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<div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Quoziente</span> </div> </a> <ul id="toc-Quoziente-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordinate_omogenee" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coordinate_omogenee"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Coordinate omogenee</span> </div> </a> <ul id="toc-Coordinate_omogenee-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Punto_all'infinito" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Punto_all'infinito"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Punto all'infinito</span> </div> </a> <ul id="toc-Punto_all'infinito-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Esempi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Esempi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Esempi</span> </div> </a> <button aria-controls="toc-Esempi-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 Esempi</span> </button> <ul id="toc-Esempi-sublist" class="vector-toc-list"> <li id="toc-Caso_reale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caso_reale"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Caso reale</span> </div> </a> <ul id="toc-Caso_reale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caso_complesso" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caso_complesso"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Caso complesso</span> </div> </a> <ul id="toc-Caso_complesso-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Campi_finiti" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Campi_finiti"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Campi finiti</span> </div> </a> <ul id="toc-Campi_finiti-sublist" class="vector-toc-list"> </ul> </li> </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">3</span> <span>Voci correlate</span> </div> </a> <ul id="toc-Voci_correlate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Altri_progetti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Altri_progetti"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Altri progetti</span> </div> </a> <ul id="toc-Altri_progetti-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">Retta proiettiva</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 10 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-10" 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">10 lingue</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Projektivn%C3%AD_p%C5%99%C3%ADmka" title="Projektivní přímka - ceco" lang="cs" hreflang="cs" data-title="Projektivní přímka" data-language-autonym="Čeština" data-language-local-name="ceco" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Projektive_Gerade" title="Projektive Gerade - tedesco" lang="de" hreflang="de" data-title="Projektive Gerade" data-language-autonym="Deutsch" data-language-local-name="tedesco" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Projective_line" title="Projective line - inglese" lang="en" hreflang="en" data-title="Projective line" 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/Recta_proyectiva" title="Recta proyectiva - spagnolo" lang="es" hreflang="es" data-title="Recta proyectiva" data-language-autonym="Español" data-language-local-name="spagnolo" 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/Droite_projective" title="Droite projective - francese" lang="fr" hreflang="fr" data-title="Droite projective" data-language-autonym="Français" data-language-local-name="francese" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B0%84%E5%BD%B1%E7%9B%B4%E7%B7%9A" title="射影直線 - giapponese" lang="ja" hreflang="ja" data-title="射影直線" data-language-autonym="日本語" data-language-local-name="giapponese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Projectieve_lijn" title="Projectieve lijn - olandese" lang="nl" hreflang="nl" data-title="Projectieve lijn" data-language-autonym="Nederlands" data-language-local-name="olandese" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Dreapt%C4%83_proiectiv%C4%83" title="Dreaptă proiectivă - rumeno" lang="ro" hreflang="ro" data-title="Dreaptă proiectivă" data-language-autonym="Română" data-language-local-name="rumeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D1%8F%D0%BC%D0%B0%D1%8F" title="Проективная прямая - russo" lang="ru" hreflang="ru" data-title="Проективная прямая" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B5%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%B0_%D0%BF%D1%80%D1%8F%D0%BC%D0%B0" title="Проективна пряма - ucraino" lang="uk" hreflang="uk" data-title="Проективна пряма" data-language-autonym="Українська" data-language-local-name="ucraino" class="interlanguage-link-target"><span>Українська</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/Q764115#sitelinks-wikipedia" title="Modifica collegamenti interlinguistici" class="wbc-editpage">Modifica collegamenti</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespace"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Retta_proiettiva" title="Vedi la voce [c]" accesskey="c"><span>Voce</span></a></li><li id="ca-talk" 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mw-list-item"><a href="/w/index.php?title=Retta_proiettiva&action=history" title="Versioni precedenti di questa pagina [h]" accesskey="h"><span>Cronologia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Strumenti" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Strumenti</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div 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class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciale:PuntanoQui/Retta_proiettiva" title="Elenco di tutte le pagine che sono collegate a questa [j]" accesskey="j"><span>Puntano qui</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciale:ModificheCorrelate/Retta_proiettiva" rel="nofollow" title="Elenco delle ultime modifiche alle pagine collegate a questa [k]" accesskey="k"><span>Modifiche correlate</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciale:PagineSpeciali" title="Elenco di tutte le pagine speciali [q]" accesskey="q"><span>Pagine speciali</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Retta_proiettiva&oldid=115551236" title="Collegamento permanente a questa versione di questa pagina"><span>Link permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Retta_proiettiva&action=info" title="Ulteriori informazioni su questa pagina"><span>Informazioni pagina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciale:Cita&page=Retta_proiettiva&id=115551236&wpFormIdentifier=titleform" title="Informazioni su come citare questa pagina"><span>Cita questa voce</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciale:UrlShortener&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FRetta_proiettiva"><span>Ottieni URL breve</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speciale:QrCode&url=https%3A%2F%2Fit.wikipedia.org%2Fwiki%2FRetta_proiettiva"><span>Scarica codice QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Stampa/esporta </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciale:Libro&bookcmd=book_creator&referer=Retta+proiettiva"><span>Crea un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciale:DownloadAsPdf&page=Retta_proiettiva&action=show-download-screen"><span>Scarica come PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Retta_proiettiva&printable=yes" title="Versione stampabile di questa pagina [p]" accesskey="p"><span>Versione stampabile</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In altri progetti </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q764115" title="Collegamento all'elemento connesso dell'archivio dati [g]" accesskey="g"><span>Elemento Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Strumenti pagine"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspetto"> <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">Aspetto</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">sposta nella barra laterale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" 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"><style data-mw-deduplicate="TemplateStyles:r133964453">.mw-parser-output .avviso .mbox-text-div>div,.mw-parser-output .avviso .mbox-text-full-div>div{font-size:90%}.mw-parser-output .avviso .mbox-image{flex-basis:52px;flex-grow:0;flex-shrink:0}.mw-parser-output .avviso .mbox-text-full-div .hide-when-compact{display:block}</style><div style="" class="ambox metadata plainlinks avviso avviso-contenuto"> <div class="avviso-immagine mbox-image noprint"><span typeof="mw:File"><a href="/wiki/File:Question_book-4.svg" class="mw-file-description" title="Niente fonti!"><img alt="Niente fonti!" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/45px-Question_book-4.svg.png" decoding="async" width="45" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/68px-Question_book-4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Question_book-4.svg/90px-Question_book-4.svg.png 2x" data-file-width="262" data-file-height="204" /></a></span></div> <div class="avviso-testo mbox-text"> <div class="mbox-text-div"><b>Questa voce o sezione  sull'argomento geometria <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">non cita le fonti necessarie</a> o quelle presenti sono insufficienti</b>. <div class="hide-when-compact"> <div class="noprint"><hr />Puoi <a class="external text" href="https://it.wikipedia.org/w/index.php?title=Retta_proiettiva&action=edit">migliorare questa voce</a> aggiungendo citazioni da <a href="/wiki/Wikipedia:Fonti_attendibili" title="Wikipedia:Fonti attendibili">fonti attendibili</a> secondo le <a href="/wiki/Wikipedia:Uso_delle_fonti" title="Wikipedia:Uso delle fonti">linee guida sull'uso delle fonti</a>. Segui i suggerimenti del <a href="/wiki/Progetto:Geometria" class="mw-redirect" title="Progetto:Geometria">progetto di riferimento</a>.</div> </div> </div> </div> </div> <p>In <a href="/wiki/Matematica" title="Matematica">matematica</a>, e più precisamente in <a href="/wiki/Geometria_proiettiva" title="Geometria proiettiva">geometria proiettiva</a>, la <b>retta proiettiva</b> è un'estensione della <a href="/wiki/Retta" title="Retta">retta</a>, ottenuta aggiungendo il "punto all'infinito". </p><p>Nel caso della <a href="/wiki/Retta_reale" class="mw-redirect" title="Retta reale">retta reale</a>, si distingue dalla <a href="/wiki/Retta_estesa" class="mw-redirect" title="Retta estesa">retta estesa</a>, che è ottenuta aggiungendo <i>due</i> punti all'infinito, uno per ogni verso: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span> 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 -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span>. </p><p>A differenza della retta estesa, che è definita soltanto per i <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a>, il concetto di retta proiettiva si applica poi su qualsiasi <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a> (ad esempio, il campo dei <a href="/wiki/Numeri_complessi" class="mw-redirect" title="Numeri complessi">complessi</a>), ed è la versione 1-dimensionale del concetto più generale di <a href="/wiki/Spazio_proiettivo" title="Spazio proiettivo">spazio proiettivo</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definizione">Definizione</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=1" title="Modifica la sezione Definizione" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=1" title="Edit section's source code: Definizione"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una definizione informale di retta proiettiva, dipendente da un campo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, potrebbe essere data aggiungendo semplicemente un punto 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span>, chiamato "infinito" o <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"></span>. Una definizione di questo tipo non mostra però come questo nuovo punto debba essere considerato nella nuova struttura: si sceglie quindi (come in tutti gli <a href="/wiki/Spazio_proiettivo" title="Spazio proiettivo">spazi proiettivi</a>) una definizione più formale ed omogenea, apparentemente molto diversa, che considera subito tutti i punti allo stesso livello. Le due descrizioni arrivano quindi a coincidere al momento in cui si deciderà che un dato punto è "quello all'infinito". </p> <div class="mw-heading mw-heading3"><h3 id="Quoziente">Quoziente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=2" title="Modifica la sezione Quoziente" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=2" title="Edit section's source code: Quoziente"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> un <a href="/wiki/Campo_(matematica)" title="Campo (matematica)">campo</a>. La <b>retta proiettiva</b> su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> è definita a partire dal piano </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 K^{2}=\{(x,y)\ |\ x,y\in K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{2}=\{(x,y)\ |\ x,y\in K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab8b1324a21e1af0b0e2b51c120262d33bc2f00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.134ex; height:3.176ex;" alt="{\displaystyle K^{2}=\{(x,y)\ |\ x,y\in K\}}"></span></dd></dl> <p>rimuovendo l'origine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> e <a href="/wiki/Insieme_quoziente" class="mw-redirect" title="Insieme quoziente">quozientando</a> per la <a href="/wiki/Relazione_d%27equivalenza" class="mw-redirect" title="Relazione d'equivalenza">relazione d'equivalenza</a> </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 (x_{1},y_{1})\sim (\lambda x_{1},\lambda y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∼</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>λ</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})\sim (\lambda x_{1},\lambda y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c348dab68073c4a41596481963148038a834c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.65ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})\sim (\lambda x_{1},\lambda y_{1})}"></span></dd></dl> <p>che identifica due punti ottenuti l'uno dall'altro tramite riscalamento per un fattore reale non nullo <span class="mwe-math-element"><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>. In altre parole, identifica tutti i punti presenti su ogni singola retta passante per l'origine, esclusa l'origine stessa. Formalmente: </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 \mathbb {P} ^{1}(K)={\big (}K^{2}\setminus \{(0,0)\}{\big )}/\sim .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{1}(K)={\big (}K^{2}\setminus \{(0,0)\}{\big )}/\sim .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799632494c3671b3842be874ed590fbe16884a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.677ex; height:3.343ex;" alt="{\displaystyle \mathbb {P} ^{1}(K)={\big (}K^{2}\setminus \{(0,0)\}{\big )}/\sim .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Coordinate_omogenee">Coordinate omogenee</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=3" title="Modifica la sezione Coordinate omogenee" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=3" title="Edit section's source code: Coordinate omogenee"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Come in ogni spazio proiettivo, ogni punto della retta proiettiva è quindi identificato da una coppia di <a href="/wiki/Coordinate_omogenee" title="Coordinate omogenee">coordinate omogenee</a> </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 P=[x_{0},x_{1}]\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=[x_{0},x_{1}]\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72dd84177a9b90a6efb71d01468e31c927c06d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:12.326ex; height:2.843ex;" alt="{\displaystyle P=[x_{0},x_{1}]\,\!}"></span></dd></dl> <p>dove si intende che moltiplicando entrambi i valori <span class="mwe-math-element"><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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> 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 x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> per un numero <span class="mwe-math-element"><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 \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedcbbf652c56a75cb890f36b4ff1f3bd984af45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.616ex; height:2.676ex;" alt="{\displaystyle \lambda \neq 0}"></span> si ottiene lo stesso punto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=[x_{0},x_{1}]=[\lambda x_{0},\lambda x_{1}].\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=[x_{0},x_{1}]=[\lambda x_{0},\lambda x_{1}].\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf341112be053bd83e229df7de3e681a792bcb72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:25.878ex; height:2.843ex;" alt="{\displaystyle P=[x_{0},x_{1}]=[\lambda x_{0},\lambda x_{1}].\,\!}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Punto_all'infinito"><span id="Punto_all.27infinito"></span>Punto all'infinito</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=4" title="Modifica la sezione Punto all'infinito" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=4" title="Edit section's source code: Punto all'infinito"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usando queste coordinate, è possibile ricavare la descrizione più familiare di retta proiettiva come unione di una retta normale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> e di un "punto all'infinito". Infatti </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 \mathbb {P} ^{1}(K)=\{[1,0]\}\cup \{[x,1]\ |\ x\in K\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mtext> </mtext> <mi>x</mi> <mo>∈</mo> <mi>K</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{1}(K)=\{[1,0]\}\cup \{[x,1]\ |\ x\in K\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c3fd9ebcc0453db256be83294cd19c0905c7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.197ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} ^{1}(K)=\{[1,0]\}\cup \{[x,1]\ |\ x\in K\}}"></span></dd></dl> <p>poiché a meno di riscalamento ogni coppia <span class="mwe-math-element"><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_{0},x_{1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{0},x_{1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec10faf54bec6a09b997bebef2b4417ec2ebc8b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.095ex; height:2.843ex;" alt="{\displaystyle [x_{0},x_{1}]}"></span> può essere espressa unicamente in uno dei modi descritti. In questa descrizione, il "punto all'infinito" è <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1,0]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [1,0]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c494ca3000263cb454b314529ef0573dde76e4a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [1,0]}"></span>. Ogni punto della retta proiettiva può però essere identificato come "punto all'infinito" in una opportuna descrizione. </p> <div class="mw-heading mw-heading2"><h2 id="Esempi">Esempi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=5" title="Modifica la sezione Esempi" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=5" title="Edit section's source code: Esempi"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Caso_reale">Caso reale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=6" title="Modifica la sezione Caso reale" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=6" title="Edit section's source code: Caso reale"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 K=\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6419d3aa99701ca996737b17a5e1174d53e6c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.842ex; height:2.176ex;" alt="{\displaystyle K=\mathbb {R} }"></span> è il campo dei <a href="/wiki/Numeri_reali" class="mw-redirect" title="Numeri reali">numeri reali</a>, la retta proiettiva è ottenuta aggiungendo un punto all'infinito alla <a href="/wiki/Retta_reale" class="mw-redirect" title="Retta reale">retta reale</a>. Dal punto di vista <a href="/wiki/Topologia" title="Topologia">topologico</a>, lo spazio che si ottiene è una <a href="/wiki/Circonferenza" title="Circonferenza">circonferenza</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Caso_complesso">Caso complesso</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=7" title="Modifica la sezione Caso complesso" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=7" title="Edit section's source code: Caso complesso"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Proiezione_stereografica.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Proiezione_stereografica.PNG/350px-Proiezione_stereografica.PNG" decoding="async" width="350" height="168" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/f/f8/Proiezione_stereografica.PNG 1.5x" data-file-width="523" data-file-height="251" /></a><figcaption>La retta proiettiva complessa è la <a href="/wiki/Sfera_di_Riemann" title="Sfera di Riemann">sfera di Riemann</a>, ottenuta aggiungendo il punto all'infinito al <a href="/wiki/Piano_complesso" title="Piano complesso">piano complesso</a>.</figcaption></figure> <p>Il caso complesso risulta essere di notevole interesse in matematica e in geometria. La retta proiettiva complessa <span class="mwe-math-element"><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 {P} ^{1}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{1}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dd640bbba6e2ba26c35d1b34bb1e00153ad982b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.962ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} ^{1}(\mathbb {C} )}"></span> è ottenuta aggiungendo un punto al <a href="/wiki/Piano_complesso" title="Piano complesso">piano complesso</a>. Topologicamente, come si evince dalla <a href="/wiki/Proiezione_stereografica" title="Proiezione stereografica">proiezione stereografica</a>, è una <a href="/wiki/Sfera" title="Sfera">sfera</a>, detta <a href="/wiki/Sfera_di_Riemann" title="Sfera di Riemann">sfera di Riemann</a>. La sfera di Riemann è un oggetto importante, che ha molti collegamenti con vari ambiti della geometria: è centrale infatti sia nella <a href="/wiki/Geometria_proiettiva" title="Geometria proiettiva">geometria proiettiva</a> che nella <a href="/wiki/Geometria_differenziale" title="Geometria differenziale">differenziale</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Campi_finiti">Campi finiti</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=8" title="Modifica la sezione Campi finiti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=8" title="Edit section's source code: Campi finiti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La definizione è ovviamente valida anche nel caso in cui il campo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> sia un <a href="/wiki/Campo_finito" title="Campo finito">campo finito</a>, con <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> elementi. In questo caso, la retta proiettiva consta 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 n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"></span> elementi. </p> <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=Retta_proiettiva&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=Retta_proiettiva&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/Spazio_proiettivo" title="Spazio proiettivo">Spazio proiettivo</a></li> <li><a href="/wiki/Piano_proiettivo" title="Piano proiettivo">Piano proiettivo</a></li> <li><a href="/wiki/Trasformazione_di_M%C3%B6bius" title="Trasformazione di Möbius">Trasformazione di Möbius</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Altri_progetti">Altri progetti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Retta_proiettiva&veaction=edit&section=10" title="Modifica la sezione Altri progetti" class="mw-editsection-visualeditor"><span>modifica</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Retta_proiettiva&action=edit&section=10" title="Edit section's source code: Altri progetti"><span>modifica wikitesto</span></a><span class="mw-editsection-bracket">]</span></span></div> <div id="interProject" class="toccolours" style="display: none; clear: both; margin-top: 2em"><p id="sisterProjects" style="background-color: #efefef; color: black; font-weight: bold; margin: 0"><span>Altri progetti</span></p><ul title="Collegamenti verso gli altri progetti Wikimedia"> <li class="" title=""><span class="plainlinks" title="commons:Projective line over a finite field"><a class="external text" href="https://commons.wikimedia.org/wiki/Projective_line_over_a_finite_field?uselang=it">Wikimedia Commons</a></span></li></ul></div> <ul><li><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/?uselang=it" title="Collabora a Wikimedia Commons"><img alt="Collabora a Wikimedia Commons" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png" decoding="async" width="18" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/27px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/36px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/?uselang=it">Wikimedia Commons</a></span> contiene immagini o altri file su <b><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Projective_line_over_a_finite_field?uselang=it">retta proiettiva</a></span></b></li></ul> <div class="noprint" style="width:100%; padding: 3px 0; display: flex; flex-wrap: wrap; row-gap: 4px; column-gap: 8px; box-sizing: border-box;"><div style="flex-grow: 1"><style data-mw-deduplicate="TemplateStyles:r140555418">.mw-parser-output .itwiki-template-occhiello{width:100%;line-height:25px;border:1px solid #CCF;background-color:#F0EEFF;box-sizing:border-box}.mw-parser-output .itwiki-template-occhiello-progetto{background-color:#FAFAFA}@media screen{html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-night .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello{background-color:#202122;border-color:#54595D}html.skin-theme-clientpref-os .mw-parser-output .itwiki-template-occhiello-progetto{background-color:#282929}}</style><div class="itwiki-template-occhiello"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal128-kmplot.svg" class="mw-file-description" title="Matematica"><img alt=" " src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/25px-Crystal128-kmplot.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/38px-Crystal128-kmplot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Crystal128-kmplot.svg/50px-Crystal128-kmplot.svg.png 2x" data-file-width="245" data-file-height="244" /></a></span> <b><a href="/wiki/Portale:Matematica" title="Portale:Matematica">Portale Matematica</a></b>: accedi alle voci di Wikipedia che trattano di matematica</div></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Estratto da "<a dir="ltr" href="https://it.wikipedia.org/w/index.php?title=Retta_proiettiva&oldid=115551236">https://it.wikipedia.org/w/index.php?title=Retta_proiettiva&oldid=115551236</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Categoria:Categorie" title="Categoria:Categorie">Categoria</a>: <ul><li><a href="/wiki/Categoria:Geometria_proiettiva" title="Categoria:Geometria proiettiva">Geometria proiettiva</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorie nascoste: <ul><li><a href="/wiki/Categoria:Senza_fonti_-_geometria" title="Categoria:Senza fonti - geometria">Senza fonti - geometria</a></li><li><a href="/wiki/Categoria:Senza_fonti_-_febbraio_2011" title="Categoria:Senza fonti - febbraio 2011">Senza fonti - febbraio 2011</a></li><li><a href="/wiki/Categoria:Collegamento_interprogetto_a_Wikimedia_Commons_presente_ma_assente_su_Wikidata" title="Categoria:Collegamento interprogetto a Wikimedia Commons presente ma assente su Wikidata">Collegamento interprogetto a Wikimedia Commons presente ma assente su Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Questa pagina è stata modificata per l'ultima volta il 16 set 2020 alle 19:59.</li> <li id="footer-info-copyright">Il testo è disponibile secondo la <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.it">licenza Creative Commons Attribuzione-Condividi allo stesso modo</a>; 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