Arc de méridien — Wikipédia

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data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Basculer la table des matières" > <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">Basculer la table des matières</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">Arc de méridien</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="Aller à un article dans une autre langue. Disponible en 8 langues." > <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-8" 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">8 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Arc_de_meridi%C3%A0" title="Arc de meridià – catalan" lang="ca" hreflang="ca" data-title="Arc de meridià" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Meridianbogen" title="Meridianbogen – allemand" lang="de" hreflang="de" data-title="Meridianbogen" data-language-autonym="Deutsch" data-language-local-name="allemand" 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/Meridian_arc" title="Meridian arc – anglais" lang="en" hreflang="en" data-title="Meridian arc" data-language-autonym="English" data-language-local-name="anglais" 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/Arco_de_meridiano" title="Arco de meridiano – espagnol" lang="es" hreflang="es" data-title="Arco de meridiano" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AD%90%E5%8D%88%E7%B7%9A%E5%BC%A7" title="子午線弧 – japonais" lang="ja" hreflang="ja" data-title="子午線弧" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Meridiano_(geodesia)" title="Meridiano (geodesia) – portugais" lang="pt" hreflang="pt" data-title="Meridiano (geodesia)" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70893996 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%94%D1%83%D0%B3%D0%B0_%D0%BC%D0%B5%D1%80%D0%B8%D0%B4%D0%B8%D0%B0%D0%BD%D0%B0" title="Дуга меридиана – russe" lang="ru" hreflang="ru" data-title="Дуга меридиана" data-language-autonym="Русский" data-language-local-name="russe" 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/Meridian_arc" title="Meridian arc – Simple English" lang="en-simple" hreflang="en-simple" data-title="Meridian arc" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</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/Q1921708#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</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="Espaces de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > 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aria-hidden="true" ><span class="vector-dropdown-label-text">Outils</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Outils</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">masquer</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Plus d’options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Arc_de_m%C3%A9ridien"><span>Lire</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&veaction=edit" title="Modifier cette page [v]" accesskey="v"><span>Modifier</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=history"><span>Voir l’historique</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Général </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Arc_de_m%C3%A9ridien" title="Liste des pages liées qui pointent sur celle-ci [j]" accesskey="j"><span>Pages liées</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Arc_de_m%C3%A9ridien" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k"><span>Suivi des pages liées</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u"><span>Téléverser un fichier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q"><span>Pages spéciales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&oldid=183883959" title="Adresse permanente de cette version de cette page"><span>Lien permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=info" title="Davantage d’informations sur cette page"><span>Informations sur la page</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Citer&page=Arc_de_m%C3%A9ridien&id=183883959&wpFormIdentifier=titleform" title="Informations sur la manière de citer cette page"><span>Citer cette page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:UrlShortener&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FArc_de_m%25C3%25A9ridien"><span>Obtenir l'URL 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de cette page [p]" accesskey="p"><span>Version imprimable</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"> Dans d’autres projets </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/Q1921708" title="Lien vers l’élément dans le dépôt de données connecté [g]" accesskey="g"><span>Élément 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="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <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">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</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">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><p>En <a href="/wiki/G%C3%A9od%C3%A9sie" title="Géodésie">géodésie</a>, la mesure d'un <b>arc de méridien</b> est la détermination la plus exacte possible de la distance entre deux points situés sur un même <a href="/wiki/M%C3%A9ridien" title="Méridien">méridien</a>, soit à la même <a href="/wiki/Longitude" title="Longitude">longitude</a>. Deux ou plusieurs déterminations de ce type dans des endroits différents précisent ensuite la forme de l'<a href="/wiki/Ellipso%C3%AFde" title="Ellipsoïde">ellipsoïde</a> de référence qui donne la meilleure approximation de la forme du <a href="/wiki/G%C3%A9o%C3%AFde" title="Géoïde">géoïde</a>. Ce processus est appelé « déterminer la <a href="/wiki/Figure_de_la_Terre" title="Figure de la Terre">figure de la Terre</a> ». Les premières mesures de la taille d'une Terre sphérique eurent besoin d'un seul <a href="/wiki/Arc_de_cercle" title="Arc de cercle">arc</a>. Les mesures les plus récentes utilisent des mesures astro-géodésiques et des méthodes de géodésie par satellite afin de déterminer l'<a href="/wiki/Mod%C3%A8le_ellipso%C3%AFdal_de_la_Terre" title="Modèle ellipsoïdal de la Terre">ellipsoïde de référence</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description_mathématique"><span id="Description_math.C3.A9matique"></span>Description mathématique</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&veaction=edit&section=1" title="Modifier la section : Description mathématique" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=edit&section=1" title="Modifier le code source de la section : Description mathématique"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un arc de <a href="/wiki/M%C3%A9ridien" title="Méridien">méridien</a> sur un ellipsoïde a la forme exacte d'une <a href="/wiki/Ellipse_(math%C3%A9matiques)" title="Ellipse (mathématiques)">ellipse</a>. Par conséquent, sa longueur entre l'<a href="/wiki/%C3%89quateur_terrestre" title="Équateur terrestre">équateur</a> et un point à la <a href="/wiki/Latitude" title="Latitude">latitude</a> φ peut être calculée comme une <a href="/wiki/Int%C3%A9grale_elliptique" title="Intégrale elliptique">intégrale elliptique</a> et approchée par une série tronquée. Le développement suivant qui fait intervenir le carré de l'<a href="/wiki/Excentricit%C3%A9_(math%C3%A9matiques)" title="Excentricité (mathématiques)">excentricité</a> <i>e</i> a été donné par <a href="/wiki/Jean-Baptiste_Joseph_Delambre" title="Jean-Baptiste Joseph Delambre">Jean-Baptiste Joseph Delambre</a> en 1799<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B\approx &\;a(1-e^{2})\left\{\left(1+{\frac {3}{4}}e^{2}+{\frac {45}{64}}e^{4}+{\frac {175}{256}}e^{6}+{\frac {11025}{16384}}e^{8}\right)\varphi \right.\\&\ -{\frac {1}{2}}\left({\frac {3}{4}}e^{2}+{\frac {15}{16}}e^{4}+{\frac {525}{512}}e^{6}+{\frac {2205}{2048}}e^{8}\right)\sin 2\varphi \\&\ +{\frac {1}{4}}\left({\frac {15}{64}}e^{4}+{\frac {105}{256}}e^{6}+{\frac {2205}{4096}}e^{8}\right)\sin 4\varphi \\&\ -{\frac {1}{6}}\left({\frac {35}{512}}e^{6}+{\frac {315}{2048}}e^{8}\right)\sin 6\varphi \\&\ +{\frac {1}{8}}\left.\left({\frac {315}{16384}}e^{8}\right)\sin 8\varphi \right\}.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>B</mi> <mo>≈</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mi>a</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>45</mn> <mn>64</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>175</mn> <mn>256</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11025</mn> <mn>16384</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>φ</mi> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mn>16</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>525</mn> <mn>512</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2205</mn> <mn>2048</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mn>64</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>105</mn> <mn>256</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2205</mn> <mn>4096</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>4</mn> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>35</mn> <mn>512</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>315</mn> <mn>2048</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>6</mn> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>315</mn> <mn>16384</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>8</mn> <mi>φ</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B\approx &\;a(1-e^{2})\left\{\left(1+{\frac {3}{4}}e^{2}+{\frac {45}{64}}e^{4}+{\frac {175}{256}}e^{6}+{\frac {11025}{16384}}e^{8}\right)\varphi \right.\\&\ -{\frac {1}{2}}\left({\frac {3}{4}}e^{2}+{\frac {15}{16}}e^{4}+{\frac {525}{512}}e^{6}+{\frac {2205}{2048}}e^{8}\right)\sin 2\varphi \\&\ +{\frac {1}{4}}\left({\frac {15}{64}}e^{4}+{\frac {105}{256}}e^{6}+{\frac {2205}{4096}}e^{8}\right)\sin 4\varphi \\&\ -{\frac {1}{6}}\left({\frac {35}{512}}e^{6}+{\frac {315}{2048}}e^{8}\right)\sin 6\varphi \\&\ +{\frac {1}{8}}\left.\left({\frac {315}{16384}}e^{8}\right)\sin 8\varphi \right\}.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b1191d929bb309c1e668bfd7c3d62ab961eab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.005ex; width:59.459ex; height:31.176ex;" alt="{\displaystyle {\begin{aligned}B\approx &\;a(1-e^{2})\left\{\left(1+{\frac {3}{4}}e^{2}+{\frac {45}{64}}e^{4}+{\frac {175}{256}}e^{6}+{\frac {11025}{16384}}e^{8}\right)\varphi \right.\\&\ -{\frac {1}{2}}\left({\frac {3}{4}}e^{2}+{\frac {15}{16}}e^{4}+{\frac {525}{512}}e^{6}+{\frac {2205}{2048}}e^{8}\right)\sin 2\varphi \\&\ +{\frac {1}{4}}\left({\frac {15}{64}}e^{4}+{\frac {105}{256}}e^{6}+{\frac {2205}{4096}}e^{8}\right)\sin 4\varphi \\&\ -{\frac {1}{6}}\left({\frac {35}{512}}e^{6}+{\frac {315}{2048}}e^{8}\right)\sin 6\varphi \\&\ +{\frac {1}{8}}\left.\left({\frac {315}{16384}}e^{8}\right)\sin 8\varphi \right\}.\\\end{aligned}}}"></span></dd></dl> <p><a href="/wiki/Friedrich_Robert_Helmert" title="Friedrich Robert Helmert">Friedrich Robert Helmert</a> a utilisé la formule suivant en 1880<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>, en posant <span class="mwe-math-element"><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={\frac {1-{\sqrt {1-e^{2}}}}{1+{\sqrt {1-e^{2}}}}}\simeq {\frac {e^{2}}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>≃</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {1-{\sqrt {1-e^{2}}}}{1+{\sqrt {1-e^{2}}}}}\simeq {\frac {e^{2}}{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b915b8a7790075e47839c0e51a7ad7d7f14f0c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.869ex; height:7.509ex;" alt="{\displaystyle n={\frac {1-{\sqrt {1-e^{2}}}}{1+{\sqrt {1-e^{2}}}}}\simeq {\frac {e^{2}}{4}}}"></span> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B\approx &\;{\frac {a}{1+n}}\left\{\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}\right)\varphi -{\frac {3}{2}}\left(n-{\frac {n^{3}}{8}}\right)\sin 2\varphi \right.\\&\ \left.+{\frac {15}{16}}\left(n^{2}-{\frac {n^{4}}{4}}\right)\sin 4\varphi -{\frac {35}{48}}n^{3}\sin 6\varphi +{\frac {315}{512}}n^{4}\sin 8\varphi \right\}.\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>B</mi> <mo>≈</mo> </mtd> <mtd> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>64</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>8</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>φ</mi> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mtext> </mtext> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mn>16</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>4</mn> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>35</mn> <mn>48</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mn>6</mn> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>315</mn> <mn>512</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mn>8</mn> <mi>φ</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B\approx &\;{\frac {a}{1+n}}\left\{\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}\right)\varphi -{\frac {3}{2}}\left(n-{\frac {n^{3}}{8}}\right)\sin 2\varphi \right.\\&\ \left.+{\frac {15}{16}}\left(n^{2}-{\frac {n^{4}}{4}}\right)\sin 4\varphi -{\frac {35}{48}}n^{3}\sin 6\varphi +{\frac {315}{512}}n^{4}\sin 8\varphi \right\}.\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5527e81e92069f021848f5dd6a10838c4bc534" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:63.066ex; height:12.676ex;" alt="{\displaystyle {\begin{aligned}B\approx &\;{\frac {a}{1+n}}\left\{\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}\right)\varphi -{\frac {3}{2}}\left(n-{\frac {n^{3}}{8}}\right)\sin 2\varphi \right.\\&\ \left.+{\frac {15}{16}}\left(n^{2}-{\frac {n^{4}}{4}}\right)\sin 4\varphi -{\frac {35}{48}}n^{3}\sin 6\varphi +{\frac {315}{512}}n^{4}\sin 8\varphi \right\}.\\\end{aligned}}}"></span></dd></dl> <p>Kazushige Kawase a donné une formule générale en 2009<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup><sup class="reference cite_virgule">,</sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite_crochet">[</span>4<span class="cite_crochet">]</span></a></sup> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\frac {a}{1+n}}\sum _{j=0}^{\infty }\left(\prod _{k=1}^{j}\varepsilon _{k}\right)^{2}\left\{\varphi +\sum _{l=1}^{2j}\left({\frac {1}{l}}-4l\right)\sin 2l\varphi \prod _{m=1}^{l}\varepsilon _{j+(-1)^{m}\lfloor m/2\rfloor }^{(-1)^{m}}\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>{</mo> <mrow> <mi>φ</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>l</mi> </mfrac> </mrow> <mo>−</mo> <mn>4</mn> <mi>l</mi> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>l</mi> <mi>φ</mi> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </munderover> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo fence="false" stretchy="false">⌊</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msubsup> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\frac {a}{1+n}}\sum _{j=0}^{\infty }\left(\prod _{k=1}^{j}\varepsilon _{k}\right)^{2}\left\{\varphi +\sum _{l=1}^{2j}\left({\frac {1}{l}}-4l\right)\sin 2l\varphi \prod _{m=1}^{l}\varepsilon _{j+(-1)^{m}\lfloor m/2\rfloor }^{(-1)^{m}}\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47086e7c423d731e9f79650e9372190ceacfa666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.733ex; height:8.176ex;" alt="{\displaystyle B={\frac {a}{1+n}}\sum _{j=0}^{\infty }\left(\prod _{k=1}^{j}\varepsilon _{k}\right)^{2}\left\{\varphi +\sum _{l=1}^{2j}\left({\frac {1}{l}}-4l\right)\sin 2l\varphi \prod _{m=1}^{l}\varepsilon _{j+(-1)^{m}\lfloor m/2\rfloor }^{(-1)^{m}}\right\},}"></span></dd></dl> <p>dans laquelle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i}=3n/2i-n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>i</mi> <mo>−</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i}=3n/2i-n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbab64d515630ec1f2c9d69048ebdec3d70cc1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.901ex; height:2.843ex;" alt="{\displaystyle \varepsilon _{i}=3n/2i-n}"></span>. </p><p>En tronquant la somme à j=2, on obtient la formule de Helmert. </p> <div class="mw-heading mw-heading2"><h2 id="Approximations">Approximations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&veaction=edit&section=2" title="Modifier la section : Approximations" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=edit&section=2" title="Modifier le code source de la section : Approximations"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La distance polaire peut être approchée par la formule de <a href="/wiki/Thomas_Muir_(math%C3%A9maticien)" title="Thomas Muir (mathématicien)">Muir</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 m_{p}=\int _{0}^{\pi /2}\!M(\varphi )\,d\varphi \;\approx {\frac {\pi }{2}}\left[{\frac {a^{3/2}+b^{3/2}}{2}}\right]^{2/3}\,\!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mi>M</mi> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>φ</mi> <mspace width="thickmathspace" /> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{p}=\int _{0}^{\pi /2}\!M(\varphi )\,d\varphi \;\approx {\frac {\pi }{2}}\left[{\frac {a^{3/2}+b^{3/2}}{2}}\right]^{2/3}\,\!.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0519794eb690db9354a3ecfb500c0fd12398b544" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.387ex; height:7.009ex;" alt="{\displaystyle m_{p}=\int _{0}^{\pi /2}\!M(\varphi )\,d\varphi \;\approx {\frac {\pi }{2}}\left[{\frac {a^{3/2}+b^{3/2}}{2}}\right]^{2/3}\,\!.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&veaction=edit&section=3" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=edit&section=3" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text">Delambre, J. B. J. (1799): <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DBAOAAAAQAAJ&pg=PA72"><i>Méthodes Analytiques pour la Détermination d'un Arc du Méridien</i>; précédées d'un mémoire sur le même sujet par A. M. Legendre</a>, De L'Imprimerie de Crapelet, Paris, 72–73</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><abbr class="abbr indicateur-langue" title="Langue : allemand">(de)</abbr> Helmert, F. R. (1880): <a rel="nofollow" class="external text" href="http://ia301506.us.archive.org/2/items/diemathematisch01helmgoog/diemathematisch01helmgoog.pdf"><i>Die mathematischen und physikalischen Theorieen der höheren Geodäsie</i>, Einleitung und 1 Teil</a>, Druck und Verlag von B. G. Teubner, Leipzig, 44–48</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><abbr class="abbr indicateur-langue" title="Langue : japonais">(ja)</abbr> 河瀬和重 (Kawase, K.) (2009): <a rel="nofollow" class="external text" href="http://www.gsi.go.jp/common/000054736.pdf">緯度を与えて赤道からの子午線弧長を求める一般的な計算式 (A General Formula for Meridional Distance from the Equator to Given Latitude)</a>, 国土地理院時報 (Journal of the Geographical Survey Institute), <b>119</b>, 45–55</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink noprint"><a href="#cite_ref-4">↑</a> </span><span class="reference-text"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Kawase, K. (2011): <a rel="nofollow" class="external text" href="http://www.gsi.go.jp/common/000062452.pdf">A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection</a>, Bulletin of the <a href="/wiki/Institut_d%27%C3%A9tudes_g%C3%A9ographiques_du_Japon" title="Institut d'études géographiques du Japon">Geospatial Information Authority of Japan</a>, <b>59</b>, 1–13</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&veaction=edit&section=4" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=edit&section=4" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&veaction=edit&section=5" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Arc_de_m%C3%A9ridien&action=edit&section=5" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arc_g%C3%A9od%C3%A9sique_de_Struve" title="Arc géodésique de Struve">Arc géodésique de Struve</a></li> <li><a href="/wiki/Exp%C3%A9ditions_g%C3%A9od%C3%A9siques_fran%C3%A7aises" title="Expéditions géodésiques françaises">Expéditions géodésiques françaises</a></li> <li><a href="/wiki/Figure_de_la_Terre_et_m%C3%A9ridienne_de_Delambre_et_M%C3%A9chain" title="Figure de la Terre et méridienne de Delambre et Méchain">Figure de la Terre et méridienne de Delambre et Méchain</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:G%C3%A9od%C3%A9sie_et_g%C3%A9ophysique" title="Portail de la géodésie et de la géophysique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Jordens_inre.svg/23px-Jordens_inre.svg.png" decoding="async" width="23" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Jordens_inre.svg/35px-Jordens_inre.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Jordens_inre.svg/46px-Jordens_inre.svg.png 2x" data-file-width="800" data-file-height="833" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:G%C3%A9od%C3%A9sie_et_g%C3%A9ophysique" title="Portail:Géodésie et géophysique">Portail de la géodésie et de la géophysique</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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