経緯度 - Wikipedia

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for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">個人用ツール</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="利用者メニュー" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ja.wikipedia.org&uselang=ja"><span>寄付</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%82%A2%E3%82%AB%E3%82%A6%E3%83%B3%E3%83%88%E4%BD%9C%E6%88%90&returnto=%E7%B5%8C%E7%B7%AF%E5%BA%A6" title="アカウントを作成してログインすることをお勧めしますが、必須ではありません"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>アカウント作成</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%83%AD%E3%82%B0%E3%82%A4%E3%83%B3&returnto=%E7%B5%8C%E7%B7%AF%E5%BA%A6" title="ログインすることを推奨します。ただし、必須ではありません。 [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>ログイン</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> ログアウトした編集者のページ <a href="/wiki/Wikipedia:%E3%82%A6%E3%82%A3%E3%82%AD%E3%83%9A%E3%83%87%E3%82%A3%E3%82%A2%E3%81%B8%E3%82%88%E3%81%86%E3%81%93%E3%81%9D" aria-label="編集の詳細"><span>もっと詳しく</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E8%87%AA%E5%88%86%E3%81%AE%E6%8A%95%E7%A8%BF%E8%A8%98%E9%8C%B2" title="このIPアドレスからなされた編集の一覧 [y]" accesskey="y"><span>投稿記録</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%88%E3%83%BC%E3%82%AF%E3%83%9A%E3%83%BC%E3%82%B8" title="このIPアドレスからなされた編集についての議論 [n]" accesskey="n"><span>トーク</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="サイト"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目次" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">非表示</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">ページ先頭</div> </a> </li> <li id="toc-地理経緯度" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#地理経緯度"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>地理経緯度</span> </div> </a> <ul id="toc-地理経緯度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-経緯度の歴史" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#経緯度の歴史"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>経緯度の歴史</span> </div> </a> <button aria-controls="toc-経緯度の歴史-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>経緯度の歴史サブセクションを切り替えます</span> </button> <ul id="toc-経緯度の歴史-sublist" class="vector-toc-list"> <li id="toc-天文経緯度" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#天文経緯度"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>天文経緯度</span> </div> </a> <ul id="toc-天文経緯度-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-地理経緯度_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#地理経緯度_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>地理経緯度</span> </div> </a> <ul id="toc-地理経緯度_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-地理経緯度の変換式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#地理経緯度の変換式"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>地理経緯度の変換式</span> </div> </a> <button aria-controls="toc-地理経緯度の変換式-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>地理経緯度の変換式サブセクションを切り替えます</span> </button> <ul id="toc-地理経緯度の変換式-sublist" class="vector-toc-list"> <li id="toc-回転楕円体面に沿う最短距離の式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#回転楕円体面に沿う最短距離の式"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>回転楕円体面に沿う最短距離の式</span> </div> </a> <ul id="toc-回転楕円体面に沿う最短距離の式-sublist" class="vector-toc-list"> <li id="toc-微小量" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#微小量"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>微小量</span> </div> </a> <ul id="toc-微小量-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-短距離近似式" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#短距離近似式"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>短距離近似式</span> </div> </a> <ul id="toc-短距離近似式-sublist" class="vector-toc-list"> <li id="toc-ガウスの平均緯度法（中間緯度法）" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#ガウスの平均緯度法（中間緯度法）"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2.1</span> <span>ガウスの平均緯度法（中間緯度法）</span> </div> </a> <ul id="toc-ガウスの平均緯度法（中間緯度法）-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </li> <li id="toc-経度・緯度を並べる順序" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#経度・緯度を並べる順序"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>経度・緯度を並べる順序</span> </div> </a> <button aria-controls="toc-経度・緯度を並べる順序-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>経度・緯度を並べる順序サブセクションを切り替えます</span> </button> <ul id="toc-経度・緯度を並べる順序-sublist" class="vector-toc-list"> <li id="toc-地図投影法の表式における_'"`UNIQ--postMath-0000002F-QINU`"'_平面座標の取り方" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#地図投影法の表式における_'"`UNIQ--postMath-0000002F-QINU`"'_平面座標の取り方"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>地図投影法の表式における '"`UNIQ--postMath-0000002F-QINU`"' 平面座標の取り方</span> </div> </a> <ul id="toc-地図投影法の表式における_'"`UNIQ--postMath-0000002F-QINU`"'_平面座標の取り方-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-方位角との対応関係" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#方位角との対応関係"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>方位角との対応関係</span> </div> </a> <ul id="toc-方位角との対応関係-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-右手系経緯度の採用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#右手系経緯度の採用"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>右手系経緯度の採用</span> </div> </a> <button aria-controls="toc-右手系経緯度の採用-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>右手系経緯度の採用サブセクションを切り替えます</span> </button> <ul id="toc-右手系経緯度の採用-sublist" class="vector-toc-list"> <li id="toc-polygonの頂点配列が時計周り順" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#polygonの頂点配列が時計周り順"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>polygonの頂点配列が時計周り順</span> </div> </a> <ul id="toc-polygonの頂点配列が時計周り順-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-左手系経緯度の採用" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#左手系経緯度の採用"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>左手系経緯度の採用</span> </div> </a> <ul id="toc-左手系経緯度の採用-sublist" class="vector-toc-list"> <li id="toc-左手系地図投影法の採用" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#左手系地図投影法の採用"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>左手系地図投影法の採用</span> </div> </a> <ul id="toc-左手系地図投影法の採用-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-脚注" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#脚注"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>脚注</span> </div> </a> <ul id="toc-脚注-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-関連項目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>関連項目</span> </div> </a> <ul id="toc-関連項目-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="目次" 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="目次の表示・非表示を切り替え" > <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">目次の表示・非表示を切り替え</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">経緯度</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="特定の記事の別の言語版に移動します。利用可能な言語5件" > <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-5" 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">5の言語版</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Geographic_coordinate_conversion" title="英語: Geographic coordinate conversion" lang="en" hreflang="en" data-title="Geographic coordinate conversion" data-language-autonym="English" data-language-local-name="英語" 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/Conversi%C3%B3n_de_coordenadas_geogr%C3%A1ficas" title="スペイン語: Conversión de coordenadas geográficas" lang="es" hreflang="es" data-title="Conversión de coordenadas geográficas" data-language-autonym="Español" data-language-local-name="スペイン語" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B2%BD%EC%9C%84%EB%8F%84" title="韓国語: 경위도" lang="ko" hreflang="ko" data-title="경위도" data-language-autonym="한국어" data-language-local-name="韓国語" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B5%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%BE%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5_%D0%B3%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D1%85_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="ロシア語: Преобразование геодезических систем координат" lang="ru" hreflang="ru" data-title="Преобразование геодезических систем координат" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%B6%93%E7%B7%AF%E5%BA%A6" title="中国語: 經緯度" lang="zh" hreflang="zh" data-title="經緯度" data-language-autonym="中文" data-language-local-name="中国語" class="interlanguage-link-target"><span>中文</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/Q11606510#sitelinks-wikipedia" title="言語間リンクを編集" class="wbc-editpage">リンクを編集</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="名前空間"> <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/%E7%B5%8C%E7%B7%AF%E5%BA%A6" title="本文を閲覧 [c]" accesskey="c"><span>ページ</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E3%83%8E%E3%83%BC%E3%83%88:%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&redlink=1" rel="discussion" class="new" title="「本文ページについての議論」 (存在しないページ) [t]" accesskey="t"><span>ノート</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" 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<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">表示</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">非表示</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">出典: フリー百科事典『ウィキペディア（Wikipedia）』</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="ja" dir="ltr"><p><b>経緯度</b>（けいいど、<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英語</a>: <span lang="en">longitude and latitude</span>）とは、<a href="/wiki/%E7%B5%8C%E5%BA%A6" title="経度">経度</a>（<span lang="en">longitude</span>）および<a href="/wiki/%E7%B7%AF%E5%BA%A6" title="緯度">緯度</a>（<span lang="en">latitude</span>）を指し、<a href="/wiki/%E5%9C%B0%E7%90%83" title="地球">地球</a>を含む<a href="/wiki/%E5%A4%A9%E4%BD%93" title="天体">天体</a>表面上で位置（点）を示すための座標表現である。本稿では<a href="/wiki/%E5%9C%B0%E7%90%86%E5%BA%A7%E6%A8%99%E7%B3%BB" title="地理座標系">地理座標系</a>で用いられる経緯度を説明する。 </p><p>基本的に、その<a href="/wiki/%E5%A4%A9%E4%BD%93" title="天体">天体</a>の表面点の垂直ベクトルを考え、その向きを<a href="/wiki/%E7%90%83%E9%9D%A2%E5%BA%A7%E6%A8%99" class="mw-redirect" title="球面座標">球面座標</a>（<a href="/wiki/%E8%A7%92%E5%BA%A6" title="角度">角度</a>）で表現する<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。 </p> <figure typeof="mw:File/Thumb"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Latitude_and_longitude_graticule_on_an_ellipsoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/170px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png" decoding="async" width="170" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/255px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Latitude_and_longitude_graticule_on_an_ellipsoid.svg/340px-Latitude_and_longitude_graticule_on_an_ellipsoid.svg.png 2x" data-file-width="825" data-file-height="750" /></a><figcaption>経度（<span class="mwe-math-element"><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>）、緯度（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>）、および垂直線（赤）。</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:ECEF_ENU_Longitude_Latitude_right-hand-rule.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/ECEF_ENU_Longitude_Latitude_right-hand-rule.svg/250px-ECEF_ENU_Longitude_Latitude_right-hand-rule.svg.png" decoding="async" width="250" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/ECEF_ENU_Longitude_Latitude_right-hand-rule.svg/375px-ECEF_ENU_Longitude_Latitude_right-hand-rule.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e1/ECEF_ENU_Longitude_Latitude_right-hand-rule.svg/500px-ECEF_ENU_Longitude_Latitude_right-hand-rule.svg.png 2x" data-file-width="538" data-file-height="500" /></a><figcaption>ECEF直交座標・地理座標・局所座標の関係（回転楕円体面上）。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y,Z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,Y,Z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15fcf4aac62f9533d646603bdc5a9cf76ce95c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.311ex; height:2.843ex;" alt="{\displaystyle (X,Y,Z)}"></span>および<a href="/wiki/%E6%96%B9%E4%BD%8D%E8%A7%92" title="方位角">方位角</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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>の取り方は<a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</a>。</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="地理経緯度"><span id=".E5.9C.B0.E7.90.86.E7.B5.8C.E7.B7.AF.E5.BA.A6"></span>地理経緯度</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=1" title="節を編集: 地理経緯度"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>経緯度は基本的にその地表点の垂直ベクトルに基づき、そのベクトルの方向を<a href="/wiki/%E7%90%83%E9%9D%A2%E5%BA%A7%E6%A8%99" class="mw-redirect" title="球面座標">球面座標</a>で<a href="/wiki/%E8%A7%92%E5%BA%A6" title="角度">角度</a>表現したものである。 </p> <dl><dd>{経度<span class="mwe-math-element"><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>、緯度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>｝⇔｛局所垂直ベクトル<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\cos \phi \cos \lambda ,\,\cos \phi \sin \lambda ,\,\sin \phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>λ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>λ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cos \phi \cos \lambda ,\,\cos \phi \sin \lambda ,\,\sin \phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f34959689234793b04be55e39f8a7f379d52f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.272ex; height:2.843ex;" alt="{\displaystyle (\cos \phi \cos \lambda ,\,\cos \phi \sin \lambda ,\,\sin \phi )}"></span>｝。</dd></dl> <p>地理座標系で用いられる地理経緯度（geographic longitude and latitude）<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>は、地球を<a href="/wiki/%E5%9B%9E%E8%BB%A2%E6%A5%95%E5%86%86%E4%BD%93" title="回転楕円体">回転楕円体</a>と見なし、その面の<a href="/wiki/%E6%B3%95%E7%B7%9A" class="mw-redirect" title="法線">法線</a>ベクトル方向に基づく<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="経緯度の歴史"><span id=".E7.B5.8C.E7.B7.AF.E5.BA.A6.E3.81.AE.E6.AD.B4.E5.8F.B2"></span>経緯度の歴史</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=2" title="節を編集: 経緯度の歴史"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="天文経緯度"><span id=".E5.A4.A9.E6.96.87.E7.B5.8C.E7.B7.AF.E5.BA.A6"></span>天文経緯度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=3" title="節を編集: 天文経緯度"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>歴史的には、地表の<a href="/wiki/%E9%89%9B%E7%9B%B4%E7%B7%9A" class="mw-redirect" title="鉛直線">鉛直線</a>に基づく垂直方向（<a href="/wiki/%E5%A4%A9%E9%A0%82" title="天頂">天頂</a>）が<a href="/wiki/%E5%A4%A9%E7%90%83" title="天球">天球</a>のどこを指すかによって決めた天文経緯度（astronomical longitude and latitude）が使われてきた。これは地球の重力の鉛直線偏差の影響（加えて地球の<a href="/wiki/%E6%A5%B5%E9%81%8B%E5%8B%95" title="極運動">極運動</a>の影響）を被っている。従って、距離・面積との関係も簡素にならない。 </p> <div class="mw-heading mw-heading3"><h3 id="地理経緯度_2"><span id=".E5.9C.B0.E7.90.86.E7.B5.8C.E7.B7.AF.E5.BA.A6_2"></span>地理経緯度</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=4" title="節を編集: 地理経緯度"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>地理学・測地学の発展とともに、経緯度原点を国内に設け、その地点の天文経緯度を原点として位置づけ、接する<a href="/wiki/%E6%BA%96%E6%8B%A0%E6%A5%95%E5%86%86%E4%BD%93" class="mw-redirect" title="準拠楕円体">準拠楕円体</a>に基づく地理経緯度を用いる方式が行われた（地域的<a href="/wiki/%E6%B8%AC%E5%9C%B0%E7%B3%BB" title="測地系">測地系</a>）。 </p><p>さらに近年は全地球的な<a href="/wiki/%E6%BA%96%E6%8B%A0%E6%A5%95%E5%86%86%E4%BD%93" class="mw-redirect" title="準拠楕円体">準拠楕円体</a>に基づく方式の採用が増えている（全地球的<a href="/wiki/%E6%B8%AC%E5%9C%B0%E7%B3%BB" title="測地系">測地系</a>）。 </p> <div class="mw-heading mw-heading2"><h2 id="地理経緯度の変換式"><span id=".E5.9C.B0.E7.90.86.E7.B5.8C.E7.B7.AF.E5.BA.A6.E3.81.AE.E5.A4.89.E6.8F.9B.E5.BC.8F"></span>地理経緯度の変換式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=5" title="節を編集: 地理経緯度の変換式"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>地理座標（経度<span class="mwe-math-element"><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>、緯度<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>、高度（<a href="/wiki/%E6%A5%95%E5%86%86%E4%BD%93%E9%AB%98" class="mw-redirect" title="楕円体高">楕円体高</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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>）とECEF直交座標系<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>との変換、および微小量の式は下記となる（<a href="/wiki/%E5%9C%B0%E7%90%83%E6%A5%95%E5%86%86%E4%BD%93" title="地球楕円体">地球楕円体</a>の<a href="/wiki/%E9%95%B7%E5%8D%8A%E5%BE%84" class="mw-redirect" title="長半径">長半径</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>、<a href="/wiki/%E9%9B%A2%E5%BF%83%E7%8E%87" title="離心率">離心率</a><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e={\sqrt {f(2-f)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\sqrt {f(2-f)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f0c43f810b87d6a30cc7d2e5668b81727a9028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.875ex; height:4.843ex;" alt="{\displaystyle e={\sqrt {f(2-f)}}}"></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{cases}x=\left(N(\phi )+h\right)\cos {\phi }\cos {\lambda },\\y=\left(N(\phi )+h\right)\cos {\phi }\sin {\lambda },\\z=\left(N(\phi )(1-e^{2})+h\right)\sin {\phi },\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <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> <mo>+</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x=\left(N(\phi )+h\right)\cos {\phi }\cos {\lambda },\\y=\left(N(\phi )+h\right)\cos {\phi }\sin {\lambda },\\z=\left(N(\phi )(1-e^{2})+h\right)\sin {\phi },\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6863c444354a0ebdd508bb978366cc1acd0f0a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.183ex; height:9.176ex;" alt="{\displaystyle {\begin{cases}x=\left(N(\phi )+h\right)\cos {\phi }\cos {\lambda },\\y=\left(N(\phi )+h\right)\cos {\phi }\sin {\lambda },\\z=\left(N(\phi )(1-e^{2})+h\right)\sin {\phi },\end{cases}}}"></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\begin{pmatrix}dx\\dy\\dz\\\end{pmatrix}}&={\begin{pmatrix}-\sin \lambda &-\sin \phi \cos \lambda &\cos \phi \cos \lambda \\\cos \lambda &-\sin \phi \sin \lambda &\cos \phi \sin \lambda \\0&\cos \phi &\sin \phi \\\end{pmatrix}}{\begin{pmatrix}dE\\dN\\dU\\\end{pmatrix}},\\{\begin{pmatrix}dE\\dN\\dU\\\end{pmatrix}}&={\begin{pmatrix}\left(N(\phi )+h\right)\cos \phi &0&0\\0&M(\phi )+h&0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}d\lambda \\d\phi \\dh\\\end{pmatrix}},\\N(\phi )&\triangleq {\frac {a}{\sqrt {1-e^{2}\sin ^{2}\phi }}},\\M(\phi )&\triangleq {\frac {a(1-e^{2})}{\left(1-e^{2}\sin ^{2}\phi \right)^{3/2}}}={\frac {\{N(\phi )\}^{3}(1-e^{2})}{a^{2}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>z</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>λ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>λ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>cos</mi> <mo>⁡</mo> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>λ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>λ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>N</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>U</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>N</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>U</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>h</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>h</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </msqrt> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">{</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <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> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{pmatrix}dx\\dy\\dz\\\end{pmatrix}}&={\begin{pmatrix}-\sin \lambda &-\sin \phi \cos \lambda &\cos \phi \cos \lambda \\\cos \lambda &-\sin \phi \sin \lambda &\cos \phi \sin \lambda \\0&\cos \phi &\sin \phi \\\end{pmatrix}}{\begin{pmatrix}dE\\dN\\dU\\\end{pmatrix}},\\{\begin{pmatrix}dE\\dN\\dU\\\end{pmatrix}}&={\begin{pmatrix}\left(N(\phi )+h\right)\cos \phi &0&0\\0&M(\phi )+h&0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}d\lambda \\d\phi \\dh\\\end{pmatrix}},\\N(\phi )&\triangleq {\frac {a}{\sqrt {1-e^{2}\sin ^{2}\phi }}},\\M(\phi )&\triangleq {\frac {a(1-e^{2})}{\left(1-e^{2}\sin ^{2}\phi \right)^{3/2}}}={\frac {\{N(\phi )\}^{3}(1-e^{2})}{a^{2}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/479f1c6b52a3c0c01c56eb6fe2c9815a67f58f8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.838ex; width:59.137ex; height:34.843ex;" alt="{\displaystyle {\begin{aligned}{\begin{pmatrix}dx\\dy\\dz\\\end{pmatrix}}&={\begin{pmatrix}-\sin \lambda &-\sin \phi \cos \lambda &\cos \phi \cos \lambda \\\cos \lambda &-\sin \phi \sin \lambda &\cos \phi \sin \lambda \\0&\cos \phi &\sin \phi \\\end{pmatrix}}{\begin{pmatrix}dE\\dN\\dU\\\end{pmatrix}},\\{\begin{pmatrix}dE\\dN\\dU\\\end{pmatrix}}&={\begin{pmatrix}\left(N(\phi )+h\right)\cos \phi &0&0\\0&M(\phi )+h&0\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}d\lambda \\d\phi \\dh\\\end{pmatrix}},\\N(\phi )&\triangleq {\frac {a}{\sqrt {1-e^{2}\sin ^{2}\phi }}},\\M(\phi )&\triangleq {\frac {a(1-e^{2})}{\left(1-e^{2}\sin ^{2}\phi \right)^{3/2}}}={\frac {\{N(\phi )\}^{3}(1-e^{2})}{a^{2}}}.\end{aligned}}}"></span></dd></dl> <p>微小量三成分はどれも互いに直交方向となる。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe239e1050529410001cc1c0b3245945bc69709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=0}"></span>では回転楕円体となり、また<a href="/wiki/%E5%AD%90%E5%8D%88%E7%B7%9A%E5%BC%A7" title="子午線弧">子午線弧</a>（<a href="/wiki/%E7%B5%8C%E7%B7%9A" class="mw-redirect" title="経線">経線</a>弧）の<a href="/wiki/%E6%9B%B2%E7%8E%87%E5%8D%8A%E5%BE%84" class="mw-redirect" title="曲率半径">曲率半径</a>は<span class="mwe-math-element"><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(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2734370563a5315e7db26954e2846e80c8bdc9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle M(\phi )}"></span>、<a href="/wiki/%E5%8D%AF%E9%85%89%E7%B7%9A" title="卯酉線">卯酉線</a>弧は<span class="mwe-math-element"><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(\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9358fddbf2bfa4718163e933bc3e86ecd7b1cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.258ex; height:2.843ex;" alt="{\displaystyle N(\phi )}"></span>（<a href="/wiki/%E7%B7%AF%E7%B7%9A" title="緯線">緯線</a>弧は<span class="mwe-math-element"><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(\phi )\cos \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\phi )\cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27134f133e891061f8f944de0ddafd07663770b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.529ex; height:2.843ex;" alt="{\displaystyle N(\phi )\cos \phi }"></span>）となる<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>。 </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span>から<span class="mwe-math-element"><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 ,\,\phi ,\,h)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>ϕ</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\,\phi ,\,h)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f094a84701568e300f4233a516f1ba304b0f264a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.731ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\,\phi ,\,h)}"></span>を求める変換計算については上記から導かれる <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> の方程式を解く必要がある<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading3"><h3 id="回転楕円体面に沿う最短距離の式"><span id=".E5.9B.9E.E8.BB.A2.E6.A5.95.E5.86.86.E4.BD.93.E9.9D.A2.E3.81.AB.E6.B2.BF.E3.81.86.E6.9C.80.E7.9F.AD.E8.B7.9D.E9.9B.A2.E3.81.AE.E5.BC.8F"></span>回転楕円体面に沿う最短距離の式</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=6" title="節を編集: 回転楕円体面に沿う最短距離の式"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="rellink" style="margin-bottom: 0.5em; padding-left: 2em; font-size: 90%;" role="note">→詳細は「<a href="https://en.wikipedia.org/wiki/Geographical_distance#Ellipsoidal_Earth_projected_to_a_plane" class="extiw" title="en:Geographical distance">地理上の距離（英語版）</a>」を参照</div> <div class="mw-heading mw-heading4"><h4 id="微小量"><span id=".E5.BE.AE.E5.B0.8F.E9.87.8F"></span>微小量</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=7" title="節を編集: 微小量"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>回転楕円体面に沿う最短距離（<a href="/wiki/%E6%B8%AC%E5%9C%B0%E7%B7%9A" title="測地線">測地線</a>距離）<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> の微小量は上記から得られる（<a href="/wiki/%E5%BE%AE%E5%88%86%E5%B9%BE%E4%BD%95%E5%AD%A6" title="微分幾何学">微分幾何学</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 h=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe239e1050529410001cc1c0b3245945bc69709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.176ex;" alt="{\displaystyle h=0}"></span>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d868ee7902023d29169252eb059f8faff9f08fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.044ex; height:2.176ex;" alt="{\displaystyle U=0}"></span>）の下で、 </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 ds={\sqrt {dE^{2}+dN^{2}}}={\sqrt {\left(N\left(\phi \right)\cos \phi \,d\lambda \right)^{2}+\left(M\left(\phi \right)d\phi \right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>d</mi> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds={\sqrt {dE^{2}+dN^{2}}}={\sqrt {\left(N\left(\phi \right)\cos \phi \,d\lambda \right)^{2}+\left(M\left(\phi \right)d\phi \right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dedf8b56c75b6bf85466ecd8d267bcdefd31d6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:56.549ex; height:4.843ex;" alt="{\displaystyle ds={\sqrt {dE^{2}+dN^{2}}}={\sqrt {\left(N\left(\phi \right)\cos \phi \,d\lambda \right)^{2}+\left(M\left(\phi \right)d\phi \right)^{2}}}.}"></span></dd></dl> <p>ただし、両極が特異点となる。 </p> <div class="mw-heading mw-heading4"><h4 id="短距離近似式"><span id=".E7.9F.AD.E8.B7.9D.E9.9B.A2.E8.BF.91.E4.BC.BC.E5.BC.8F"></span>短距離近似式</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=8" title="節を編集: 短距離近似式"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>二点間<a href="/wiki/%E6%B8%AC%E5%9C%B0%E7%B7%9A" title="測地線">測地線</a>距離<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7783a6d29d2ca4d9f1e0f501c4c6483fc058bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.026ex; height:2.176ex;" alt="{\displaystyle \Delta s}"></span>は、短距離の場合には、簡素な近似形を導出できる。<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \lambda =\lambda _{1}-\lambda _{2},\ \Delta \phi =\phi _{1}-\phi _{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>=</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \lambda =\lambda _{1}-\lambda _{2},\ \Delta \phi =\phi _{1}-\phi _{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb00499f2ad5819a2662bf699b3519136780d0af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.45ex; height:2.509ex;" alt="{\displaystyle \Delta \lambda =\lambda _{1}-\lambda _{2},\ \Delta \phi =\phi _{1}-\phi _{2},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{\textrm {m}}={\frac {\phi _{1}+\phi _{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{\textrm {m}}={\frac {\phi _{1}+\phi _{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072ab27578db5aadb02d612e16ac2e1acddd0b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.641ex; height:5.343ex;" alt="{\displaystyle \phi _{\textrm {m}}={\frac {\phi _{1}+\phi _{2}}{2}}}"></span>とおいて、短距離条件は、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta \phi |\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta \phi |\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f24ce08b6af015e0bb1150f58dfd2a502bea23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.392ex; height:2.843ex;" alt="{\displaystyle |\Delta \phi |\ll 1}"></span>かつ<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cos \phi _{\textrm {m}}\Delta \lambda |\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cos \phi _{\textrm {m}}\Delta \lambda |\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97822da0bf3dc8f66668a2feb98626b120966061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.233ex; height:2.843ex;" alt="{\displaystyle |\cos \phi _{\textrm {m}}\Delta \lambda |\ll 1}"></span>と表される。 </p><p>これに従うと<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd7783a6d29d2ca4d9f1e0f501c4c6483fc058bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.026ex; height:2.176ex;" alt="{\displaystyle \Delta s}"></span> の近似式が導出される（誤差は <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta s_{\text{approx}}-\Delta s_{\text{true}}|\lessapprox {\frac {\Delta s^{3}}{24a^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>approx</mtext> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>true</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⪅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>24</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta s_{\text{approx}}-\Delta s_{\text{true}}|\lessapprox {\frac {\Delta s^{3}}{24a^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e790b606e777e894fae26f09d5fa7b6abfb2032d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.108ex; height:6.009ex;" alt="{\displaystyle |\Delta s_{\text{approx}}-\Delta s_{\text{true}}|\lessapprox {\frac {\Delta s^{3}}{24a^{2}}}}"></span>; 30kmでは1ppmの精度）。 </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 \Delta s={\sqrt {\left(2N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\right)^{2}+\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \,\cos {\frac {\Delta \lambda }{2}}\right)^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s={\sqrt {\left(2N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\right)^{2}+\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \,\cos {\frac {\Delta \lambda }{2}}\right)^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f013c72a1a89c0dfe1b2822c54c4c5411e36e86a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:63.011ex; height:7.676ex;" alt="{\displaystyle \Delta s={\sqrt {\left(2N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\sin {\frac {\Delta \lambda }{2}}\right)^{2}+\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \,\cos {\frac {\Delta \lambda }{2}}\right)^{2}}}}"></span>.</dd></dl> <p>他の計算式としては、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta \phi |\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta \phi |\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f24ce08b6af015e0bb1150f58dfd2a502bea23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.392ex; height:2.843ex;" alt="{\displaystyle |\Delta \phi |\ll 1}"></span>かつ<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta \lambda |\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta \lambda |\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d80bf933c9f7c07fa06b2f2c040d4903c1e363d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.361ex; height:2.843ex;" alt="{\displaystyle |\Delta \lambda |\ll 1}"></span>と仮定すると（上記の微小量式を率直に一次式（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>→<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span>）と見なすことに相当）、より簡素な下記の近似計算式が導出される <sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>（平面法などと呼ばれることがある<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>）。 </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 \Delta s={\sqrt {\left(N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\Delta \lambda \right)^{2}+\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s={\sqrt {\left(N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\Delta \lambda \right)^{2}+\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd80e0e995af044e056f6fa7354bbaef462fbb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:46.406ex; height:4.843ex;" alt="{\displaystyle \Delta s={\sqrt {\left(N\left(\phi _{\textrm {m}}\right)\cos \phi _{\textrm {m}}\Delta \lambda \right)^{2}+\left(M\left(\phi _{\textrm {m}}\right)\Delta \phi \right)^{2}}}.}"></span></dd></dl> <p>しかしながら、この <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta \lambda |\ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta \lambda |\ll 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d80bf933c9f7c07fa06b2f2c040d4903c1e363d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.361ex; height:2.843ex;" alt="{\displaystyle |\Delta \lambda |\ll 1}"></span> は高緯度（および極近傍）では必ずしも適切な短距離条件とは言えず、それによる三角関数の近似を行ったことから両極に特異性を生じさせるなど難点を持つが<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>、高緯度（および極近傍）を除けば短距離近似として妥当であり多用される。 </p><p>さらに中長距離へ近似精度を改善した計算法も歴史的に多くの研究者によって開発されている。それらは高次の級数計算もしくは反復を含んでいることが多い<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading5"><h5 id="ガウスの平均緯度法（中間緯度法）"><span id=".E3.82.AC.E3.82.A6.E3.82.B9.E3.81.AE.E5.B9.B3.E5.9D.87.E7.B7.AF.E5.BA.A6.E6.B3.95.EF.BC.88.E4.B8.AD.E9.96.93.E7.B7.AF.E5.BA.A6.E6.B3.95.EF.BC.89"></span>ガウスの平均緯度法（中間緯度法）</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=9" title="節を編集: ガウスの平均緯度法（中間緯度法）"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>二点間測地線計算の球面近似の一種で<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> 、近似精度が改善される（ Rapp (1991)<sup id="cite_ref-rapp91_12-0" class="reference"><a href="#cite_note-rapp91-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> §6.4 ）（誤差は <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Delta s_{\text{approx}}-\Delta s_{\text{true}}|\lessapprox {\frac {\Delta s^{3}}{400a^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>approx</mtext> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>true</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⪅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>400</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Delta s_{\text{approx}}-\Delta s_{\text{true}}|\lessapprox {\frac {\Delta s^{3}}{400a^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e9a2e81b101cf189076ab6b45184ed76b13b09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.27ex; height:6.009ex;" alt="{\displaystyle |\Delta s_{\text{approx}}-\Delta s_{\text{true}}|\lessapprox {\frac {\Delta s^{3}}{400a^{2}}}}"></span>; 100kmでは1ppmの精度）。 </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 \Delta s=2N\left(\phi _{\textrm {m}}\right)\arcsin {\sqrt {\left(\sin {\frac {\Delta \lambda }{2}}\cos \phi _{\textrm {m}}\right)^{2}+\left(\cos {\frac {\Delta \lambda }{2}}\sin {\frac {M\left(\phi _{\textrm {m}}\right)\Delta \phi }{2N\left(\phi _{\textrm {m}}\right)}}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>s</mi> <mo>=</mo> <mn>2</mn> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>λ</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>M</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> </mrow> <mrow> <mn>2</mn> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta s=2N\left(\phi _{\textrm {m}}\right)\arcsin {\sqrt {\left(\sin {\frac {\Delta \lambda }{2}}\cos \phi _{\textrm {m}}\right)^{2}+\left(\cos {\frac {\Delta \lambda }{2}}\sin {\frac {M\left(\phi _{\textrm {m}}\right)\Delta \phi }{2N\left(\phi _{\textrm {m}}\right)}}\right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28016675f04937d1a58116da326f5089a6d52d86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:73.699ex; height:7.676ex;" alt="{\displaystyle \Delta s=2N\left(\phi _{\textrm {m}}\right)\arcsin {\sqrt {\left(\sin {\frac {\Delta \lambda }{2}}\cos \phi _{\textrm {m}}\right)^{2}+\left(\cos {\frac {\Delta \lambda }{2}}\sin {\frac {M\left(\phi _{\textrm {m}}\right)\Delta \phi }{2N\left(\phi _{\textrm {m}}\right)}}\right)^{2}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="経度・緯度を並べる順序"><span id=".E7.B5.8C.E5.BA.A6.E3.83.BB.E7.B7.AF.E5.BA.A6.E3.82.92.E4.B8.A6.E3.81.B9.E3.82.8B.E9.A0.86.E5.BA.8F"></span>経度・緯度を並べる順序</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=10" title="節を編集: 経度・緯度を並べる順序"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>並べる順序には、異なる慣行が存在する。正負については、<a href="/wiki/%E6%9D%B1%E7%B5%8C" class="mw-redirect" title="東経">東経</a>を正の<a href="/wiki/%E7%B5%8C%E5%BA%A6" title="経度">経度</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 \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>、<a href="/wiki/%E5%8C%97%E7%B7%AF" class="mw-redirect" title="北緯">北緯</a>を正の<a href="/wiki/%E7%B7%AF%E5%BA%A6" title="緯度">緯度</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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>、<a href="/wiki/%E5%8D%97%E7%B7%AF" class="mw-redirect" title="南緯">南緯</a>向きを正の<a href="/wiki/%E4%BD%99%E7%B7%AF%E5%BA%A6" class="mw-redirect" title="余緯度">余緯度</a>とする。 </p> <ul><li><a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</a>では：（<a href="/wiki/%E7%B5%8C%E5%BA%A6" title="経度">経度</a>、<a href="/wiki/%E7%B7%AF%E5%BA%A6" title="緯度">緯度</a>、及び<a href="/wiki/%E9%AB%98%E5%BA%A6" title="高度">高度</a>）の順とする<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>。</li> <li>これに対して<a href="/wiki/%E5%B7%A6%E6%89%8B%E7%B3%BB" class="mw-redirect" title="左手系">左手系</a><sup id="cite_ref-lhs_15-0" class="reference"><a href="#cite_note-lhs-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>では：（緯度、経度、及び高度）の順とする。局所座標系（地平面）の <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 方向が北・緯度座標、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> 方向が東・経度座標となる。</li></ul> <div class="mw-heading mw-heading3"><h3 id="地図投影法の表式における_'"`UNIQ--postMath-0000002F-QINU`"'_平面座標の取り方"><span id=".E5.9C.B0.E5.9B.B3.E6.8A.95.E5.BD.B1.E6.B3.95.E3.81.AE.E8.A1.A8.E5.BC.8F.E3.81.AB.E3.81.8A.E3.81.91.E3.82.8B_.7F.27.22.60UNIQ--postMath-0000002F-QINU.60.22.27.7F_.E5.B9.B3.E9.9D.A2.E5.BA.A7.E6.A8.99.E3.81.AE.E5.8F.96.E3.82.8A.E6.96.B9"></span>地図投影法の表式における <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,\ y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,\ y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/714e4c4e5f89dfc0168fee91c8d64b729e528bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.1ex; height:2.009ex;" alt="{\displaystyle x,\ y}"></span> 平面座標の取り方</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=11" title="節を編集: 地図投影法の表式における '"`UNIQ--postMath-0000002F-QINU`"' 平面座標の取り方"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E5%9C%B0%E5%9B%B3%E5%AD%A6" title="地図学">地図学</a>における<a href="/wiki/%E5%9C%B0%E5%9B%B3%E6%8A%95%E5%BD%B1%E6%B3%95" class="mw-redirect" title="地図投影法">地図投影法</a>の表式で <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,\ y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mtext> </mtext> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,\ y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/714e4c4e5f89dfc0168fee91c8d64b729e528bae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.1ex; height:2.009ex;" alt="{\displaystyle x,\ y}"></span> 平面座標の取り方は<a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</a>で表されることが多い。 </p> <ul><li><a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>方向を右横方向、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>方向を上縦方向</li> <li><a href="/wiki/%E5%B7%A6%E6%89%8B%E7%B3%BB" class="mw-redirect" title="左手系">左手系</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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>方向を上縦方向、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>方向を右横方向<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="方位角との対応関係"><span id=".E6.96.B9.E4.BD.8D.E8.A7.92.E3.81.A8.E3.81.AE.E5.AF.BE.E5.BF.9C.E9.96.A2.E4.BF.82"></span>方位角との対応関係</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=12" title="節を編集: 方位角との対応関係"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="rellink" style="margin-bottom: 0.5em; padding-left: 2em; font-size: 90%;" role="note">→詳細は「<a href="/wiki/%E6%96%B9%E4%BD%8D%E8%A7%92" title="方位角">方位角</a>」を参照</div> <p><a href="/wiki/%E6%96%B9%E4%BD%8D%E8%A7%92" title="方位角">方位角</a>は上記と対応した関係が存在する： </p> <ul><li><a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</a>（<a href="/wiki/%E5%8F%8D%E6%99%82%E8%A8%88%E5%9B%9E%E3%82%8A" class="mw-redirect" title="反時計回り">反時計回り</a>）：（<b>東</b>→北→西→南）<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/%E5%B7%A6%E6%89%8B%E7%B3%BB" class="mw-redirect" title="左手系">左手系</a><sup id="cite_ref-lhs_15-1" class="reference"><a href="#cite_note-lhs-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>（<a href="/wiki/%E6%99%82%E8%A8%88%E5%9B%9E%E3%82%8A" class="mw-redirect" title="時計回り">時計回り</a>）：（<b>北</b>→東→南→西）</li></ul> <p>方位角を <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>として、局所座標系（地平面）の単位円は <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)=(\cos \theta ,\sin \theta ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)=(\cos \theta ,\sin \theta ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb96f1499a2274f4f4f02c09f1f3b24d140abf7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.51ex; height:2.843ex;" alt="{\displaystyle (x,y,z)=(\cos \theta ,\sin \theta ,0)}"></span> となる。 </p> <div class="mw-heading mw-heading2"><h2 id="右手系経緯度の採用"><span id=".E5.8F.B3.E6.89.8B.E7.B3.BB.E7.B5.8C.E7.B7.AF.E5.BA.A6.E3.81.AE.E6.8E.A1.E7.94.A8"></span>右手系経緯度の採用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=13" title="節を編集: 右手系経緯度の採用"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>下記では右手系経緯度が採用されている。 </p> <ul><li><a href="/wiki/OpenLayers" title="OpenLayers">OpenLayers</a></li> <li><a href="/w/index.php?title=MapboxGL&action=edit&redlink=1" class="new" title="「MapboxGL」 (存在しないページ)">MapboxGL</a></li> <li><a href="/wiki/KML" title="KML">KML</a></li> <li><a href="/wiki/GeoJSON" title="GeoJSON">GeoJSON</a></li> <li><a href="/wiki/Well-known_text" title="Well-known text">Well-known text</a></li> <li><a href="/wiki/MySQL" title="MySQL">MySQL</a></li> <li><a href="/wiki/MongoDB" title="MongoDB">MongoDB</a></li> <li><a href="/wiki/Redis" title="Redis">Redis</a></li> <li><a href="https://en.wikipedia.org/wiki/Oracle_Spatial_and_Graph" class="extiw" title="en:Oracle Spatial and Graph">Oracle Spatial</a></li> <li><a href="/wiki/Solr" class="mw-redirect" title="Solr">Solr</a></li> <li><a href="/wiki/Elasticsearch" title="Elasticsearch">Elasticsearch</a></li> <li><a href="/w/index.php?title=Geocouch&action=edit&redlink=1" class="new" title="「Geocouch」 (存在しないページ)">Geocouch</a></li> <li><a href="/w/index.php?title=OSRM&action=edit&redlink=1" class="new" title="「OSRM」 (存在しないページ)">OSRM</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="polygonの頂点配列が時計周り順"><span id="polygon.E3.81.AE.E9.A0.82.E7.82.B9.E9.85.8D.E5.88.97.E3.81.8C.E6.99.82.E8.A8.88.E5.91.A8.E3.82.8A.E9.A0.86"></span><a href="/wiki/%E3%83%9D%E3%83%AA%E3%82%B4%E3%83%B3" title="ポリゴン">polygon</a>の頂点配列が<a href="/wiki/%E6%99%82%E8%A8%88%E5%91%A8%E3%82%8A" class="mw-redirect" title="時計周り">時計周り</a>順</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=14" title="節を編集: polygonの頂点配列が時計周り順"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>右手系経緯度を採用しているもののうち、<a href="/wiki/%E3%83%9D%E3%83%AA%E3%82%B4%E3%83%B3" title="ポリゴン">polygon</a>の頂点配列順については<a href="/wiki/%E6%99%82%E8%A8%88%E5%91%A8%E3%82%8A" class="mw-redirect" title="時計周り">時計周り</a>順（左手系）を採用しているものがある： </p> <ul><li><a href="/wiki/%E3%82%B7%E3%82%A7%E3%83%BC%E3%83%97%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB" title="シェープファイル">Shapefile</a></li> <li><a href="/wiki/ArcGIS" title="ArcGIS">ArcGIS</a> API for JavaScript</li> <li><a href="/wiki/D3.js" title="D3.js">D3.js</a></li> <li><a href="/wiki/PostGIS" title="PostGIS">PostGIS</a></li> <li><a href="/wiki/SpatiaLite" title="SpatiaLite">SpatiaLite</a></li> <li><a href="/wiki/TopoJSON" class="mw-redirect" title="TopoJSON">TopoJSON</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="左手系経緯度の採用"><span id=".E5.B7.A6.E6.89.8B.E7.B3.BB.E7.B5.8C.E7.B7.AF.E5.BA.A6.E3.81.AE.E6.8E.A1.E7.94.A8"></span>左手系経緯度の採用</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=15" title="節を編集: 左手系経緯度の採用"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>下記では左手系経緯度（緯度、経度の順）が採用されている。 </p> <ul><li><a href="/wiki/Leaflet" title="Leaflet">Leaflet</a></li> <li><a href="/wiki/Google_Maps" class="mw-redirect" title="Google Maps">Google Maps</a> API</li> <li>Apple <a href="/w/index.php?title=MapKit&action=edit&redlink=1" class="new" title="「MapKit」 (存在しないページ)">MapKit</a></li> <li><a href="/w/index.php?title=ArangoDB&action=edit&redlink=1" class="new" title="「ArangoDB」 (存在しないページ)">ArangoDB</a></li> <li><a href="/wiki/GeoRSS" title="GeoRSS">GeoRSS</a></li> <li><a href="https://en.wikipedia.org/wiki/Open_Geospatial_Consortium" class="extiw" title="en:Open Geospatial Consortium">Open Geospatial Consortium</a> (OGC)の <a href="https://en.wikipedia.org/wiki/Spatial_reference_system" class="extiw" title="en:Spatial reference system">Spatial Reference System</a> (SRS)<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading4"><h4 id="左手系地図投影法の採用"><span id=".E5.B7.A6.E6.89.8B.E7.B3.BB.E5.9C.B0.E5.9B.B3.E6.8A.95.E5.BD.B1.E6.B3.95.E3.81.AE.E6.8E.A1.E7.94.A8"></span>左手系地図投影法の採用</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=16" title="節を編集: 左手系地図投影法の採用"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>下記では左手系の地図投影法を採用し、平面座標の<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>軸は右横方向が正、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>軸は下縦方向が正としている<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>。 </p> <ul><li><a href="https://en.wikipedia.org/wiki/Vector_tiles" class="extiw" title="en:Vector tiles">Mapbox Vector Tiles</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="脚注"><span id=".E8.84.9A.E6.B3.A8"></span>脚注</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=17" title="節を編集: 脚注"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text"><a href="/wiki/%E5%A4%A9%E4%BD%93" title="天体">天体</a>が球体であれば、球面上の垂直ベクトルは中心を通るので、地理経緯度は<a href="/w/index.php?title=%E5%9C%B0%E5%BF%83%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&redlink=1" class="new" title="「地心経緯度」 (存在しないページ)">地心経緯度</a>に等しい。</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">地理経緯度は測地経緯度、測地学的経緯度（geodetic longitude and latitude）とも呼ばれる。</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text">扁長もしくは<a href="https://en.wikipedia.org/wiki/Oblate_spheroidal_coordinates" class="extiw" title="en:Oblate spheroidal coordinates">扁平楕円体座標系</a>とは異なる。</span> </li> <li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text"><a href="/wiki/%E3%83%A0%E3%83%BC%E3%83%8B%E3%82%A8%E3%81%AE%E5%AE%9A%E7%90%86" title="ムーニエの定理">ムーニエの定理</a>も参照。</span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">微分関係式は、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d[N(\phi )\cos \phi ]}{d\phi }}=-M(\phi )\sin \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d[N(\phi )\cos \phi ]}{d\phi }}=-M(\phi )\sin \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f66bdbf4b6bf637e02ec238a05b9aad8afbb67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.434ex; height:6.176ex;" alt="{\displaystyle {\frac {d[N(\phi )\cos \phi ]}{d\phi }}=-M(\phi )\sin \phi }"></span></span> </li> <li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="reference-text">解くべき <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> の方程式は <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}&{\frac {p}{\cos \phi }}-{\frac {z}{\sin \phi }}-e^{2}N(\phi )=0,\\&p={\sqrt {x^{2}+y^{2}}}\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 /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>p</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\frac {p}{\cos \phi }}-{\frac {z}{\sin \phi }}-e^{2}N(\phi )=0,\\&p={\sqrt {x^{2}+y^{2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87272c6370290f9daed5aa283fe8d5a11e5ea9c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:29.92ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}&{\frac {p}{\cos \phi }}-{\frac {z}{\sin \phi }}-e^{2}N(\phi )=0,\\&p={\sqrt {x^{2}+y^{2}}}\end{aligned}}}"></span></dd></dl> で、またこれは変数 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\frac {p}{z}}\tan \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>z</mi> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {p}{z}}\tan \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfe7be1f506e23e68990a2f2a47aebf68ce0982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.962ex; height:4.843ex;" alt="{\displaystyle \kappa ={\frac {p}{z}}\tan \phi }"></span>についての<a href="/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F" title="方程式">方程式</a>に帰着できる： <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 \kappa -1-{\frac {e^{2}a\kappa }{\sqrt {p^{2}+(1-e^{2})z^{2}\kappa ^{2}}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> <mi>κ</mi> </mrow> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <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> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa -1-{\frac {e^{2}a\kappa }{\sqrt {p^{2}+(1-e^{2})z^{2}\kappa ^{2}}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43dc4de269caf74188ee9cc44fd644d1aa62e8b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.155ex; height:7.009ex;" alt="{\displaystyle \kappa -1-{\frac {e^{2}a\kappa }{\sqrt {p^{2}+(1-e^{2})z^{2}\kappa ^{2}}}}=0}"></span></dd></dl> 解き方は<a href="https://en.wikipedia.org/wiki/Geographic_coordinate_conversion#From_ECEF_to_geodetic_coordinates" class="extiw" title="en:Geographic coordinate conversion">Geographic_coordinate_conversion#From_ECEF_to_geodetic_coordinates</a>等を参照のこと。また <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=e^{-2}(\kappa ^{-1}-(1-e^{2})){\sqrt {p^{2}+z^{2}\kappa ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <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> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=e^{-2}(\kappa ^{-1}-(1-e^{2})){\sqrt {p^{2}+z^{2}\kappa ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bafd660809eeea712388ec6dc41abf9c85596e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:36.051ex; height:4.843ex;" alt="{\displaystyle h=e^{-2}(\kappa ^{-1}-(1-e^{2})){\sqrt {p^{2}+z^{2}\kappa ^{2}}}}"></span></span> </li> <li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text"><cite class="citation web" style="font-style:normal">Williams, E. (2013年). “<a rel="nofollow" class="external text" href="https://edwilliams.org/avform147.htm#flat">Aviation Formulary.</a>”. <span title="">2024年6月23日</span>閲覧。</cite></span> </li> <li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text">日本では「Hubeny の（簡易）式」などと呼ばれることもある（ただしその名称は適切ではない）。</span> </li> <li id="cite_note-9"><b><a href="#cite_ref-9">^</a></b> <span class="reference-text">180度経線に対しても特異性を持つが、対処は容易である。</span> </li> <li id="cite_note-10"><b><a href="#cite_ref-10">^</a></b> <span class="reference-text">例えば「ガウスの平均（中間）緯度法」の式を級数展開したものとして、 Hubeny, K. (1954). Entwicklung der Gauss'schen Mittelbreitenformeln, Österreichische Zeitschrift für Vermessungswesen, Hubeny, K. (1959). Weiterentwicklung der Gauss'schen Mittelbreitenformeln. Zeitschrift für Vermessungswesen.</span> </li> <li id="cite_note-11"><b><a href="#cite_ref-11">^</a></b> <span class="reference-text">したがって「<a href="/wiki/%E7%90%83%E9%9D%A2%E4%B8%89%E8%A7%92%E6%B3%95#haversine_半正矢関数" title="球面三角法">haversine関数</a>を用いる<a href="/wiki/%E5%A4%A7%E5%86%86%E8%B7%9D%E9%9B%A2#数式" title="大円距離">大円距離計算</a>」（円の弦長に基づき弧長を求める）を<a href="/wiki/%E5%9B%9E%E8%BB%A2%E6%A5%95%E5%86%86%E4%BD%93" title="回転楕円体">回転楕円体</a>（<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f134ae48822cc0ef5227f676b01c08d82029a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.344ex; height:2.676ex;" alt="{\displaystyle e\neq 0}"></span>）へ拡張した形となっている。</span> </li> <li id="cite_note-rapp91-12"><b><a href="#cite_ref-rapp91_12-0">^</a></b> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r101121245">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:var(--color-success,#3a3);margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFRapp1991" class="citation report cs1">Rapp, R, H (1991). Geometric Geodesy, Part I (Report). Ohio Start Univ. <a href="/wiki/Hdl_(%E8%AD%98%E5%88%A5%E5%AD%90)" class="mw-redirect" title="Hdl (識別子)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1811%2F24333">1811/24333</a>。</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Geometric+Geodesy%2C+Part+I&rft.pub=Ohio+Start+Univ.&rft.date=1991&rft_id=info%3Ahdl%2F1811%2F24333&rft.aulast=Rapp&rft.aufirst=R%2C+H&rfr_id=info%3Asid%2Fja.wikipedia.org%3A%E7%B5%8C%E7%B7%AF%E5%BA%A6" class="Z3988"></span></span> </li> <li id="cite_note-13"><b><a href="#cite_ref-13">^</a></b> <span class="reference-text"><a href="/w/index.php?title=%E5%92%8C%E6%BC%A2&action=edit&redlink=1" class="new" title="「和漢」 (存在しないページ)">和漢</a>の用例でも、この（<a href="/wiki/%E7%B5%8C%E5%BA%A6" title="経度">経度</a>・<a href="/wiki/%E7%B7%AF%E5%BA%A6" title="緯度">緯度</a>）の順である「経緯度」である（例えば「<a href="/wiki/%E6%97%A5%E6%9C%AC%E7%B5%8C%E7%B7%AF%E5%BA%A6%E5%8E%9F%E7%82%B9" title="日本経緯度原点">日本経緯度原点</a>」、「<a href="/wiki/%E7%B5%8C%E7%B7%AF%E7%B7%9A" class="mw-redirect" title="経緯線">経緯線</a>」）。</span> </li> <li id="cite_note-14"><b><a href="#cite_ref-14">^</a></b> <span class="reference-text"><a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</a>の別慣行の変数及び順序は：（<a href="/wiki/%E4%BD%99%E7%B7%AF%E5%BA%A6" class="mw-redirect" title="余緯度">余緯度</a>、<a href="/wiki/%E7%B5%8C%E5%BA%A6" title="経度">経度</a>、及び<a href="/wiki/%E9%AB%98%E5%BA%A6" title="高度">高度</a>）。数学・物理学における<a href="/wiki/%E7%90%83%E9%9D%A2%E5%BA%A7%E6%A8%99%E7%B3%BB" title="球面座標系">球面座標系</a>の標準はこれに当たる。</span> </li> <li id="cite_note-lhs-15">^ <a href="#cite_ref-lhs_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-lhs_15-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text">この<a href="/wiki/%E5%B7%A6%E6%89%8B%E7%B3%BB" class="mw-redirect" title="左手系">左手系</a>の使用は一般的には非推奨とされている。ただし<a href="/wiki/%E6%B8%AC%E9%87%8F" title="測量">測量</a>、<a href="/wiki/%E8%88%AA%E6%B5%B7%E8%A1%93" title="航海術">航海術</a>や<a href="/wiki/%E5%9C%B0%E7%90%86%E5%AD%A6" title="地理学">地理学</a>などの分野はこの左手系の使用は極めて標準的である。</span> </li> <li id="cite_note-16"><b><a href="#cite_ref-16">^</a></b> <span class="reference-text">左手系の別慣行では、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>方向を右横方向、<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>方向を下縦方向にとる。</span> </li> <li id="cite_note-17"><b><a href="#cite_ref-17">^</a></b> <span class="reference-text"><a href="/wiki/%E5%B9%B3%E9%9D%A2%E7%9B%B4%E8%A7%92%E5%BA%A7%E6%A8%99%E7%B3%BB" title="平面直角座標系">平面直角座標系</a>（日本の規格）では左手系である。</span> </li> <li id="cite_note-18"><b><a href="#cite_ref-18">^</a></b> <span class="reference-text"><a href="/wiki/%E5%8F%B3%E6%89%8B%E7%B3%BB" title="右手系">右手系</a>の別慣行では：（<b>南</b>→東→北→西）</span> </li> <li id="cite_note-19"><b><a href="#cite_ref-19">^</a></b> <span class="reference-text">OGCによるSRS/CRS の定義では大多数の測地系は axis order を左手系経緯度と定義する。</span> </li> <li id="cite_note-20"><b><a href="#cite_ref-20">^</a></b> <span class="reference-text">他に<a href="/wiki/Scalable_Vector_Graphics" title="Scalable Vector Graphics">SVG</a>フォーマットでは左手系座標が採用されている。</span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%B5%8C%E7%B7%AF%E5%BA%A6&action=edit&section=18" title="節を編集: 関連項目"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E7%B5%8C%E7%B7%AF%E7%B7%9A" class="mw-redirect" title="経緯線">経緯線</a></li> <li><a href="/wiki/%E6%97%A5%E6%9C%AC%E7%B5%8C%E7%B7%AF%E5%BA%A6%E5%8E%9F%E7%82%B9" title="日本経緯度原点">日本経緯度原点</a></li> <li><a href="/wiki/IERS%E5%9F%BA%E6%BA%96%E5%AD%90%E5%8D%88%E7%B7%9A" title="IERS基準子午線">IERS基準子午線</a></li> <li><a href="/wiki/%E5%9C%B0%E7%90%86%E5%BA%A7%E6%A8%99%E7%B3%BB" title="地理座標系">地理座標系</a></li> <li><a href="/wiki/%E3%83%A1%E3%83%AB%E3%82%AB%E3%83%88%E3%83%AB%E5%9B%B3%E6%B3%95" title="メルカトル図法">メルカトル図法</a></li> <li><a href="/wiki/%E5%B9%B3%E9%9D%A2%E7%9B%B4%E8%A7%92%E5%BA%A7%E6%A8%99%E7%B3%BB" title="平面直角座標系">平面直角座標系</a>（日本の規格。<a href="/wiki/%E5%B7%A6%E6%89%8B%E7%B3%BB" class="mw-redirect" title="左手系">左手系</a>）</li> <li><a href="/wiki/ISO_6709" title="ISO 6709">ISO 6709</a> (座標の文字列表現の国際標準。原則として<a href="/wiki/%E5%B7%A6%E6%89%8B%E7%B3%BB" class="mw-redirect" title="左手系">左手系</a>）</li> <li><a href="/wiki/%E5%9B%BD%E5%AE%B6%E5%BA%A7%E6%A8%99" title="国家座標">国家座標</a></li> <li><a href="/wiki/Vincenty%E6%B3%95" title="Vincenty法">Vincenty法</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r96783592">.mw-parser-output .asbox{position:relative;overflow:hidden}.mw-parser-output .asbox table{background:transparent}.mw-parser-output .asbox p{margin:0}.mw-parser-output .asbox p+p{margin-top:0.25em}.mw-parser-output .asbox{font-size:90%}.mw-parser-output .asbox-note{font-size:90%}.mw-parser-output .asbox .navbar{position:absolute;top:-0.90em;right:1em;display:none}</style><div role="note" class="metadata plainlinks asbox stub"><table role="presentation"><tbody><tr class="noresize"><td><span typeof="mw:File"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Globe_and_books_mgx.svg" class="mw-file-description"><img alt="スタブアイコン" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Globe_and_books_mgx.svg/30px-Globe_and_books_mgx.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Globe_and_books_mgx.svg/45px-Globe_and_books_mgx.svg.png 1.5x, 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