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[o]" accesskey="o" class=""><span>Đăng nhập</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Thêm tùy chọn" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Công cụ cá nhân" > <label id="vector-user-links-dropdown-label" 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">Công cụ cá nhân</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Bảng chọn thành viên" > <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_vi.wikipedia.org&uselang=vi"><span>Quyên góp</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:M%E1%BB%9F_t%C3%A0i_kho%E1%BA%A3n&returnto=%C4%90%C6%B0%E1%BB%9Dng+tr%E1%BA%AFc+%C4%91%E1%BB%8Ba" title="Bạn được khuyến khích mở tài khoản và đăng nhập; tuy nhiên, không bắt buộc phải có tài khoản"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Tạo tài khoản</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C4%83ng_nh%E1%BA%ADp&returnto=%C4%90%C6%B0%E1%BB%9Dng+tr%E1%BA%AFc+%C4%91%E1%BB%8Ba" title="Đăng nhập sẽ có lợi hơn, tuy nhiên không bắt buộc. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Đăng nhập</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"> Trang dành cho người dùng chưa đăng nhập <a href="/wiki/Tr%E1%BB%A3_gi%C3%BAp:Gi%E1%BB%9Bi_thi%E1%BB%87u" aria-label="Tìm hiểu thêm về sửa đổi"><span>tìm hiểu thêm</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/%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C3%B3ng_g%C3%B3p_c%E1%BB%A7a_t%C3%B4i" title="Danh sách các sửa đổi được thực hiện qua địa chỉ IP này [y]" accesskey="y"><span>Đóng góp</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BA%A3o_lu%E1%BA%ADn_t%C3%B4i" title="Thảo luận với địa chỉ IP này [n]" accesskey="n"><span>Thảo luận cho địa chỉ IP này</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="Trang Web"> <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="Nội dung" 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">Nội dung</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ẩn</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">Đầu</div> </a> </li> <li id="toc-Đường_trắc_địa_a-phin" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Đường_trắc_địa_a-phin"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Đường trắc địa a-phin</span> </div> </a> <button aria-controls="toc-Đường_trắc_địa_a-phin-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>Hiện/ẩn mục Đường trắc địa a-phin</span> </button> <ul id="toc-Đường_trắc_địa_a-phin-sublist" class="vector-toc-list"> <li id="toc-Đường_dòng_trắc_địa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Đường_dòng_trắc_địa"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Đường dòng trắc địa</span> </div> </a> <ul id="toc-Đường_dòng_trắc_địa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tham_khảo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tham_khảo"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Tham khảo</span> </div> </a> <ul id="toc-Tham_khảo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thư_mục" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Thư_mục"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Thư mục</span> </div> </a> <ul id="toc-Thư_mục-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="Nội dung" 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="Đóng mở mục lục" > <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">Đóng mở mục lục</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">Đường trắc địa</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="Xem bài viết trong ngôn ngữ khác. Bài có sẵn trong 42 ngôn ngữ" > <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-42" 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">42 ngôn ngữ</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D9%8A%D9%88%D8%AF%D9%8A%D8%B3%D9%8A" title="جيوديسي – Tiếng Ả Rập" lang="ar" hreflang="ar" data-title="جيوديسي" data-language-autonym="العربية" data-language-local-name="Tiếng Ả Rập" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geodesik" title="Geodesik – Tiếng Indonesia" lang="id" hreflang="id" data-title="Geodesik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Tiếng Indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AD%E0%A7%82%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%B0%E0%A7%87%E0%A6%96%E0%A6%BE" title="ভূমিতিক বক্ররেখা – Tiếng Bangla" lang="bn" hreflang="bn" data-title="ভূমিতিক বক্ররেখা" data-language-autonym="বাংলা" data-language-local-name="Tiếng Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geod%C3%A8sica" title="Geodèsica – Tiếng Catalan" lang="ca" hreflang="ca" data-title="Geodèsica" data-language-autonym="Català" data-language-local-name="Tiếng Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D0%BB%D0%BB%D0%B5_%D0%B9%C4%95%D1%80" title="Геодезилле йĕр – Tiếng Chuvash" lang="cv" hreflang="cv" data-title="Геодезилле йĕр" data-language-autonym="Чӑвашла" data-language-local-name="Tiếng Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Geodetika" title="Geodetika – Tiếng Séc" lang="cs" hreflang="cs" data-title="Geodetika" data-language-autonym="Čeština" data-language-local-name="Tiếng Séc" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Geod%C3%A4te" title="Geodäte – Tiếng Đức" lang="de" hreflang="de" data-title="Geodäte" data-language-autonym="Deutsch" data-language-local-name="Tiếng Đức" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Geodeetiline_joon" title="Geodeetiline joon – Tiếng Estonia" lang="et" hreflang="et" data-title="Geodeetiline joon" data-language-autonym="Eesti" data-language-local-name="Tiếng Estonia" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CF%89%CE%B4%CE%B1%CE%B9%CF%83%CE%B9%CE%B1%CE%BA%CE%AE" title="Γεωδαισιακή – Tiếng Hy Lạp" lang="el" hreflang="el" data-title="Γεωδαισιακή" data-language-autonym="Ελληνικά" data-language-local-name="Tiếng Hy Lạp" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Geodesic" title="Geodesic – Tiếng Anh" lang="en" hreflang="en" data-title="Geodesic" data-language-autonym="English" data-language-local-name="Tiếng Anh" 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/L%C3%ADnea_geod%C3%A9sica" title="Línea geodésica – Tiếng Tây Ban Nha" lang="es" hreflang="es" data-title="Línea geodésica" data-language-autonym="Español" data-language-local-name="Tiếng Tây Ban Nha" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Geodezia_linio" title="Geodezia linio – Tiếng Quốc Tế Ngữ" lang="eo" hreflang="eo" data-title="Geodezia linio" data-language-autonym="Esperanto" data-language-local-name="Tiếng Quốc Tế Ngữ" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%98%D8%A6%D9%88%D8%AF%D8%B2%DB%8C%DA%A9" title="ژئودزیک – Tiếng Ba Tư" lang="fa" hreflang="fa" data-title="ژئودزیک" data-language-autonym="فارسی" data-language-local-name="Tiếng Ba Tư" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/G%C3%A9od%C3%A9sique" title="Géodésique – Tiếng Pháp" lang="fr" hreflang="fr" data-title="Géodésique" data-language-autonym="Français" data-language-local-name="Tiếng Pháp" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Geodasach" title="Geodasach – Tiếng Ireland" lang="ga" hreflang="ga" data-title="Geodasach" data-language-autonym="Gaeilge" data-language-local-name="Tiếng Ireland" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Xeod%C3%A9sica" title="Xeodésica – Tiếng Galician" lang="gl" hreflang="gl" data-title="Xeodésica" data-language-autonym="Galego" data-language-local-name="Tiếng Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B8%A1%EC%A7%80%EC%84%A0" title="측지선 – Tiếng Hàn" lang="ko" hreflang="ko" data-title="측지선" data-language-autonym="한국어" data-language-local-name="Tiếng Hàn" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%A5%D5%B8%D5%A4%D5%A5%D5%A6%D5%AB%D5%AF_%D5%A3%D5%AE%D5%A5%D6%80" title="Գեոդեզիկ գծեր – Tiếng Armenia" lang="hy" hreflang="hy" data-title="Գեոդեզիկ գծեր" data-language-autonym="Հայերեն" data-language-local-name="Tiếng Armenia" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geodetica" title="Geodetica – Tiếng Italy" lang="it" hreflang="it" data-title="Geodetica" data-language-autonym="Italiano" data-language-local-name="Tiếng Italy" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%99%D7%9C%D7%94_%D7%92%D7%90%D7%95%D7%93%D7%96%D7%99%D7%AA" title="מסילה גאודזית – Tiếng Do Thái" lang="he" hreflang="he" data-title="מסילה גאודזית" data-language-autonym="עברית" data-language-local-name="Tiếng Do Thái" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%8F%D0%BB%D1%8B%D2%9B_%D1%81%D1%8B%D0%B7%D1%8B%D2%9B%D1%82%D0%B0%D1%80" title="Геодезиялық сызықтар – Tiếng Kazakh" lang="kk" hreflang="kk" data-title="Геодезиялық сызықтар" data-language-autonym="Қазақша" data-language-local-name="Tiếng Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Geodetesch_Linn" title="Geodetesch Linn – Tiếng Luxembourg" lang="lb" hreflang="lb" data-title="Geodetesch Linn" data-language-autonym="Lëtzebuergesch" data-language-local-name="Tiếng Luxembourg" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Geodezin%C4%97_kreiv%C4%97" title="Geodezinė kreivė – Tiếng Litva" lang="lt" hreflang="lt" data-title="Geodezinė kreivė" data-language-autonym="Lietuvių" data-language-local-name="Tiếng Litva" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Geodetikus_vonalak" title="Geodetikus vonalak – Tiếng Hungary" lang="hu" hreflang="hu" data-title="Geodetikus vonalak" data-language-autonym="Magyar" data-language-local-name="Tiếng Hungary" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geodeet_(wiskunde)" title="Geodeet (wiskunde) – Tiếng Hà Lan" lang="nl" hreflang="nl" data-title="Geodeet (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Tiếng Hà Lan" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B8%AC%E5%9C%B0%E7%B7%9A" title="測地線 – Tiếng Nhật" lang="ja" hreflang="ja" data-title="測地線" data-language-autonym="日本語" data-language-local-name="Tiếng Nhật" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Geodetisk_kurve" title="Geodetisk kurve – Tiếng Na Uy (Bokmål)" lang="nb" hreflang="nb" data-title="Geodetisk kurve" data-language-autonym="Norsk bokmål" data-language-local-name="Tiếng Na Uy (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Geodetisk_kurve" title="Geodetisk kurve – Tiếng Na Uy (Nynorsk)" lang="nn" hreflang="nn" data-title="Geodetisk kurve" data-language-autonym="Norsk nynorsk" data-language-local-name="Tiếng Na Uy (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Geodezik_chiziq" title="Geodezik chiziq – Tiếng Uzbek" lang="uz" hreflang="uz" data-title="Geodezik chiziq" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Tiếng Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Linia_geodezyjna" title="Linia geodezyjna – Tiếng Ba Lan" lang="pl" hreflang="pl" data-title="Linia geodezyjna" data-language-autonym="Polski" data-language-local-name="Tiếng Ba Lan" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Geod%C3%A9sica" title="Geodésica – Tiếng Bồ Đào Nha" lang="pt" hreflang="pt" data-title="Geodésica" data-language-autonym="Português" data-language-local-name="Tiếng Bồ Đào Nha" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geodezic%C4%83" title="Geodezică – Tiếng Romania" lang="ro" hreflang="ro" data-title="Geodezică" data-language-autonym="Română" data-language-local-name="Tiếng Romania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F" title="Геодезическая – Tiếng Nga" lang="ru" hreflang="ru" data-title="Геодезическая" data-language-autonym="Русский" data-language-local-name="Tiếng Nga" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Geodesic" title="Geodesic – Simple English" lang="en-simple" hreflang="en-simple" data-title="Geodesic" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Geodetka" title="Geodetka – Tiếng Slovenia" lang="sl" hreflang="sl" data-title="Geodetka" data-language-autonym="Slovenščina" data-language-local-name="Tiếng Slovenia" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%98%D1%81%D0%BA%D0%B0_%D0%BB%D0%B8%D0%BD%D0%B8%D1%98%D0%B0" title="Геодезијска линија – Tiếng Serbia" lang="sr" hreflang="sr" data-title="Геодезијска линија" data-language-autonym="Српски / srpski" data-language-local-name="Tiếng Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Geodezijska_linija" title="Geodezijska linija – Tiếng Serbo-Croatia" lang="sh" hreflang="sh" data-title="Geodezijska linija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Tiếng Serbo-Croatia" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Geodeesi" title="Geodeesi – Tiếng Phần Lan" lang="fi" hreflang="fi" data-title="Geodeesi" data-language-autonym="Suomi" data-language-local-name="Tiếng Phần Lan" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Jeodezik" title="Jeodezik – Tiếng Thổ Nhĩ Kỳ" lang="tr" hreflang="tr" data-title="Jeodezik" data-language-autonym="Türkçe" data-language-local-name="Tiếng Thổ Nhĩ Kỳ" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D0%B5%D0%BE%D0%B4%D0%B5%D0%B7%D0%B8%D1%87%D0%BD%D0%B0_%D0%BB%D1%96%D0%BD%D1%96%D1%8F" title="Геодезична лінія – Tiếng Ukraina" lang="uk" hreflang="uk" data-title="Геодезична лінія" data-language-autonym="Українська" data-language-local-name="Tiếng Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E6%B8%AC%E5%9C%B0%E7%B7%9A" title="測地線 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="測地線" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B5%8B%E5%9C%B0%E7%BA%BF" title="测地线 – Tiếng Trung" lang="zh" hreflang="zh" data-title="测地线" data-language-autonym="中文" data-language-local-name="Tiếng Trung" 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/Q213488#sitelinks-wikipedia" title="Sửa liên kết giữa ngôn ngữ" class="wbc-editpage">Sửa liên kết</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="Không gian tên"> <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/%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba" title="Xem bài viết [c]" accesskey="c"><span>Bài viết</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Th%E1%BA%A3o_lu%E1%BA%ADn:%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit&redlink=1" rel="discussion" class="new" title="Thảo luận về trang này (trang không tồn tại) [t]" accesskey="t"><span>Thảo luận</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Thay đổi biến thể ngôn ngữ" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tiếng Việt</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Giao diện"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba"><span>Đọc</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&veaction=edit" title="Sửa đổi trang này [v]" accesskey="v"><span>Sửa đổi</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit" title="Sửa đổi mã nguồn của trang này [e]" accesskey="e"><span>Sửa mã nguồn</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=history" title="Các phiên bản cũ của trang này [h]" accesskey="h"><span>Xem lịch sử</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Công cụ trang"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Công cụ" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Công cụ</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Công cụ</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ẩn</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Thêm tùy chọn" > <div class="vector-menu-heading"> Tác vụ </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba"><span>Đọc</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&veaction=edit" title="Sửa đổi trang này [v]" accesskey="v"><span>Sửa đổi</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit" title="Sửa đổi mã nguồn của trang này [e]" accesskey="e"><span>Sửa mã nguồn</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=history"><span>Xem lịch sử</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Chung </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Li%C3%AAn_k%E1%BA%BFt_%C4%91%E1%BA%BFn_%C4%91%C3%A2y/%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba" title="Các trang liên kết đến đây [j]" accesskey="j"><span>Các liên kết đến đây</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Thay_%C4%91%E1%BB%95i_li%C3%AAn_quan/%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba" rel="nofollow" title="Thay đổi gần đây của các trang liên kết đến đây [k]" accesskey="k"><span>Thay đổi liên quan</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Trang_%C4%91%E1%BA%B7c_bi%E1%BB%87t" title="Một danh sách chứa tất cả trang đặc biệt [q]" accesskey="q"><span>Trang đặc biệt</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&oldid=67157414" title="Liên kết thường trực đến phiên bản này của trang"><span>Liên kết thường trực</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=info" title="Thêm chi tiết về trang này"><span>Thông tin trang</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:Tr%C3%ADch_d%E1%BA%ABn&page=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&id=67157414&wpFormIdentifier=titleform" title="Hướng dẫn cách trích dẫn trang này"><span>Trích dẫn trang này</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:UrlShortener&url=https%3A%2F%2Fvi.wikipedia.org%2Fwiki%2F%25C4%2590%25C6%25B0%25E1%25BB%259Dng_tr%25E1%25BA%25AFc_%25C4%2591%25E1%25BB%258Ba"><span>Lấy URL ngắn gọn</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:QrCode&url=https%3A%2F%2Fvi.wikipedia.org%2Fwiki%2F%25C4%2590%25C6%25B0%25E1%25BB%259Dng_tr%25E1%25BA%25AFc_%25C4%2591%25E1%25BB%258Ba"><span>Tải mã QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> In và xuất </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:S%C3%A1ch&bookcmd=book_creator&referer=%C4%90%C6%B0%E1%BB%9Dng+tr%E1%BA%AFc+%C4%91%E1%BB%8Ba"><span>Tạo một quyển sách</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%C4%90%E1%BA%B7c_bi%E1%BB%87t:DownloadAsPdf&page=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=show-download-screen"><span>Tải dưới dạng PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&printable=yes" title="Bản để in ra của trang [p]" accesskey="p"><span>Bản để in ra</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Tại dự án khác </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Geodesic_(mathematics)" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q213488" title="Liên kết đến khoản mục kết nối trong kho dữ liệu [g]" accesskey="g"><span>Khoản mục Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Công cụ trang"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Giao diện"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Giao diện</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ẩn</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">Bách khoa toàn thư mở 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="vi" dir="ltr"><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/T%E1%BA%ADp_tin:Spherical_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/150px-Spherical_triangle.svg.png" decoding="async" width="150" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/225px-Spherical_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Spherical_triangle.svg/300px-Spherical_triangle.svg.png 2x" data-file-width="356" data-file-height="358" /></a><figcaption>Tam giác trắc địa trên mặt cầu.Các đường trắc địa là các cung tròn lớn.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/T%E1%BA%ADp_tin:Duong_dong_trac_dia_cua_duong_tron.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Duong_dong_trac_dia_cua_duong_tron.png/220px-Duong_dong_trac_dia_cua_duong_tron.png" decoding="async" width="220" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Duong_dong_trac_dia_cua_duong_tron.png/330px-Duong_dong_trac_dia_cua_duong_tron.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Duong_dong_trac_dia_cua_duong_tron.png/440px-Duong_dong_trac_dia_cua_duong_tron.png 2x" data-file-width="557" data-file-height="493" /></a><figcaption>Một phần của hai đường dòng trắc địa trên phân thớ tiếp xúc của đường tròn. Lưu ý rằng ta thu được đường trắc địa bằng phép chiếu xuống đường tròn.</figcaption></figure> <p>Trong <a href="/wiki/H%C3%ACnh_h%E1%BB%8Dc_Riemann" title="Hình học Riemann">hình học Riemann</a>, <b>đường trắc địa</b> (hay <b>cung trắc địa</b>) là một đường cong có <a href="/w/index.php?title=%C4%90%E1%BB%99_cong_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit&redlink=1" class="new" title="Độ cong trắc địa (trang không tồn tại)">độ cong trắc địa</a> bằng không tại mọi điểm, và cũng là đường có độ dài cực trị trong các đường nối hai điểm trên một <a href="/wiki/%C4%90a_t%E1%BA%A1p_Riemann" title="Đa tạp Riemann">đa tạp Riemann</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Đường trắc địa trên <a href="/wiki/M%E1%BA%B7t_c%E1%BA%A7u" title="Mặt cầu">mặt cầu</a> là các (đoạn của) <a href="/w/index.php?title=V%C3%B2ng_tr%C3%B2n_l%E1%BB%9Bn&action=edit&redlink=1" class="new" title="Vòng tròn lớn (trang không tồn tại)">vòng tròn lớn</a>, trên <a href="/wiki/M%E1%BA%B7t_tr%E1%BB%A5" title="Mặt trụ">mặt trụ</a> <a href="/wiki/M%E1%BA%B7t_tr%C3%B2n_xoay" title="Mặt tròn xoay">tròn xoay</a> là các (đoạn của) đường xoắn ốc, trên mặt phẳng là các đoạn/đường thẳng. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Đường_trắc_địa_a-phin"><span id=".C4.90.C6.B0.E1.BB.9Dng_tr.E1.BA.AFc_.C4.91.E1.BB.8Ba_a-phin"></span>Đường trắc địa a-phin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&veaction=edit&section=1" title="Sửa đổi phần “Đường trắc địa a-phin”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit&section=1" title="Sửa mã nguồn tại đề mục: Đường trắc địa a-phin"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Xét một đa tạp trơn <i>M</i> với một <a href="/wiki/Li%C3%AAn_k%E1%BA%BFt_(ph%C3%A2n_th%E1%BB%9B_v%C3%A9c_t%C6%A1)" title="Liên kết (phân thớ véc tơ)">liên kết a-phin</a> ∇. Một đường trắc địa a-phin (đối với liên kết ∇) được định nghĩa là một đường cong γ(<i>t</i>) sao cho <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 \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/998d786a63d948705e7212f53917b10213171b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.584ex; height:3.176ex;" alt="{\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}"></span>.<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> </p><p>Sử dụng một hệ tọa độ địa phương, ta có thể viết <b>phương trình trắc địa</b> </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 {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45585f6a1e2cf2568e1d332016998dfcfdf43f6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.799ex; height:6.176ex;" alt="{\displaystyle {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ }"></span></dd></dl> <p>với <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 \gamma ^{\mu }=x^{\mu }\circ \gamma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>∘</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{\mu }=x^{\mu }\circ \gamma (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89889b02694dae9b5f264b3d005100d8063d2890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.261ex; height:2.843ex;" alt="{\displaystyle \gamma ^{\mu }=x^{\mu }\circ \gamma (t)}"></span> và <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 \Gamma _{\mu \nu }^{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{\mu \nu }^{\lambda }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ed711a5120b51b00a8bb4bdd696b6f38655ee2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.547ex; height:3.176ex;" alt="{\displaystyle \Gamma _{\mu \nu }^{\lambda }}"></span>là các kí hiệu Christoffel của liên kết ∇. </p><p>Đường trắc địa giống như là quĩ đạo của một hạt chuyển động tự do trên đa tạp. </p><p>Trong trường hợp liên kết Levi-Civita, ta thu được đường trắc địa theo nghĩa Riemann. </p> <div class="mw-heading mw-heading3"><h3 id="Đường_dòng_trắc_địa"><span id=".C4.90.C6.B0.E1.BB.9Dng_d.C3.B2ng_tr.E1.BA.AFc_.C4.91.E1.BB.8Ba"></span>Đường dòng trắc địa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&veaction=edit&section=2" title="Sửa đổi phần “Đường dòng trắc địa”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit&section=2" title="Sửa mã nguồn tại đề mục: Đường dòng trắc địa"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Xét đa tạp <i>M</i> được trang bị một liên kết a-phin. <b>Đường dòng trắc địa</b> (<i>geodesic flow</i>) của đa tạp <i>M</i> là một <b>R</b>-tác động địa phương trên <a href="/wiki/Ph%C3%A2n_th%E1%BB%9B_ti%E1%BA%BFp_tuy%E1%BA%BFn" title="Phân thớ tiếp tuyến">phân thớ tiếp xúc</a> <i>TM</i> sao cho </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 G^{t}(V)={\dot {\gamma }}_{V}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{t}(V)={\dot {\gamma }}_{V}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba0cb3eb1fec6233de4124531c25c1b57652f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.755ex; height:3.009ex;" alt="{\displaystyle G^{t}(V)={\dot {\gamma }}_{V}(t)}"></span></dd></dl> <p>với <i>t</i> ∈ <b>R</b>, <i>V</i> ∈ <i>TM</i> và <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 \gamma _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e564896baedd0fbdb5645daa0e467d4c7efb412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.7ex; height:2.176ex;" alt="{\displaystyle \gamma _{V}}"></span> là đường trắc địa (cục bộ) với điều kiện ban đầu <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 {\dot {\gamma }}_{V}(0)=V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\gamma }}_{V}(0)=V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903b229c4679111a13875213b13b0c2b9c3262e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.616ex; height:2.843ex;" alt="{\displaystyle {\dot {\gamma }}_{V}(0)=V}"></span>. Nói riêng, <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6767de5d80e3c90879b1c52fcac3070109a6dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.653ex; height:2.509ex;" alt="{\displaystyle G^{t}}"></span></i>(<i>V</i>) = exp(<i>tV</i>) là ánh xạ mũ của véc-tơ <i>tV</i>. Hình chiếu của một đường dòng trắc địa lên <i>M</i> là một đường trắc địa trên <i>M</i>.<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><p>Trên một đa tạp Riemann, đường dòng trắc địa được đồng nhất với đường dòng Hamilton trên phân thớ đối tiếp xúc. Đường dòng trắc địa bảo toàn metric, tức là </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 g(G^{t}(V),G^{t}(V))=g(V,V).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(G^{t}(V),G^{t}(V))=g(V,V).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9387876b13605e6f4f5e98ce29f38347920b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.124ex; height:3.009ex;" alt="{\displaystyle g(G^{t}(V),G^{t}(V))=g(V,V).\,}"></span></dd></dl> <p>Hệ quả là, nếu <i>V</i> là véc-tơ đơn vị, <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 \gamma _{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e564896baedd0fbdb5645daa0e467d4c7efb412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.7ex; height:2.176ex;" alt="{\displaystyle \gamma _{V}}"></span> có tốc độ không đổi bằng <i>1</i>, do đó đường dòng trắc địa luôn nằm trong phân thớ tiếp xúc đơn vị. </p><p>Đường dòng trắc địa cũng là đường dòng của một <a href="/wiki/Tr%C6%B0%E1%BB%9Dng_v%C3%A9c_t%C6%A1" class="mw-redirect" title="Trường véc tơ">trường véc-tơ</a> trên đa tạp <i>TM</i>. Trường véc-tơ ấy được gọi là <b>tia trắc địa</b> (<i>geodesic spray</i>). Ta cũng có một chùm trắc địa trên phân thớ tiếp xúc đơn vị <i>S(TM)</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Tham_khảo"><span id="Tham_kh.E1.BA.A3o"></span>Tham khảo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&veaction=edit&section=3" title="Sửa đổi phần “Tham khảo”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit&section=3" title="Sửa mã nguồn tại đề mục: Tham khảo"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r71728118">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Đoàn Quỳnh (2000), tr. 342</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Đoàn Quỳnh (2000), tr. 332</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text">Jost (2002), tr. 47</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Thư_mục"><span id="Th.C6.B0_m.E1.BB.A5c"></span>Thư mục</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&veaction=edit&section=4" title="Sửa đổi phần “Thư mục”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%C4%90%C6%B0%E1%BB%9Dng_tr%E1%BA%AFc_%C4%91%E1%BB%8Ba&action=edit&section=4" title="Sửa mã nguồn tại đề mục: Thư mục"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Đoàn Quỳnh, 2000, Hình học vi phân, Nhà xuất bản giáo dục</li> <li>Jost, Jürgen, 2002, Riemannian Geometry and Geometric Analysis, Berlin, New York: Springer-Verlag, <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/9783540426271" class="internal mw-magiclink-isbn">ISBN 978-3-540-42627-1</a></li> <li>Lee, John, 1997, Introduction to Riemannian Manifolds, <a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Ngu%E1%BB%93n_s%C3%A1ch/9783319917559" class="internal mw-magiclink-isbn">ISBN 978-3-319-91755-9</a><div class="boilerplate metadata plainlinks" id="stub"><table class="metadata plainlinks stub" style="background: transparent;" role="presentation"><tbody><tr> <td><span typeof="mw:File"><a href="/wiki/Th%E1%BB%83_lo%E1%BA%A1i:S%C6%A1_khai_to%C3%A1n_h%E1%BB%8Dc" title="Thể loại:Sơ khai toán học"><img alt="Hình tượng sơ khai" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/45px-E-to-the-i-pi.svg.png" decoding="async" width="45" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/68px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/90px-E-to-the-i-pi.svg.png 2x" data-file-width="512" data-file-height="453" /></a></span></td> <td><i>Bài viết liên quan đến <a href="/wiki/To%C3%A1n_h%E1%BB%8Dc" title="Toán học">toán học</a> này vẫn còn <a href="/wiki/Wikipedia:B%C3%A0i_s%C6%A1_khai" title="Wikipedia:Bài sơ khai">sơ khai</a>. 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