Universal Transverse Mercator coordinate system

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<span>Definitions</span> </div> </a> <button aria-controls="toc-Definitions-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>Toggle Definitions subsection</span> </button> <ul id="toc-Definitions-sublist" class="vector-toc-list"> <li id="toc-UTM_zone" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#UTM_zone"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>UTM zone</span> </div> </a> <ul id="toc-UTM_zone-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exceptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exceptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Exceptions</span> </div> </a> <ul id="toc-Exceptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Overlapping_grids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Overlapping_grids"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Overlapping grids</span> </div> </a> <ul id="toc-Overlapping_grids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Latitude_bands" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Latitude_bands"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Latitude bands</span> </div> </a> <ul id="toc-Latitude_bands-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Locating_a_position_using_UTM_coordinates" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Locating_a_position_using_UTM_coordinates"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Locating a position using UTM coordinates</span> </div> </a> <button aria-controls="toc-Locating_a_position_using_UTM_coordinates-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>Toggle Locating a position using UTM coordinates subsection</span> </button> <ul id="toc-Locating_a_position_using_UTM_coordinates-sublist" class="vector-toc-list"> <li id="toc-Simplified_formulae" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simplified_formulae"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Simplified formulae</span> </div> </a> <ul id="toc-Simplified_formulae-sublist" class="vector-toc-list"> <li id="toc-From_latitude,_longitude_(φ,_λ)_to_UTM_coordinates_(E,_N)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#From_latitude,_longitude_(φ,_λ)_to_UTM_coordinates_(E,_N)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>From latitude, longitude (<i>φ</i>, <i>λ</i>) to UTM coordinates (E, N)</span> </div> </a> <ul id="toc-From_latitude,_longitude_(φ,_λ)_to_UTM_coordinates_(E,_N)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-From_UTM_coordinates_(E,_N,_Zone,_Hemi)_to_latitude,_longitude_(φ,_λ)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#From_UTM_coordinates_(E,_N,_Zone,_Hemi)_to_latitude,_longitude_(φ,_λ)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>From UTM coordinates (E, N, Zone, Hemi) to latitude, longitude (<i>φ</i>, <i>λ</i>)</span> </div> </a> <ul id="toc-From_UTM_coordinates_(E,_N,_Zone,_Hemi)_to_latitude,_longitude_(φ,_λ)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Universal Transverse Mercator coordinate system</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="Go to an article in another language. Available in 33 languages" > <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-33" 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">33 languages</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/%D9%86%D8%B8%D8%A7%D9%85_%D8%A5%D8%B3%D9%82%D8%A7%D8%B7_%D9%85%D8%B1%D9%83%D8%A7%D8%AA%D9%88%D8%B1_%D8%A7%D9%84%D9%85%D8%B3%D8%AA%D8%B9%D8%B1%D8%B6_%D8%A7%D9%84%D8%B9%D8%A7%D9%84%D9%85%D9%8A" title="نظام إسقاط مركاتور المستعرض العالمي – Arabic" lang="ar" hreflang="ar" data-title="نظام إسقاط مركاتور المستعرض العالمي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Sistema_de_Coordenaes_Universal_Tresversal_Mercator" title="Sistema de Coordenaes Universal Tresversal Mercator – Asturian" lang="ast" hreflang="ast" data-title="Sistema de Coordenaes Universal Tresversal Mercator" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Projecci%C3%B3_Universal_Transversa_de_Mercator" title="Projecció Universal Transversa de Mercator – Catalan" lang="ca" hreflang="ca" data-title="Projecció Universal Transversa de Mercator" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/UTM" title="UTM – Czech" lang="cs" hreflang="cs" data-title="UTM" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Universal_Transverse_Mercator" title="Universal Transverse Mercator – Danish" lang="da" hreflang="da" data-title="Universal Transverse Mercator" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/UTM-Koordinatensystem" title="UTM-Koordinatensystem – German" lang="de" hreflang="de" data-title="UTM-Koordinatensystem" data-language-autonym="Deutsch" data-language-local-name="German" 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/Mercatori_universaalne_p%C3%B5ikprojektsioon" title="Mercatori universaalne põikprojektsioon – Estonian" lang="et" hreflang="et" data-title="Mercatori universaalne põikprojektsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_de_coordenadas_universal_transversal_de_Mercator" title="Sistema de coordenadas universal transversal de Mercator – Spanish" lang="es" hreflang="es" data-title="Sistema de coordenadas universal transversal de Mercator" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%AE%D8%AA%D8%B5%D8%A7%D8%AA_%D8%AC%D9%87%D8%A7%D9%86%DB%8C_%D9%85%D8%B1%DA%A9%D8%A7%D8%AA%D9%88%D8%B1_%D9%85%D8%B9%DA%A9%D9%88%D8%B3" title="دستگاه مختصات جهانی مرکاتور معکوس – Persian" lang="fa" hreflang="fa" data-title="دستگاه مختصات جهانی مرکاتور معکوس" data-language-autonym="فارسی" data-language-local-name="Persian" 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/Transverse_universelle_de_Mercator" title="Transverse universelle de Mercator – French" lang="fr" hreflang="fr" data-title="Transverse universelle de Mercator" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/UTM_%EC%A2%8C%ED%91%9C%EA%B3%84" title="UTM 좌표계 – Korean" lang="ko" hreflang="ko" data-title="UTM 좌표계" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ilo mw-list-item"><a href="https://ilo.wikipedia.org/wiki/Sistema_ti_nagsasabtan_ti_UTM" title="Sistema ti nagsasabtan ti UTM – Iloko" lang="ilo" hreflang="ilo" data-title="Sistema ti nagsasabtan ti UTM" data-language-autonym="Ilokano" data-language-local-name="Iloko" class="interlanguage-link-target"><span>Ilokano</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Proiezione_universale_trasversa_di_Mercatore" title="Proiezione universale trasversa di Mercatore – Italian" lang="it" hreflang="it" data-title="Proiezione universale trasversa di Mercatore" data-language-autonym="Italiano" data-language-local-name="Italian" 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%A8%D7%A9%D7%AA_UTM" title="רשת UTM – Hebrew" lang="he" hreflang="he" data-title="רשת UTM" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/UTM-Koordinatesystem" title="UTM-Koordinatesystem – Luxembourgish" lang="lb" hreflang="lb" data-title="UTM-Koordinatesystem" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/UTM" title="UTM – Hungarian" lang="hu" hreflang="hu" data-title="UTM" data-language-autonym="Magyar" data-language-local-name="Hungarian" 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/Universele_transversale_mercatorprojectie" title="Universele transversale mercatorprojectie – Dutch" lang="nl" hreflang="nl" data-title="Universele transversale mercatorprojectie" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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/%E3%83%A6%E3%83%8B%E3%83%90%E3%83%BC%E3%82%B5%E3%83%AB%E6%A8%AA%E3%83%A1%E3%83%AB%E3%82%AB%E3%83%88%E3%83%AB%E5%9B%B3%E6%B3%95" title="ユニバーサル横メルカトル図法 – Japanese" lang="ja" hreflang="ja" data-title="ユニバーサル横メルカトル図法" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/UTM-Koordinatensystem" title="UTM-Koordinatensystem – Northern Frisian" lang="frr" hreflang="frr" data-title="UTM-Koordinatensystem" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/UTM-koordinater" title="UTM-koordinater – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="UTM-koordinater" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian 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/UTM-koordinatar" title="UTM-koordinatar – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="UTM-koordinatar" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian 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/Universal_transvers_Mercator_proektsiyasi" title="Universal transvers Mercator proektsiyasi – Uzbek" lang="uz" hreflang="uz" data-title="Universal transvers Mercator proektsiyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="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/Uk%C5%82ad_UTM" title="Układ UTM – Polish" lang="pl" hreflang="pl" data-title="Układ UTM" data-language-autonym="Polski" data-language-local-name="Polish" 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/Universal_Transversa_de_Mercator" title="Universal Transversa de Mercator – Portuguese" lang="pt" hreflang="pt" data-title="Universal Transversa de Mercator" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D0%BE%D0%BF%D0%B5%D1%80%D0%B5%D1%87%D0%BD%D0%B0%D1%8F_%D0%BF%D1%80%D0%BE%D0%B5%D0%BA%D1%86%D0%B8%D1%8F_%D0%9C%D0%B5%D1%80%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%B0" title="Универсальная поперечная проекция Меркатора – Russian" lang="ru" hreflang="ru" data-title="Универсальная поперечная проекция Меркатора" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D0%B7%D0%B0%D0%BB%D0%BD%D0%B8_%D0%A2%D1%80%D0%B0%D0%BD%D1%81%D0%B2%D0%B5%D1%80%D0%B7%D0%B0%D0%BB%D0%BD%D0%B8_%D0%9C%D0%B5%D1%80%D0%BA%D0%B0%D1%82%D0%BE%D1%80%D0%BE%D0%B2_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B8_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Универзални Трансверзални Меркаторов координатни систем – Serbian" lang="sr" hreflang="sr" data-title="Универзални Трансверзални Меркаторов координатни систем" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/UTM-projektio" title="UTM-projektio – Finnish" lang="fi" hreflang="fi" data-title="UTM-projektio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Universal_Transverse_Mercator" title="Universal Transverse Mercator – Swedish" lang="sv" hreflang="sv" data-title="Universal Transverse Mercator" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AF%E0%AF%82%E0%AE%A9%E0%AE%BF%E0%AE%B5%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%9A%E0%AE%B2%E0%AF%8D_%E0%AE%9F%E0%AF%8D%E0%AE%B0%E0%AE%BE%E0%AE%A9%E0%AF%8D%E0%AE%B8%E0%AF%8D%E0%AE%B5%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%B8%E0%AF%8D_%E0%AE%AE%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%95%E0%AF%87%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%B0%E0%AF%8D_%E0%AE%AA%E0%AF%8D%E0%AE%B0%E0%AE%BE%E0%AE%9C%E0%AF%86%E0%AE%95%E0%AF%8D%E0%AE%9A%E0%AE%A9%E0%AF%8D_(UTM)" title="யூனிவெர்சல் ட்ரான்ஸ்வெர்ஸ் மெர்கேட்டர் ப்ராஜெக்சன் (UTM) – Tamil" lang="ta" hreflang="ta" data-title="யூனிவெர்சல் ட்ரான்ஸ்வெர்ஸ் மெர்கேட்டர் ப்ராஜெக்சன் (UTM)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%9E%E0%B8%B4%E0%B8%81%E0%B8%B1%E0%B8%94%E0%B8%81%E0%B8%A3%E0%B8%B4%E0%B8%94%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%A2%E0%B8%B9%E0%B8%97%E0%B8%B5%E0%B9%80%E0%B8%AD%E0%B9%87%E0%B8%A1" title="ระบบพิกัดกริดแบบยูทีเอ็ม – Thai" lang="th" hreflang="th" data-title="ระบบพิกัดกริดแบบยูทีเอ็ม" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Evrensel_Enlem_Merkat%C3%B6r%C3%BC_koordinat_sistemi" title="Evrensel Enlem Merkatörü koordinat sistemi – Turkish" lang="tr" hreflang="tr" data-title="Evrensel Enlem Merkatörü 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style="padding-bottom:0.4em;"><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Globe_Atlantic.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/120px-Globe_Atlantic.svg.png" decoding="async" width="120" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/180px-Globe_Atlantic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Globe_Atlantic.svg/240px-Globe_Atlantic.svg.png 2x" data-file-width="717" data-file-height="721" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Fundamentals</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Geodesy" title="Geodesy">Geodesy</a></li> <li><a href="/wiki/Geodynamics" title="Geodynamics">Geodynamics</a></li> <li><a href="/wiki/Geomatics" title="Geomatics">Geomatics</a></li> <li><a href="/wiki/History_of_geodesy" title="History of geodesy">History</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Concepts</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Geographical_distance" title="Geographical distance">Geographical distance</a></li> <li><a href="/wiki/Geoid" title="Geoid">Geoid</a></li> <li><a href="/wiki/Figure_of_the_Earth" title="Figure of the Earth">Figure of the Earth</a> <small>(<a href="/wiki/Earth_radius" title="Earth radius">radius</a> and <a href="/wiki/Earth%27s_circumference" title="Earth's circumference">circumference</a>)</small></li> <li><a href="/wiki/Geodetic_coordinates" title="Geodetic coordinates">Geodetic coordinates</a></li> <li><a href="/wiki/Geodetic_datum" title="Geodetic datum">Geodetic datum</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Horizontal_position_representation" title="Horizontal position representation">Horizontal position representation</a></li> <li><span class="nowrap"><a href="/wiki/Latitude" title="Latitude">Latitude</a> / <a href="/wiki/Longitude" title="Longitude">Longitude</a></span></li> <li><a href="/wiki/Map_projection" title="Map projection">Map projection</a></li> <li><a href="/wiki/Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">Reference ellipsoid</a></li> <li><a href="/wiki/Satellite_geodesy" title="Satellite geodesy">Satellite geodesy</a></li> <li><a href="/wiki/Spatial_reference_system" title="Spatial reference system">Spatial reference system</a></li> <li><a href="/wiki/Spatial_relation" title="Spatial relation">Spatial relations</a></li> <li><a href="/wiki/Vertical_position" title="Vertical position">Vertical positions</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Technologies</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Satellite_navigation" title="Satellite navigation">Global Nav. Sat. Systems (GNSSs)</a></li> <li><a href="/wiki/Global_Positioning_System" title="Global Positioning System">Global Pos. System (GPS)</a></li> <li><a href="/wiki/GLONASS" title="GLONASS">GLONASS <span style="font-size:85%;">(Russia)</span></a></li> <li><a href="/wiki/BeiDou" title="BeiDou">BeiDou (BDS) <span style="font-size:85%;">(China)</span></a></li> <li><a href="/wiki/Galileo_(satellite_navigation)" title="Galileo (satellite navigation)">Galileo <span style="font-size:85%;">(Europe)</span></a></li> <li><a href="/wiki/Indian_Regional_Navigation_Satellite_System" title="Indian Regional Navigation Satellite System">NAVIC <span style="font-size:85%;">(India)</span></a></li> <li><a href="/wiki/Quasi-Zenith_Satellite_System" title="Quasi-Zenith Satellite System">Quasi-Zenith Sat. Sys. (QZSS) <span style="font-size:85%;">(Japan)</span></a></li> <li><a href="/wiki/Discrete_Global_Grid" class="mw-redirect" title="Discrete Global Grid">Discrete Global Grid and Geocoding</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed plainlist"><div class="sidebar-list-title" style="color: var(--color-base)">Standards (history)</div><div class="sidebar-list-content mw-collapsible-content hlist"><table class="nowrap" style="width:100%;border-collapse:collapse;border-spacing:0px 0px;border:none;line-height:1.2em;"><tbody><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Sea_Level_Datum_of_1929" class="mw-redirect" title="Sea Level Datum of 1929">NGVD 29</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Sea Level Datum 1929</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/OSGB36" class="mw-redirect" title="OSGB36">OSGB36</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Ordnance Survey Great Britain 1936</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/SK-42_reference_system" title="SK-42 reference system">SK-42</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Systema Koordinat 1942 goda</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/ED50" title="ED50">ED50</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> European Datum 1950</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/South_American_Datum#SAD69" title="South American Datum">SAD69</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> South American Datum 1969</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/GRS_80" class="mw-redirect" title="GRS 80">GRS 80</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Geodetic Reference System 1980</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/ISO_6709" title="ISO 6709">ISO 6709</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Geographic point coord. 1983</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/North_American_Datum#North_American_Datum_of_1983" title="North American Datum">NAD 83</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> North American Datum 1983</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/World_Geodetic_System" title="World Geodetic System">WGS 84</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> World Geodetic System 1984</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/North_American_Vertical_Datum_of_1988" title="North American Vertical Datum of 1988">NAVD 88</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> N. American Vertical Datum 1988</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/European_Terrestrial_Reference_System_1989" title="European Terrestrial Reference System 1989">ETRS89</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> European Terrestrial Ref. Sys. 1989</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Restrictions_on_geographic_data_in_China" title="Restrictions on geographic data in China">GCJ-02</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Chinese obfuscated datum 2002</td></tr><tr style="vertical-align:top"><td style="text-align:left;padding-left:0.5em;"> <a href="/wiki/Geo_URI_scheme" title="Geo URI scheme">Geo URI</a></td><td style="text-align:right;font-size:90%;padding-right:0.55em;"> Internet link to a point 2010</td></tr></tbody></table> <ul><li><a href="/wiki/International_Terrestrial_Reference_System" class="mw-redirect" title="International Terrestrial Reference System">International Terrestrial Reference System</a></li> <li><a href="/wiki/SRID" class="mw-redirect" title="SRID">Spatial Reference System Identifier (SRID)</a></li> <li><a class="mw-selflink selflink">Universal Transverse Mercator (UTM)</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Geodesy" title="Template:Geodesy"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Geodesy" title="Template talk:Geodesy"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geodesy" title="Special:EditPage/Template:Geodesy"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>The <b>Universal Transverse Mercator</b> (<b>UTM</b>) is a <a href="/wiki/Map_projection" title="Map projection">map projection</a> system for assigning <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">coordinates</a> to locations on the surface of the <a href="/wiki/Earth" title="Earth">Earth</a>. Like the traditional method of <a href="/wiki/Latitude" title="Latitude">latitude</a> and <a href="/wiki/Longitude" title="Longitude">longitude</a>, it is a <a href="/wiki/Horizontal_position_representation" title="Horizontal position representation">horizontal position representation</a>, which means it ignores <a href="/wiki/Altitude" title="Altitude">altitude</a> and treats the earth surface as a perfect <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a>. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the <i>x</i>, <i>y</i> coordinate in that plane. The projection from <a href="/wiki/Spheroid" title="Spheroid">spheroid</a> to a UTM zone is some <a href="/wiki/Parameter" title="Parameter">parameterization</a> of the <a href="/wiki/Transverse_Mercator" class="mw-redirect" title="Transverse Mercator">transverse Mercator</a> projection. The parameters vary by nation or region or mapping system. </p><p>Most zones in UTM span 6 degrees of <a href="/wiki/Longitude" title="Longitude">longitude</a>, and each has a designated central meridian. The scale factor at the central meridian is specified to be 0.9996 of true scale for most UTM systems in use.<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><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> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Universal_Transverse_Mercator_zones.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Universal_Transverse_Mercator_zones.svg/300px-Universal_Transverse_Mercator_zones.svg.png" decoding="async" width="300" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Universal_Transverse_Mercator_zones.svg/450px-Universal_Transverse_Mercator_zones.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Universal_Transverse_Mercator_zones.svg/600px-Universal_Transverse_Mercator_zones.svg.png 2x" data-file-width="512" data-file-height="267" /></a><figcaption>UTM zones on an equirectangular world map with irregular zones in red and New York City's zone highlighted</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/National_Oceanic_and_Atmospheric_Administration" title="National Oceanic and Atmospheric Administration">National Oceanic and Atmospheric Administration</a> (NOAA) website states that the system was developed by the <a href="/wiki/United_States_Army_Corps_of_Engineers" title="United States Army Corps of Engineers">United States Army Corps of Engineers</a>, starting in the early 1940s.<sup id="cite_ref-Dracup_3-0" class="reference"><a href="#cite_note-Dracup-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, a series of aerial photos found in the Bundesarchiv-Militärarchiv (the military section of the <a href="/wiki/German_Federal_Archives" title="German Federal Archives">German Federal Archives</a>) apparently dating from 1943–1944 bear the inscription UTMREF followed by grid letters and digits, and projected according to the transverse Mercator,<sup id="cite_ref-Buchroithner_4-0" class="reference"><a href="#cite_note-Buchroithner-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> a finding that would indicate that something called the UTM Reference system was developed in the 1942–43 time frame by the <a href="/wiki/Wehrmacht" title="Wehrmacht">Wehrmacht</a>. It was probably carried out by the Abteilung für Luftbildwesen (Department for Aerial Photography). From 1947 onward the US Army employed a very similar system, but with the now-standard 0.9996 scale factor at the central meridian as opposed to the German 1.0.<sup id="cite_ref-Buchroithner_4-1" class="reference"><a href="#cite_note-Buchroithner-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For areas within the <a href="/wiki/Contiguous_United_States" title="Contiguous United States">contiguous United States</a> the <a href="/wiki/Alexander_Ross_Clarke" title="Alexander Ross Clarke">Clarke</a> <a href="/wiki/Earth_ellipsoid#Historical_Earth_ellipsoids" title="Earth ellipsoid">Ellipsoid</a> of 1866<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> was used. For the remaining areas of Earth, including <a href="/wiki/Hawaii" title="Hawaii">Hawaii</a>, the <a href="/wiki/International_Ellipsoid" class="mw-redirect" title="International Ellipsoid">International Ellipsoid</a><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> was used. The <a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a> WGS84 ellipsoid is now generally used to model the Earth in the UTM coordinate system, which means current UTM northing at a given point can differ up to 200 meters from the old. For different geographic regions, other <a href="/wiki/Geodetic_datum" title="Geodetic datum">datum</a> systems can be used. </p><p>Prior to the development of the Universal Transverse Mercator coordinate system, several European nations demonstrated the utility of grid-based conformal maps by mapping their territory during the <a href="/wiki/Interwar_period" title="Interwar period">interwar period</a>. Calculating the distance between two points on these maps could be performed more easily in the field (using the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>) than was possible using the trigonometric formulas required under the graticule-based system of <a href="/wiki/Geographic_coordinate_system" title="Geographic coordinate system">latitude and longitude</a>. In the post-war years, these concepts were extended into the Universal Transverse Mercator/<a href="/wiki/Universal_Polar_Stereographic" class="mw-redirect" title="Universal Polar Stereographic">Universal Polar Stereographic</a> (UTM/UPS) coordinate system, which is a global (or universal) system of grid-based maps. </p><p>The transverse Mercator projection is a variant of the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>, which was originally developed by the <a href="/wiki/Flemish_people" title="Flemish people">Flemish</a> geographer and cartographer <a href="/wiki/Gerardus_Mercator" title="Gerardus Mercator">Gerardus Mercator</a>, in 1570. This projection is <a href="/wiki/Conformal_map_projection#Conformal" title="Conformal map projection">conformal</a>, which means it preserves angles and therefore shapes across small regions. However, it distorts distance and area. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=2" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="UTM_zone">UTM zone</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=3" title="Edit section: UTM zone"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Utm-zones-USA.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Utm-zones-USA.svg/220px-Utm-zones-USA.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Utm-zones-USA.svg/330px-Utm-zones-USA.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Utm-zones-USA.svg/440px-Utm-zones-USA.svg.png 2x" data-file-width="720" data-file-height="540" /></a><figcaption> Simplified view of <a href="/wiki/Contiguous_United_States" title="Contiguous United States">contiguous US</a> UTM zones, projected with Lambert conformal conic.</figcaption></figure> <p>The UTM system divides the Earth into 60 zones, each 6° of longitude in width. Zone 1 covers longitude 180° to 174° W; zone numbering increases eastward to zone 60, which covers longitude 174°E to 180°. The polar regions south of <a href="/wiki/80th_parallel_south" title="80th parallel south">80°S</a> and north of <a href="/wiki/84th_parallel_north" title="84th parallel north">84°N</a> are excluded, and instead covered by the <a href="/wiki/Universal_polar_stereographic_coordinate_system" title="Universal polar stereographic coordinate system">universal polar stereographic (UPS)</a> coordinate system. </p><p>Each of the 60 zones uses a <a href="/wiki/Transverse_Mercator" class="mw-redirect" title="Transverse Mercator">transverse Mercator</a> projection that can map a region of large north-south extent with low distortion. By using narrow zones of 6° of longitude (up to 668 km) in width, and reducing the <a href="/wiki/Scale_(map)" title="Scale (map)">scale</a> factor along the central <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridian</a> to 0.9996 (a reduction of 1:2500), the amount of distortion is held below 1 part in 1,000 inside each zone. Distortion of scale increases to 1.0010 at the zone boundaries along the <a href="/wiki/Equator" title="Equator">equator</a>. </p><p>In each zone the scale factor of the central meridian reduces the diameter of the transverse cylinder to produce a secant projection with two <a href="/wiki/Standard_line" class="mw-redirect" title="Standard line">standard lines</a>, or lines of true scale, about 180 km on each side of, and about parallel to, the central meridian (Arc cos 0.9996 = 1.62° at the Equator). The scale is less than 1 inside the standard lines and greater than 1 outside them, but the overall distortion is minimized. </p> <div class="mw-heading mw-heading3"><h3 id="Exceptions">Exceptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=4" title="Edit section: Exceptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The UTM zones are uniform across the globe, except in two areas. On the southwest coast of <a href="/wiki/Norway" title="Norway">Norway</a>, zone 32 is extended 3° further west, and zone 31 is correspondingly shrunk to cover only open water. Also, in the region around <a href="/wiki/Svalbard" title="Svalbard">Svalbard</a>, the zones 32, 34 and 36 are not used, while zones 31 (9° wide), 33 (12° wide), 35 (12° wide), and 37 (9° wide) are extended to cover the gaps. </p> <div class="mw-heading mw-heading3"><h3 id="Overlapping_grids">Overlapping grids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=5" title="Edit section: Overlapping grids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Modified_UTM_Zones.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Modified_UTM_Zones.png/220px-Modified_UTM_Zones.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Modified_UTM_Zones.png/330px-Modified_UTM_Zones.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Modified_UTM_Zones.png/440px-Modified_UTM_Zones.png 2x" data-file-width="1557" data-file-height="1558" /></a><figcaption>Universal Transverse Mercator (UTM) Grid Zones 31N thru 37N differ from the standard 6° wide by 84° zone for the northern hemisphere, in part to accommodate the western part of the Kingdom of Norway. For more on its history, see Clifford J. Mugnier's article on Grids & Datums of The Kingdom of Norway that appeared in the October 1999 issue of PE&RS <a rel="nofollow" class="external free" href="http://www.asprs.org/a/resources/grids/10-99-norway.pdf">http://www.asprs.org/a/resources/grids/10-99-norway.pdf</a></figcaption></figure> <p>Distortion of scale increases in each UTM zone as the boundaries between the UTM zones are approached. However, it is often convenient or necessary to measure a series of locations on a single grid when some are located in two adjacent zones. Around the boundaries of large scale maps (1:100,000 or larger) coordinates for both adjoining UTM zones are usually printed within a minimum distance of 40 km on either side of a zone boundary. Ideally, the coordinates of each position should be measured on the grid for the zone in which they are located, but because the scale factor is still relatively small near zone boundaries, it is possible to overlap measurements into an adjoining zone for some distance when necessary. </p> <div class="mw-heading mw-heading3"><h3 id="Latitude_bands">Latitude bands</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=6" title="Edit section: Latitude bands"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Latitude bands are not a part of UTM,<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> but rather a part of the <a href="/wiki/Military_grid_reference_system" class="mw-redirect" title="Military grid reference system">military grid reference system</a> (MGRS).<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> They are however sometimes included in UTM notation. Including latitude bands in UTM notation can lead to ambiguous coordinates—as the letter "S" either refers to the southern hemisphere or a latitude band in the northern hemisphere—and should therefore be avoided. </p> <div class="mw-heading mw-heading2"><h2 id="Locating_a_position_using_UTM_coordinates">Locating a position using UTM coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=7" title="Edit section: Locating a position using UTM coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A position on the Earth is given by the UTM zone number and hemisphere designator and the <a href="/wiki/Easting_and_northing" class="mw-redirect" title="Easting and northing">easting and northing</a> planar coordinate pair in that zone. </p><p>The <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">point of origin</a> of each UTM zone is the intersection of the equator and the zone's central meridian. To avoid dealing with negative numbers, a false Easting of <span class="nowrap"><span data-sort-value="2994500000000000000♠"></span>−500<span style="margin-left:.25em;">000</span></span> meters is added to the central meridian. Thus a point that has an easting of <span class="nowrap"><span data-sort-value="7005400000000000000♠"></span>400<span style="margin-left:.25em;">000</span></span> meters is about 100 km west of the central meridian. For most such points, the true distance would be slightly more than 100 km as measured on the surface of the Earth because of the distortion of the projection. UTM eastings range from about <span class="nowrap"><span data-sort-value="7005166000000000000♠"></span>166<span style="margin-left:.25em;">000</span></span> meters to <span class="nowrap"><span data-sort-value="7005834000000000000♠"></span>834<span style="margin-left:.25em;">000</span></span> meters at the equator. </p><p>In the northern hemisphere positions are measured northward from zero at the equator. The maximum "northing" value is about <span class="nowrap"><span data-sort-value="7006930000000000000♠"></span>9<span style="margin-left:.25em;">300</span><span style="margin-left:.25em;">000</span></span> meters at latitude 84 degrees North, the north end of the UTM zones. The southern hemisphere's northing at the equator is set at <span class="nowrap"><span data-sort-value="7007100000000000000♠"></span>10<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span></span> meters. Northings decrease southward from these <span class="nowrap"><span data-sort-value="7007100000000000000♠"></span>10<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">000</span></span> meters to about <span class="nowrap"><span data-sort-value="7006110000000000000♠"></span>1<span style="margin-left:.25em;">100</span><span style="margin-left:.25em;">000</span></span> meters at 80 degrees South, the south end of the UTM zones. Therefore, no point has a negative northing value. </p><p>For example, the <a href="/wiki/CN_Tower" title="CN Tower">CN Tower</a> is at <span class="geo-inline"><style data-mw-deduplicate="TemplateStyles:r1156832818">.mw-parser-output .geo-default,.mw-parser-output .geo-dms,.mw-parser-output .geo-dec{display:inline}.mw-parser-output .geo-nondefault,.mw-parser-output .geo-multi-punct,.mw-parser-output .geo-inline-hidden{display:none}.mw-parser-output .longitude,.mw-parser-output .latitude{white-space:nowrap}</style><span class="plainlinks nourlexpansion"><a class="external text" href="https://geohack.toolforge.org/geohack.php?pagename=Universal_Transverse_Mercator_coordinate_system&params=43_38_33.24_N_79_23_13.7_W_&title=CN+Tower"><span class="geo-default"><span class="geo-dms" title="Maps, aerial photos, and other data for this location"><span class="latitude">43°38′33.24″N</span> <span class="longitude">79°23′13.7″W</span></span></span><span class="geo-multi-punct"> / </span><span class="geo-nondefault"><span class="vcard"><span class="geo-dec" title="Maps, aerial photos, and other data for this location">43.6425667°N 79.387139°W</span><span style="display:none"> / <span class="geo">43.6425667; -79.387139</span></span><span style="display:none"> (<span class="fn org">CN Tower</span>)</span></span></span></a></span></span>, which is in UTM zone 17, and the grid position is <span class="nowrap"><span data-sort-value="7005630084000000000♠"></span>630<span style="margin-left:.25em;">084</span> m</span> east, <span class="nowrap"><span data-sort-value="7006483343800000000♠"></span>4<span style="margin-left:.25em;">833</span><span style="margin-left:.25em;">438</span> m</span> north. Two points in Zone 17 have these coordinates, one in the northern hemisphere and one in the south; the non-ambiguous format is to specify the full zone and hemisphere designator, that is, "17N 630084 4833438". </p> <div class="mw-heading mw-heading3"><h3 id="Simplified_formulae">Simplified formulae</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=8" title="Edit section: Simplified formulae"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These formulae are truncated version of <a href="/w/index.php?title=Transverse_Mercator:_flattening_series&action=edit&redlink=1" class="new" title="Transverse Mercator: flattening series (page does not exist)">Transverse Mercator: flattening series</a>, which were originally derived by <a href="/wiki/Johann_Heinrich_Louis_Kr%C3%BCger" title="Johann Heinrich Louis Krüger">Johann Heinrich Louis Krüger</a> in 1912.<sup id="cite_ref-kruger_9-0" class="reference"><a href="#cite_note-kruger-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> They are accurate to around a <a href="/wiki/Millimeter" class="mw-redirect" title="Millimeter">millimeter</a> within <span class="nowrap"><span data-sort-value="7006300000000000000♠"></span>3000 km</span> of the central meridian.<sup id="cite_ref-karney_10-0" class="reference"><a href="#cite_note-karney-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Concise commentaries for their derivation have also been given.<sup id="cite_ref-kawase2012_11-0" class="reference"><a href="#cite_note-kawase2012-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kawase2011_12-0" class="reference"><a href="#cite_note-kawase2011-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/WGS_84" class="mw-redirect" title="WGS 84">WGS 84</a> <a href="/wiki/Spatial_reference_system" title="Spatial reference system">spatial reference system</a> describes Earth as an <a href="/wiki/Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate spheroid</a> along north-south axis with an <a href="/wiki/Equatorial_radius" class="mw-redirect" title="Equatorial radius">equatorial radius</a> of <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=6378.137}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>6378.137</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=6378.137}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/474ba27784128a19328eb5c3c526c82f385326a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.112ex; height:2.176ex;" alt="{\displaystyle a=6378.137}"></span> km and an inverse <a href="/wiki/Flattening" title="Flattening">flattening</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f=298.257\,223\,563}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> <mo>=</mo> <mn>298.257</mn> <mspace width="thinmathspace" /> <mn>223</mn> <mspace width="thinmathspace" /> <mn>563</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f=298.257\,223\,563}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b862d95e1612dea59122ed9876e5cd9347e5608c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.072ex; height:2.843ex;" alt="{\displaystyle 1/f=298.257\,223\,563}"></span>. Let's take a point of latitude <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 \,\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9883b5284ba056e46161bc398e97c766e3fefed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.907ex; height:2.176ex;" alt="{\displaystyle \,\varphi }"></span> and of longitude <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"> <mspace width="thinmathspace" /> <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/a4fb547bf232593dc98e480402a274f1746d8a4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.742ex; height:2.176ex;" alt="{\displaystyle \,\lambda }"></span> and compute its UTM coordinates as well as <a href="/wiki/Transverse_Mercator_projection#Point_scale" title="Transverse Mercator projection">point scale factor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c7c89fc898ac42f121705843458e24c50d5ff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:1.598ex; height:2.176ex;" alt="{\displaystyle k\,\!}"></span> and <a href="/wiki/Transverse_Mercator_projection#Convergence" title="Transverse Mercator projection">meridian convergence</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 \gamma \,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma \,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b0952c818e8f7b4bf1f3fd3034baa6fe397910" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:1.649ex; height:2.176ex;" alt="{\displaystyle \gamma \,\!}"></span> using a reference meridian of longitude <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 _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa5ad1eb6cdaf3d8dfd77991ee9ce7bdf169184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \lambda _{0}}"></span>. By convention, in the <a href="/wiki/Northern_hemisphere" class="mw-redirect" title="Northern hemisphere">northern hemisphere</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 N_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56156e8c58ddc8fe8ecc77197f6c08719084d0f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.181ex; height:2.509ex;" alt="{\displaystyle N_{0}=0}"></span> km and in the <a href="/wiki/Southern_hemisphere" class="mw-redirect" title="Southern hemisphere">southern hemisphere</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 N_{0}=10000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>10000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}=10000}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b6754f119ef2047679bd620ca98e286a2704452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.831ex; height:2.509ex;" alt="{\displaystyle N_{0}=10000}"></span> km. By convention also <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{0}=0.9996}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.9996</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{0}=0.9996}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d72c7ad5aabad44da8aa17125631b4eadf1341dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.823ex; height:2.509ex;" alt="{\displaystyle k_{0}=0.9996}"></span> and <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_{0}=500}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>500</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}=500}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/672f015b25de8f36d512e4b6b059562ad05c346d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.355ex; height:2.509ex;" alt="{\displaystyle E_{0}=500}"></span> km. </p><p>In the following formulas, the distances are in <a href="/wiki/Kilometer" class="mw-redirect" title="Kilometer">kilometers</a>. First, here are some preliminary values: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}n&={\frac {f}{2-f}},&A&={\frac {a}{1+n}}\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}+\cdots \right),\\{}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>f</mi> <mrow> <mn>2</mn> <mo>−</mo> <mi>f</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mi>A</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mn>64</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}n&={\frac {f}{2-f}},&A&={\frac {a}{1+n}}\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}+\cdots \right),\\{}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/011a02773679a947178d6786cafd6e6a38f535f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:51.549ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}n&={\frac {f}{2-f}},&A&={\frac {a}{1+n}}\left(1+{\frac {n^{2}}{4}}+{\frac {n^{4}}{64}}+\cdots \right),\\{}\end{aligned}}}"></span></dd> <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}\alpha _{1}&={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {5}{16}}n^{3},&\alpha _{2}&={\frac {13}{48}}n^{2}-{\frac {3}{5}}n^{3},&\alpha _{3}&={\frac {61}{240}}n^{3},\\[12pt]\beta _{1}&={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {37}{96}}n^{3},&\beta _{2}&={\frac {1}{48}}n^{2}+{\frac {1}{15}}n^{3},&\beta _{3}&={\frac {17}{480}}n^{3},\\[12pt]\delta _{1}&=2n-{\frac {2}{3}}n^{2}-2n^{3},&\delta _{2}&={\frac {7}{3}}n^{2}-{\frac {8}{5}}n^{3},&\delta _{3}&={\frac {56}{15}}n^{3}.\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="1.5em 1.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>16</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>13</mn> <mn>48</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>61</mn> <mn>240</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>37</mn> <mn>96</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>48</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>15</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>17</mn> <mn>480</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>5</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>56</mn> <mn>15</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha _{1}&={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {5}{16}}n^{3},&\alpha _{2}&={\frac {13}{48}}n^{2}-{\frac {3}{5}}n^{3},&\alpha _{3}&={\frac {61}{240}}n^{3},\\[12pt]\beta _{1}&={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {37}{96}}n^{3},&\beta _{2}&={\frac {1}{48}}n^{2}+{\frac {1}{15}}n^{3},&\beta _{3}&={\frac {17}{480}}n^{3},\\[12pt]\delta _{1}&=2n-{\frac {2}{3}}n^{2}-2n^{3},&\delta _{2}&={\frac {7}{3}}n^{2}-{\frac {8}{5}}n^{3},&\delta _{3}&={\frac {56}{15}}n^{3}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98239306388c8687badf08861cb0199f6e390ae7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.104ex; margin-bottom: -0.234ex; width:68.868ex; height:21.843ex;" alt="{\displaystyle {\begin{aligned}\alpha _{1}&={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {5}{16}}n^{3},&\alpha _{2}&={\frac {13}{48}}n^{2}-{\frac {3}{5}}n^{3},&\alpha _{3}&={\frac {61}{240}}n^{3},\\[12pt]\beta _{1}&={\frac {1}{2}}n-{\frac {2}{3}}n^{2}+{\frac {37}{96}}n^{3},&\beta _{2}&={\frac {1}{48}}n^{2}+{\frac {1}{15}}n^{3},&\beta _{3}&={\frac {17}{480}}n^{3},\\[12pt]\delta _{1}&=2n-{\frac {2}{3}}n^{2}-2n^{3},&\delta _{2}&={\frac {7}{3}}n^{2}-{\frac {8}{5}}n^{3},&\delta _{3}&={\frac {56}{15}}n^{3}.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="From_latitude,_longitude_(φ,_λ)_to_UTM_coordinates_(E,_N)"><span id="From_latitude.2C_longitude_.28.CF.86.2C_.CE.BB.29_to_UTM_coordinates_.28E.2C_N.29"></span>From latitude, longitude (<i>φ</i>, <i>λ</i>) to UTM coordinates (E, N)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=9" title="Edit section: From latitude, longitude (φ, λ) to UTM coordinates (E, N)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>First we compute some intermediate values: </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 t=\sinh \left(\tanh ^{-1}\left(\sin \varphi \right)-{\frac {2{\sqrt {n}}}{1+n}}\tanh ^{-1}\left({\frac {2{\sqrt {n}}}{1+n}}\sin \varphi \right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>n</mi> </msqrt> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=\sinh \left(\tanh ^{-1}\left(\sin \varphi \right)-{\frac {2{\sqrt {n}}}{1+n}}\tanh ^{-1}\left({\frac {2{\sqrt {n}}}{1+n}}\sin \varphi \right)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1949d6f5f26b7ba11105420b318699aee6157287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.962ex; height:6.509ex;" alt="{\displaystyle t=\sinh \left(\tanh ^{-1}\left(\sin \varphi \right)-{\frac {2{\sqrt {n}}}{1+n}}\tanh ^{-1}\left({\frac {2{\sqrt {n}}}{1+n}}\sin \varphi \right)\right),}"></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 \xi '=\tan ^{-1}\left({\frac {t}{\cos(\lambda -\lambda _{0})}}\right),\,\,\,\eta '=\tanh ^{-1}\left({\frac {\sin(\lambda -\lambda _{0})}{\sqrt {1+t^{2}}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi '=\tan ^{-1}\left({\frac {t}{\cos(\lambda -\lambda _{0})}}\right),\,\,\,\eta '=\tanh ^{-1}\left({\frac {\sin(\lambda -\lambda _{0})}{\sqrt {1+t^{2}}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e747cc8bb9a9b52ba1d28745d4976bbde3b79fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.863ex; height:7.509ex;" alt="{\displaystyle \xi '=\tan ^{-1}\left({\frac {t}{\cos(\lambda -\lambda _{0})}}\right),\,\,\,\eta '=\tanh ^{-1}\left({\frac {\sin(\lambda -\lambda _{0})}{\sqrt {1+t^{2}}}}\right),}"></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 \sigma =1+\sum _{j=1}^{3}2j\alpha _{j}\cos(2j\xi ')\cosh(2j\eta '),\,\,\,\tau =\sum _{j=1}^{3}2j\alpha _{j}\sin(2j\xi ')\sinh(2j\eta ').}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mn>2</mn> <mi>j</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>τ</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mn>2</mn> <mi>j</mi> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =1+\sum _{j=1}^{3}2j\alpha _{j}\cos(2j\xi ')\cosh(2j\eta '),\,\,\,\tau =\sum _{j=1}^{3}2j\alpha _{j}\sin(2j\xi ')\sinh(2j\eta ').}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cadf51d56ecdecfa9a92c1fec870d3d2bed2a518" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:71.033ex; height:7.509ex;" alt="{\displaystyle \sigma =1+\sum _{j=1}^{3}2j\alpha _{j}\cos(2j\xi ')\cosh(2j\eta '),\,\,\,\tau =\sum _{j=1}^{3}2j\alpha _{j}\sin(2j\xi ')\sinh(2j\eta ').}"></span></dd></dl> <p>The final formulae are: </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 E=E_{0}+k_{0}A\left(\eta '+\sum _{j=1}^{3}\alpha _{j}\cos(2j\xi ')\sinh(2j\eta ')\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=E_{0}+k_{0}A\left(\eta '+\sum _{j=1}^{3}\alpha _{j}\cos(2j\xi ')\sinh(2j\eta ')\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e915eb806215986e71aaecdd214acbd863aa163" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:49.903ex; height:7.676ex;" alt="{\displaystyle E=E_{0}+k_{0}A\left(\eta '+\sum _{j=1}^{3}\alpha _{j}\cos(2j\xi ')\sinh(2j\eta ')\right),}"></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 N=N_{0}+k_{0}A\left(\xi '+\sum _{j=1}^{3}\alpha _{j}\sin(2j\xi ')\cosh(2j\eta ')\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>j</mi> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=N_{0}+k_{0}A\left(\xi '+\sum _{j=1}^{3}\alpha _{j}\sin(2j\xi ')\cosh(2j\eta ')\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea10f5cc54abcfcbe2354e187377c79baa168fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:50.202ex; height:7.676ex;" alt="{\displaystyle N=N_{0}+k_{0}A\left(\xi '+\sum _{j=1}^{3}\alpha _{j}\sin(2j\xi ')\cosh(2j\eta ')\right),}"></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 k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\sigma ^{2}+\tau ^{2}}{t^{2}+\cos ^{2}(\lambda -\lambda _{0})}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\sigma ^{2}+\tau ^{2}}{t^{2}+\cos ^{2}(\lambda -\lambda _{0})}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8fde484b4044bd21889a35e4d9e3b28e7b7b953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.905ex; height:8.176ex;" alt="{\displaystyle k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\sigma ^{2}+\tau ^{2}}{t^{2}+\cos ^{2}(\lambda -\lambda _{0})}}}},}"></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 \gamma =\tan ^{-1}\left({\frac {\tau {\sqrt {1+t^{2}}}+\sigma t\tan(\lambda -\lambda _{0})}{\sigma {\sqrt {1+t^{2}}}-\tau t\tan(\lambda -\lambda _{0})}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>σ</mi> <mi>t</mi> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>τ</mi> <mi>t</mi> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\tan ^{-1}\left({\frac {\tau {\sqrt {1+t^{2}}}+\sigma t\tan(\lambda -\lambda _{0})}{\sigma {\sqrt {1+t^{2}}}-\tau t\tan(\lambda -\lambda _{0})}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d28662595e2664ddba3a02e770a590f9145b61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.811ex; height:7.509ex;" alt="{\displaystyle \gamma =\tan ^{-1}\left({\frac {\tau {\sqrt {1+t^{2}}}+\sigma t\tan(\lambda -\lambda _{0})}{\sigma {\sqrt {1+t^{2}}}-\tau t\tan(\lambda -\lambda _{0})}}\right).}"></span></dd></dl> <p>where <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> is Easting, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is Northing, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the Scale Factor, and <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 }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the Grid Convergence. </p> <div class="mw-heading mw-heading4"><h4 id="From_UTM_coordinates_(E,_N,_Zone,_Hemi)_to_latitude,_longitude_(φ,_λ)"><span id="From_UTM_coordinates_.28E.2C_N.2C_Zone.2C_Hemi.29_to_latitude.2C_longitude_.28.CF.86.2C_.CE.BB.29"></span>From UTM coordinates (E, N, Zone, Hemi) to latitude, longitude (<i>φ</i>, <i>λ</i>)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=10" title="Edit section: From UTM coordinates (E, N, Zone, Hemi) to latitude, longitude (φ, λ)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Note: Hemi = +1 for Northern, Hemi = −1 for Southern </p><p>First let's compute some intermediate values: </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 \xi ={\frac {N-N_{0}}{k_{0}A}},\,\,\,\eta ={\frac {E-E_{0}}{k_{0}A}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>η</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi ={\frac {N-N_{0}}{k_{0}A}},\,\,\,\eta ={\frac {E-E_{0}}{k_{0}A}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc75d1c6d6f79976bba6e7a90fbc788da772373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.121ex; height:5.843ex;" alt="{\displaystyle \xi ={\frac {N-N_{0}}{k_{0}A}},\,\,\,\eta ={\frac {E-E_{0}}{k_{0}A}},}"></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 \xi '=\xi -\sum _{j=1}^{3}\beta _{j}\sin \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\eta '=\eta -\sum _{j=1}^{3}\beta _{j}\cos \left(2j\xi \right)\sinh \left(2j\eta \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>ξ</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mi>η</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>η</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi '=\xi -\sum _{j=1}^{3}\beta _{j}\sin \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\eta '=\eta -\sum _{j=1}^{3}\beta _{j}\cos \left(2j\xi \right)\sinh \left(2j\eta \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d51bf7661a2569bbc58a289f6d0f24a96aead9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:68.616ex; height:7.509ex;" alt="{\displaystyle \xi '=\xi -\sum _{j=1}^{3}\beta _{j}\sin \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\eta '=\eta -\sum _{j=1}^{3}\beta _{j}\cos \left(2j\xi \right)\sinh \left(2j\eta \right),}"></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 \sigma '=1-\sum _{j=1}^{3}2j\beta _{j}\cos \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\tau '=\sum _{j=1}^{3}2j\beta _{j}\sin \left(2j\xi \right)\sinh \left(2j\eta \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mo>′</mo> </msup> <mo>=</mo> <mn>1</mn> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mn>2</mn> <mi>j</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mi>cosh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <msup> <mi>τ</mi> <mo>′</mo> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mn>2</mn> <mi>j</mi> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>ξ</mi> </mrow> <mo>)</mo> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma '=1-\sum _{j=1}^{3}2j\beta _{j}\cos \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\tau '=\sum _{j=1}^{3}2j\beta _{j}\sin \left(2j\xi \right)\sinh \left(2j\eta \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44192551487f4a06176ea39936e0119bd1478cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:69.361ex; height:7.509ex;" alt="{\displaystyle \sigma '=1-\sum _{j=1}^{3}2j\beta _{j}\cos \left(2j\xi \right)\cosh \left(2j\eta \right),\,\,\,\tau '=\sum _{j=1}^{3}2j\beta _{j}\sin \left(2j\xi \right)\sinh \left(2j\eta \right),}"></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 \chi =\sin ^{-1}\left({\frac {\sin \xi '}{\cosh \eta '}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>χ</mi> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>ξ</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi>cosh</mi> <mo>⁡</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi =\sin ^{-1}\left({\frac {\sin \xi '}{\cosh \eta '}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73b471d9c93ce001700633826d68356e52e1b5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.295ex; height:6.176ex;" alt="{\displaystyle \chi =\sin ^{-1}\left({\frac {\sin \xi '}{\cosh \eta '}}\right).}"></span></dd></dl> <p>The final formulae are: </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 \varphi =\chi +\sum _{j=1}^{3}\delta _{j}\sin \left(2j\chi \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>χ</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>j</mi> <mi>χ</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\chi +\sum _{j=1}^{3}\delta _{j}\sin \left(2j\chi \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7210c6b096b9e505abc80525a350aefbfbb73a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.873ex; height:7.509ex;" alt="{\displaystyle \varphi =\chi +\sum _{j=1}^{3}\delta _{j}\sin \left(2j\chi \right),}"></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 \lambda _{0}=\mathrm {Z} \mathrm {o} \mathrm {n} \mathrm {e} \times 6^{\circ }-183^{\circ }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mo>×</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mo>−</mo> <msup> <mn>183</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{0}=\mathrm {Z} \mathrm {o} \mathrm {n} \mathrm {e} \times 6^{\circ }-183^{\circ }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3dbe492d8720be691f84c36531957eaa6c2177d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.242ex; height:2.676ex;" alt="{\displaystyle \lambda _{0}=\mathrm {Z} \mathrm {o} \mathrm {n} \mathrm {e} \times 6^{\circ }-183^{\circ }\,}"></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 \lambda =\lambda _{0}+\tan ^{-1}\left({\frac {\sinh \eta '}{\cos \xi '}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sinh</mi> <mo>⁡</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> </mrow> <mrow> <mi>cos</mi> <mo>⁡</mo> <msup> <mi>ξ</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =\lambda _{0}+\tan ^{-1}\left({\frac {\sinh \eta '}{\cos \xi '}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5ad51d2dba368d721e8716919866b1a7045019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.694ex; height:6.176ex;" alt="{\displaystyle \lambda =\lambda _{0}+\tan ^{-1}\left({\frac {\sinh \eta '}{\cos \xi '}}\right),}"></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 k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\cos ^{2}\xi '+\sinh ^{2}\eta '}{\sigma '^{2}+\tau '^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>A</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>n</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> </mrow> <mrow> <msup> <mi>σ</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <mo>+</mo> <msup> <mi>τ</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\cos ^{2}\xi '+\sinh ^{2}\eta '}{\sigma '^{2}+\tau '^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef250245d2e611be40acb913ca43d8a599a16401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.15ex; height:8.176ex;" alt="{\displaystyle k={\frac {k_{0}A}{a}}{\sqrt {\left\{1+\left({\frac {1-n}{1+n}}\tan \varphi \right)^{2}\right\}{\frac {\cos ^{2}\xi '+\sinh ^{2}\eta '}{\sigma '^{2}+\tau '^{2}}}}},}"></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 \gamma =\mathrm {H} \mathrm {e} \mathrm {m} \mathrm {i} \times \tan ^{-1}\left({\frac {\tau '+\sigma '\tan \xi '\tanh \eta '}{\sigma '-\tau '\tan \xi '\tanh \eta '}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> </mrow> <mo>×</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>τ</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>σ</mi> <mo>′</mo> </msup> <mi>tan</mi> <mo>⁡</mo> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mi>tanh</mi> <mo>⁡</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> </mrow> <mrow> <msup> <mi>σ</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>τ</mi> <mo>′</mo> </msup> <mi>tan</mi> <mo>⁡</mo> <msup> <mi>ξ</mi> <mo>′</mo> </msup> <mi>tanh</mi> <mo>⁡</mo> <msup> <mi>η</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\mathrm {H} \mathrm {e} \mathrm {m} \mathrm {i} \times \tan ^{-1}\left({\frac {\tau '+\sigma '\tan \xi '\tanh \eta '}{\sigma '-\tau '\tan \xi '\tanh \eta '}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2f9b3fb5308907318b077913a4bede23205068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.091ex; height:6.176ex;" alt="{\displaystyle \gamma =\mathrm {H} \mathrm {e} \mathrm {m} \mathrm {i} \times \tan ^{-1}\left({\frac {\tau '+\sigma '\tan \xi '\tanh \eta '}{\sigma '-\tau '\tan \xi '\tanh \eta '}}\right).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/European_Terrestrial_Reference_System_1989" title="European Terrestrial Reference System 1989">European Terrestrial Reference System 1989</a> (ETRS89)</li> <li><a href="/wiki/Military_grid_reference_system" class="mw-redirect" title="Military grid reference system">Military grid reference system</a>, a variant of UTM designed to simplify transfer of coordinates.</li> <li><a href="/wiki/Modified_transverse_Mercator" title="Modified transverse Mercator">Modified transverse Mercator</a>, a variation of UTM used in Canada with zones spaced 3° of longitude apart as opposed to UTM's 6°.</li> <li><a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator projection</a>, the map projection used by UTM.</li> <li><a href="/wiki/Universal_Polar_Stereographic_coordinate_system" class="mw-redirect" title="Universal Polar Stereographic coordinate system">Universal Polar Stereographic coordinate system</a>, used at the North and South poles.</li> <li><a href="/wiki/Open_Location_Code" title="Open Location Code">Open Location Code</a>, a hierarchical zoned system</li> <li><a href="/wiki/MapCode" class="mw-redirect" title="MapCode">MapCode</a>, a hierarchical zoned system</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-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.id-lock-limited a,.mw-parser-output .id-lock-registration.id-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.id-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}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.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:#085;margin-left:0.3em}.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}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://proj4.org/operations/projections/utm.html?highlight=utm">"Universal Transverse Mercator (UTM)"</a>. <i>PROJ coordinate transformation software library</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PROJ+coordinate+transformation+software+library&rft.atitle=Universal+Transverse+Mercator+%28UTM%29&rft_id=https%3A%2F%2Fproj4.org%2Foperations%2Fprojections%2Futm.html%3Fhighlight%3Dutm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged April 2023">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSnyder1987" class="citation book cs1">Snyder, John P. (1987). <i>Map projections: A working manual</i>. U.S. Government Printing Office.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Map+projections%3A+A+working+manual&rft.pub=U.S.+Government+Printing+Office&rft.date=1987&rft.aulast=Snyder&rft.aufirst=John+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-Dracup-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dracup_3-0">^</a></b></span> <span class="reference-text"><a href="/w/index.php?title=Josef_F._Dracup&action=edit&redlink=1" class="new" title="Josef F. Dracup (page does not exist)">Dracup, Josef F.</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20191218210651/https://www.history.noaa.gov/tools/surveytech.html">"NOAA History - Surveying and Mapping - Geodetic Surveys 1940 - 1990"</a>. <i>www.history.noaa.gov</i>. Archived from <a rel="nofollow" class="external text" href="http://www.history.noaa.gov/stories_tales/geod1.html">the original</a> on 2019-12-18.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.history.noaa.gov&rft.atitle=NOAA+History+-+Surveying+and+Mapping+-+Geodetic+Surveys+1940+-+1990&rft_id=http%3A%2F%2Fwww.history.noaa.gov%2Fstories_tales%2Fgeod1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-Buchroithner-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-Buchroithner_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Buchroithner_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Manfred_Buchroithner" title="Manfred Buchroithner">Buchroithner, Manfred F.</a>; Pfahlbusch, René. Geodetic grids in authoritative maps–new findings about the origin of the UTM Grid. Cartography and Geographic Information Science, 2016, <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080/15230406.2015.1128851">doi:10.1080/15230406.2015.1128851</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Equatorial radius 6,378,206.4 meters, polar radius 6,356,583.8 meters</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Equatorial radius 6,378,388 meters, reciprocal of the flattening 297 exactly</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://apps.dtic.mil/sti/tr/pdf/ADA266497.pdf">"THE UNIVERSAL GRIDS: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">National_Geospatial-Intelligence_Agency</a>. 1989-09-18. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20240213182153/https://apps.dtic.mil/sti/tr/pdf/ADA266497.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2024-02-13<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-02-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=THE+UNIVERSAL+GRIDS%3A+Universal+Transverse+Mercator+%28UTM%29+and+Universal+Polar+Stereographic+%28UPS%29&rft.pub=National_Geospatial-Intelligence_Agency&rft.date=1989-09-18&rft_id=https%3A%2F%2Fapps.dtic.mil%2Fsti%2Ftr%2Fpdf%2FADA266497.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://earth-info.nga.mil/GandG/coordsys/mmr201.pdf">"Military Map Reading 201"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/National_Geospatial-Intelligence_Agency" title="National Geospatial-Intelligence Agency">National Geospatial-Intelligence Agency</a>. 2002-05-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2009-06-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Military+Map+Reading+201&rft.pub=National+Geospatial-Intelligence+Agency&rft.date=2002-05-29&rft_id=http%3A%2F%2Fearth-info.nga.mil%2FGandG%2Fcoordsys%2Fmmr201.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-kruger-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-kruger_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrüger1912" class="citation journal cs1">Krüger, Louis (1912). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.2312/GFZ.b103-krueger28">"Konforme Abbildung des Erdellipsoids in der Ebene"</a>. Deutsches GeoForschungsZentrum GFZ. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2312%2FGFZ.b103-krueger28">10.2312/GFZ.b103-krueger28</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Konforme+Abbildung+des+Erdellipsoids+in+der+Ebene&rft.date=1912&rft_id=info%3Adoi%2F10.2312%2FGFZ.b103-krueger28&rft.aulast=Kr%C3%BCger&rft.aufirst=Louis&rft_id=https%3A%2F%2Fdx.doi.org%2F10.2312%2FGFZ.b103-krueger28&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Cite journal requires <code class="cs1-code">|journal=</code> (<a href="/wiki/Help:CS1_errors#missing_periodical" title="Help:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-karney-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-karney_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarney2011" class="citation journal cs1">Karney, Charles F. F. (2011). "Transverse Mercator with an accuracy of a few nanometers". <i>Journal of Geodesy</i>. <b>85</b> (8): 475–485. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1002.1417">1002.1417</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011JGeod..85..475K">2011JGeod..85..475K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00190-011-0445-3">10.1007/s00190-011-0445-3</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118619524">118619524</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geodesy&rft.atitle=Transverse+Mercator+with+an+accuracy+of+a+few+nanometers&rft.volume=85&rft.issue=8&rft.pages=475-485&rft.date=2011&rft_id=info%3Aarxiv%2F1002.1417&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118619524%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00190-011-0445-3&rft_id=info%3Abibcode%2F2011JGeod..85..475K&rft.aulast=Karney&rft.aufirst=Charles+F.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span></span> </li> <li id="cite_note-kawase2012-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-kawase2012_11-0">^</a></b></span> <span class="reference-text">Kawase, K. (2012): <a rel="nofollow" class="external text" href="http://www.gsi.go.jp/common/000065826.pdf">Concise Derivation of Extensive Coordinate Conversion Formulae in the Gauss-Krüger Projection</a>, Bulletin of the <a href="/wiki/Geospatial_Information_Authority_of_Japan" title="Geospatial Information Authority of Japan">Geospatial Information Authority of Japan</a>, <b>60</b>, pp 1–6</span> </li> <li id="cite_note-kawase2011-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-kawase2011_12-0">^</a></b></span> <span class="reference-text">Kawase, K. (2011): <a rel="nofollow" class="external text" href="http://www.gsi.go.jp/common/000062452.pdf">A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection</a>, Bulletin of the Geospatial Information Authority of Japan, <b>59</b>, 1–13</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://en.geofumadas.com/View-coordinates-utm-and-geogrficas-in-google-maps/">UTM coordinates in Google Maps.</a></li> <li><a rel="nofollow" class="external text" href="https://en.geofumadas.com/Understanding-projection-utm/">Understanding the UTM projection.</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Universal_Transverse_Mercator_coordinate_system&action=edit&section=14" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSnyder,_John_P.1987" class="citation book cs1">Snyder, John P. (1987). <a rel="nofollow" class="external text" href="https://pubs.er.usgs.gov/publication/pp1395"><i>Map Projections – A Working Manual. U.S. Geological Survey Professional Paper 1395</i></a>. United States Government Printing Office, Washington, D.C.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Map+Projections+%E2%80%93+A+Working+Manual.+U.S.+Geological+Survey+Professional+Paper+1395&rft.pub=United+States+Government+Printing+Office%2C+Washington%2C+D.C.&rft.date=1987&rft.au=Snyder%2C+John+P.&rft_id=https%3A%2F%2Fpubs.er.usgs.gov%2Fpublication%2Fpp1395&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniversal+Transverse+Mercator+coordinate+system" class="Z3988"></span></li></ul> <div style="clear:both;" class=""></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output 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codes</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Discrete_global_grid" title="Discrete global grid">Global</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/C-squares" title="C-squares">C-squares</a></li> <li><a href="/wiki/Geohash" title="Geohash">Geohash</a></li> <li><a href="/wiki/Geohash-36" title="Geohash-36">Geohash-36</a></li> <li><a href="/wiki/World_Geographic_Reference_System" title="World Geographic Reference System">GEOREF</a></li> <li><a href="/wiki/International_Map_of_the_World" title="International Map of the World">International Map of the World</a> indexing system</li> <li><a href="/wiki/Mapcode" title="Mapcode">Mapcode</a></li> <li><a href="/wiki/Marsden_square" title="Marsden square">Marsden square</a></li> <li><a href="/wiki/Military_Grid_Reference_System" title="Military Grid Reference System">Military Grid Reference System</a></li> <li><a href="/wiki/Natural_Area_Code" title="Natural Area Code">Natural Area Code</a></li> <li><a href="/wiki/Open_Location_Code" title="Open Location Code">Open Location Code</a></li> <li><a href="/wiki/QDGC" title="QDGC">QDGC</a></li> <li><a href="/wiki/UN/LOCODE" title="UN/LOCODE">UN/LOCODE</a></li> <li><a class="mw-selflink selflink">UTM</a></li> <li><a href="/wiki/What3words" title="What3words">what3words</a></li> <li><a href="/wiki/World_Meteorological_Organization_squares" title="World Meteorological Organization squares">WMO squares</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Regional</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ICES_Statistical_Rectangles" title="ICES Statistical Rectangles">ICES Statistical Rectangles</a> (north-east Atlantic region)</li> <li><a href="/wiki/Irish_grid_reference_system" title="Irish grid reference system">Irish grid reference system</a></li> <li><a href="/wiki/National_Topographic_System" title="National Topographic System">National Topographic System</a> (Canada)</li> <li><a href="/wiki/Ordnance_Survey_National_Grid" title="Ordnance Survey National Grid">Ordnance Survey National Grid</a> (UK)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Postal_code" title="Postal code">Postal codes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Postcodes_in_Australia" title="Postcodes in Australia">Australian post codes</a></li> <li><a href="/wiki/C%C3%B3digo_de_Endere%C3%A7amento_Postal" title="Código de Endereçamento Postal">CEP</a> (Brazil)</li> <li><a href="/wiki/Eircode" class="mw-redirect" title="Eircode">Eircodes</a> (Republic of Ireland)</li> <li><a href="/wiki/Postcodes_in_New_Zealand" title="Postcodes in New Zealand">New Zealand post codes</a></li> <li><a href="/wiki/Postal_Index_Number" title="Postal Index Number">Postal Index Number</a> (India)</li> <li><a href="/wiki/UK_post_codes" class="mw-redirect" title="UK post codes">United Kingdom post codes</a></li> <li><a href="/wiki/ZIP_Code" title="ZIP Code">ZIP Code</a> (United States)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Telephony" title="Telephony">Telephony</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_ITU_letter_codes" title="List of ITU letter codes">ITU-R country codes</a></li> <li><a href="/wiki/List_of_country_calling_codes" title="List of country calling codes">ITU-T country calling codes</a></li> <li><a href="/wiki/Mobile_country_code" title="Mobile country code">ITU-T mobile calling codes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Amateur_radio" title="Amateur radio">Amateur radio</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Maidenhead_Locator_System" title="Maidenhead Locator System">Maidenhead Locator System</a></li> <li>Historical : <a href="/wiki/QRA_locator" title="QRA locator">QRA locator</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox 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Universal Transverse Mercator coordinate system - Wikipedia