translațiile</a><sup id="cite_ref-LȚ_4-1" class="reference"><a href="#cite_note-LȚ-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup>);<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">&#91;</span>5<span class="cite-bracket">&#93;</span></a></sup></li> <li><a href="/wiki/Izometrie" title="Izometrie">Izometriile</a> conservă distanțele și unghiurile (de exemplu transformările euclidiene);<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">&#91;</span>6<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-Berger_7-0" class="reference"><a href="#cite_note-Berger-7"><span class="cite-bracket">&#91;</span>7<span class="cite-bracket">&#93;</span></a></sup></li> <li><a href="/wiki/Asem%C4%83nare_(geometrie)" title="Asemănare (geometrie)">Asemănările</a> conservă unghiurile și <a href="/wiki/Raport" title="Raport">raporturile</a> dintre distanțe (de exemplu <a href="/wiki/Scalare_(geometrie)" title="Scalare (geometrie)">scalările</a><sup id="cite_ref-LȚ_4-2" class="reference"><a href="#cite_note-LȚ-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup>);<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">&#91;</span>8<span class="cite-bracket">&#93;</span></a></sup></li> <li><a href="/w/index.php?title=Transformare_afin%C4%83&amp;action=edit&amp;redlink=1" class="new" title="Transformare afină — pagină inexistentă">Transformările afine</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q382497" class="extiw" title="d:Q382497"><span title="transformare afină la Wikidata">d</span></a>)</small></sup> conservă <a href="/wiki/Paralelism_(geometrie)" title="Paralelism (geometrie)">paralelismul</a> (de exemplu scalările, <a href="/w/index.php?title=Forfecare_(geometrie)&amp;action=edit&amp;redlink=1" class="new" title="Forfecare (geometrie) — pagină inexistentă">forfecările</a><sup id="cite_ref-LȚ_4-3" class="reference"><a href="#cite_note-LȚ-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup>);<sup id="cite_ref-Berger_7-1" class="reference"><a href="#cite_note-Berger-7"><span class="cite-bracket">&#91;</span>7<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">&#91;</span>9<span class="cite-bracket">&#93;</span></a></sup></li> <li><a href="/w/index.php?title=Omografie&amp;action=edit&amp;redlink=1" class="new" title="Omografie — pagină inexistentă">Transformările proiective</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q2112539" class="extiw" title="d:Q2112539"><span title="omografie la Wikidata">d</span></a>)</small></sup> conservă <a href="/wiki/Coliniaritate" title="Coliniaritate">coliniaritatea</a>;<sup id="cite_ref-Wilkinson_10-0" class="reference"><a href="#cite_note-Wilkinson-10"><span class="cite-bracket">&#91;</span>10<span class="cite-bracket">&#93;</span></a></sup></li></ul> <p>Fiecare dintre aceste clase o conține pe cea anterioară.<sup id="cite_ref-Wilkinson_10-1" class="reference"><a href="#cite_note-Wilkinson-10"><span class="cite-bracket">&#91;</span>10<span class="cite-bracket">&#93;</span></a></sup> </p> <ul><li><a href="/wiki/Transformare_M%C3%B6bius" title="Transformare Möbius">Transformările Möbius</a>, care utilizează coordonate în <a href="/wiki/Planul_complex" title="Planul complex">planul complex</a> (precum și <a href="/w/index.php?title=Inversiune_fa%C8%9B%C4%83_de_sfer%C4%83&amp;action=edit&amp;redlink=1" class="new" title="Inversiune față de sferă — pagină inexistentă">inversiunea față de cerc</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q17152833" class="extiw" title="d:Q17152833"><span title="inversiune față de sferă la Wikidata">d</span></a>)</small></sup>), păstrează mulțimea tuturor dreptelor și cercurilor, dar pot interschimba drepte și cercuri.</li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_identique.gif" class="mw-file-description" title="Imaginea inițială (harta Franței)"><img alt="Imaginea inițială (harta Franței)" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/France_identique.gif/108px-France_identique.gif" decoding="async" width="108" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/ef/France_identique.gif 1.5x" data-file-width="161" data-file-height="178" /></a></span></div> <div class="gallerytext"> Imaginea inițială<br />(harta Franței)</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_par_rotation_180deg.gif" class="mw-file-description" title="Izometrie"><img alt="Izometrie" src="//upload.wikimedia.org/wikipedia/commons/6/64/France_par_rotation_180deg.gif" decoding="async" width="109" height="120" class="mw-file-element" data-file-width="108" data-file-height="119" /></a></span></div> <div class="gallerytext"> Izometrie</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_par_similitude.gif" class="mw-file-description" title="Asemănare"><img alt="Asemănare" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/France_par_similitude.gif/120px-France_par_similitude.gif" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/France_par_similitude.gif/180px-France_par_similitude.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/47/France_par_similitude.gif 2x" data-file-width="200" data-file-height="200" /></a></span></div> <div class="gallerytext"> Asemănare</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_affine_(1).gif" class="mw-file-description" title="Transformare afină"><img alt="Transformare afină" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/France_affine_%281%29.gif/120px-France_affine_%281%29.gif" decoding="async" width="120" height="53" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/France_affine_%281%29.gif/180px-France_affine_%281%29.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/France_affine_%281%29.gif/240px-France_affine_%281%29.gif 2x" data-file-width="283" data-file-height="126" /></a></span></div> <div class="gallerytext"> Transformare afină</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_homographie.gif" class="mw-file-description" title="Perspectivă"><img alt="Perspectivă" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/France_homographie.gif/112px-France_homographie.gif" decoding="async" width="112" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/France_homographie.gif/169px-France_homographie.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/5/5a/France_homographie.gif 2x" data-file-width="224" data-file-height="239" /></a></span></div> <div class="gallerytext"> Perspectivă</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_circ.gif" class="mw-file-description" title="Inversiune față de cerc"><img alt="Inversiune față de cerc" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/France_circ.gif/120px-France_circ.gif" decoding="async" width="120" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/France_circ.gif/180px-France_circ.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4b/France_circ.gif 2x" data-file-width="197" data-file-height="191" /></a></span></div> <div class="gallerytext"> Inversiune față de cerc</div> </li> </ul> <ul><li><a href="/wiki/Transformare_conform%C4%83" title="Transformare conformă">Transformările conforme</a> conservă unghiurile și sunt, în primul rând, asemănări.</li> <li><a href="/wiki/Transformare_echiareal%C4%83" title="Transformare echiareală">Transformările echiareale</a>, conservă <a href="/wiki/Arie" title="Arie">ariile</a> în cazurile <a href="/wiki/Bidimensional" class="mw-redirect" title="Bidimensional">bidimensionale</a> sau <a href="/wiki/Volum_(geometrie)" title="Volum (geometrie)">volumele</a> în cazurile <a href="/wiki/Tridimensional" class="mw-redirect" title="Tridimensional">tridimensionale</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">&#91;</span>11<span class="cite-bracket">&#93;</span></a></sup> și sunt, în primul rând, transformări afine cu <a href="/wiki/Determinant_(matematic%C4%83)" title="Determinant (matematică)">determinantul</a> 1.</li> <li><a href="/w/index.php?title=Omeomorfism&amp;action=edit&amp;redlink=1" class="new" title="Omeomorfism — pagină inexistentă">Homeomorfismele</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q202906" class="extiw" title="d:Q202906"><span title="omeomorfism la Wikidata">d</span></a>)</small></sup> (transformări bicontinue) consevă <a href="/wiki/Vecin%C4%83tate_(matematic%C4%83)" title="Vecinătate (matematică)">vecinătățile</a> <a href="/wiki/Punct_(geometrie)" title="Punct (geometrie)">punctelor</a>.</li> <li><a href="/wiki/Difeomorfism" title="Difeomorfism">Difeomorfismele</a> (transformări bidiferențiabile) sunt transformări care în primul rând sunt afine; le conțin pe cele precedente drept cazuri particulare și pot fi detaliate în continuare.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">&#91;</span>12<span class="cite-bracket">&#93;</span></a></sup></li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Fconf.gif" class="mw-file-description" title="Transformare conformă"><img alt="Transformare conformă" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Fconf.gif/106px-Fconf.gif" decoding="async" width="106" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Fconf.gif/159px-Fconf.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/9/94/Fconf.gif 2x" data-file-width="164" data-file-height="186" /></a></span></div> <div class="gallerytext"> Transformare conformă</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_aire.gif" class="mw-file-description" title="Transformare echiareală"><img alt="Transformare echiareală" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/France_aire.gif/120px-France_aire.gif" decoding="async" width="120" height="84" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0a/France_aire.gif/180px-France_aire.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0a/France_aire.gif 2x" data-file-width="217" data-file-height="152" /></a></span></div> <div class="gallerytext"> Transformare echiareală</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_homothetie.gif" class="mw-file-description" title="Homeomorfism"><img alt="Homeomorfism" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/France_homothetie.gif/120px-France_homothetie.gif" decoding="async" width="120" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/France_homothetie.gif/180px-France_homothetie.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ef/France_homothetie.gif 2x" data-file-width="182" data-file-height="159" /></a></span></div> <div class="gallerytext"> Homeomorfism</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:France_diff.gif" class="mw-file-description" title="Difeomorfism"><img alt="Difeomorfism" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/France_diff.gif/120px-France_diff.gif" decoding="async" width="120" height="112" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/France_diff.gif/180px-France_diff.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/7/79/France_diff.gif 2x" data-file-width="188" data-file-height="175" /></a></span></div> <div class="gallerytext"> Difeomorfism</div> </li> </ul> <p>Transformările de același tip formează <a href="/wiki/Grup_(matematic%C4%83)" title="Grup (matematică)">grupuri</a> care pot fi subgrupuri ale altor grupuri de transformări. </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=Transformare_geometric%C4%83&amp;veaction=edit&amp;section=2" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Transformare_geometric%C4%83&amp;action=edit&amp;section=2" title="Edit section&#039;s source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> Zalman Usiskin, Anthony L. Peressini, Elena Marchisotto, <i>Mathematics for High School Teachers: An Advanced Perspective</i>, page 84.</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFVenema2006" class="citation">Venema, Gerard A. 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Chelsea. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0-8284-1087-9" title="Special:Referințe în cărți/0-8284-1087-9">0-8284-1087-9</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Geometry+and+the+Imagination&amp;rft.edition=2nd&amp;rft.pub=Chelsea&amp;rft.date=1952&amp;rft.isbn=0-8284-1087-9&amp;rft.aulast=Hilbert&amp;rft.aufirst=David&amp;rft.au=Cohn-Vossen%2C+Stephan&amp;rfr_id=info%3Asid%2Fro.wikipedia.org%3ATransformare+geometric%C4%83" class="Z3988"><span style="display:none;">&#160;</span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> John McCleary – <i>Geometry from a Differentiable Viewpoint</i>.</li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> Modenov, P. S.; Parkhomenko, A. S. (1965) . <i>Geometric Transformations</i> (2 vols.): <i>Euclidean and Affine Transformations</i>, and <i>Projective Transformations</i>. New York: Academic Press.</li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> A. N. Pressley – <i>Elementary Differential Geometry</i>.</li> <li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> Isaak Yaglom (1962, 1968, 1973, 2009) . <i>Geometric Transformations</i> (4 vols.). <a href="/wiki/Random_House" title="Random House">Random House</a> (I, II &amp; III), <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">MAA</a> (I, II, III &amp; IV).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Transformare_geometric%C4%83&amp;veaction=edit&amp;section=4" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Transformare_geometric%C4%83&amp;action=edit&amp;section=4" title="Edit section&#039;s source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint tright portal" style="border:solid #aaa 1px; margin:0.5em 0 0.5em 0.5em;"> <table style="background:var(--background-color-interactive-subtle, #f9f9f9); color:inherit; font-size:85%; line-height:110%; max-width:175px;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Nuvola_apps_edu_mathematics-p-blue.svg" class="mw-file-description"><img alt="Portal icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_mathematics-p-blue.svg/28px-Nuvola_apps_edu_mathematics-p-blue.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_mathematics-p-blue.svg/42px-Nuvola_apps_edu_mathematics-p-blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_mathematics-p-blue.svg/56px-Nuvola_apps_edu_mathematics-p-blue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </td> <td style="padding: 0 0.2em; vertical-align: middle; font-style: italic; font-weight: bold"><b><a href="/wiki/Portal:Matematic%C4%83" title="Portal:Matematică">Portal Matematică </a></b> </td></tr> </tbody></table></div> <ul><li><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Commons-logo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Materiale media legate de <span class="plainlinks"><b><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Transformations_(geometry)?uselang=ro">transformare geometrică</a></b></span> la <a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons">Wikimedia Commons</a></li></ul> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐d465dfd78‐r87qr Cached time: 20241126140352 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.164 seconds Real time usage: 0.312 seconds Preprocessor visited node count: 725/1000000 Post‐expand include size: 21091/2097152 bytes Template argument size: 194/2097152 bytes Highest expansion depth: 9/100 Expensive parser function count: 1/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 36259/5000000 bytes Lua time usage: 0.084/10.000 seconds Lua memory usage: 3423986/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 226.108 1 -total 41.86% 94.639 4 Format:Ill-wd 17.23% 38.969 2 Format:Citation 8.13% 18.384 5 Format:Cite_web 6.98% 15.772 1 Format:Commonscat-inline 3.50% 7.910 1 Format:Portal 2.35% 5.303 1 Format:Cite_book 2.01% 4.553 1 Format:Portal/core 1.12% 2.530 18 Format:En_icon 0.82% 1.858 1 Format:Portal/Imagine/Matematică --> <!-- Saved in parser cache with key rowiki:pcache:2900346:|#|:idhash:canonical and timestamp 20241126140352 and revision id 16488448. 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