Tissot's indicatrix

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_distance_on_the_ellipsoid"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Differential distance on the ellipsoid</span> </div> </a> <ul id="toc-Differential_distance_on_the_ellipsoid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transforming_the_element_of_distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transforming_the_element_of_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Transforming the element of distance</span> </div> </a> <ul id="toc-Transforming_the_element_of_distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Numerical_computation_and_SVD" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numerical_computation_and_SVD"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Numerical 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Available in 14 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-14" 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">14 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B8%D0%BF%D1%81%D0%B0_%D0%BD%D0%B0_%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D0%B8%D1%82%D0%B5" title="Елипса на деформациите – Bulgarian" lang="bg" hreflang="bg" data-title="Елипса на деформациите" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Indicatriu_de_Tissot" title="Indicatriu de Tissot – Catalan" lang="ca" hreflang="ca" data-title="Indicatriu de Tissot" 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/Tissotova_indikatrix" title="Tissotova indikatrix – Czech" lang="cs" hreflang="cs" data-title="Tissotova indikatrix" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Tissotsche_Indikatrix" title="Tissotsche Indikatrix – German" lang="de" hreflang="de" data-title="Tissotsche Indikatrix" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Indicatriz_de_Tissot" title="Indicatriz de Tissot – Spanish" lang="es" hreflang="es" data-title="Indicatriz de Tissot" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Indicatrice_de_Tissot" title="Indicatrice de Tissot – French" lang="fr" hreflang="fr" data-title="Indicatrice de Tissot" 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/%ED%8B%B0%EC%86%8C_%ED%83%80%EC%9B%90" title="티소 타원 – Korean" lang="ko" hreflang="ko" data-title="티소 타원" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Indicatore_di_Tissot" title="Indicatore di Tissot – Italian" lang="it" hreflang="it" data-title="Indicatore di Tissot" 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%9E%D7%97%D7%95%D7%95%D7%9F_%D7%98%D7%99%D7%A1%D7%95" title="מחוון טיסו – Hebrew" lang="he" hreflang="he" data-title="מחוון טיסו" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Indicatrix_van_Tissot" title="Indicatrix van Tissot – Dutch" lang="nl" hreflang="nl" data-title="Indicatrix van Tissot" 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%86%E3%82%A4%E3%82%BD%E3%83%BC%E3%81%AE%E6%8C%87%E7%A4%BA%E6%A5%95%E5%86%86" title="テイソーの指示楕円 – Japanese" lang="ja" hreflang="ja" data-title="テイソーの指示楕円" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenia_Tissota" title="Twierdzenia Tissota – Polish" lang="pl" hreflang="pl" data-title="Twierdzenia Tissota" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D0%B8%D0%BF%D1%81_%D0%B8%D1%81%D0%BA%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D1%8F" title="Эллипс искажения – Russian" lang="ru" hreflang="ru" data-title="Эллипс искажения" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D1%96%D0%BF%D1%81_%D1%81%D0%BF%D0%BE%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Еліпс спотворення – Ukrainian" lang="uk" hreflang="uk" data-title="Еліпс спотворення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q534527#sitelinks-wikipedia" title="Edit interlanguage links" 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id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Characterization of distortion in map projections</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tissot_behrmann.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Tissot_behrmann.png/250px-Tissot_behrmann.png" decoding="async" width="250" height="106" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Tissot_behrmann.png/375px-Tissot_behrmann.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Tissot_behrmann.png/500px-Tissot_behrmann.png 2x" data-file-width="1193" data-file-height="504" /></a><figcaption>The <a href="/wiki/Behrmann_projection" title="Behrmann projection">Behrmann projection</a> with Tissot's indicatrices</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tissot_mercator.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Tissot_mercator.png/250px-Tissot_mercator.png" decoding="async" width="250" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Tissot_mercator.png/375px-Tissot_mercator.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Tissot_mercator.png/500px-Tissot_mercator.png 2x" data-file-width="777" data-file-height="720" /></a><figcaption>The <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a> with Tissot's indicatrices</figcaption></figure> <p>In <a href="/wiki/Cartography" title="Cartography">cartography</a>, a <b>Tissot's indicatrix</b> (<b>Tissot indicatrix</b>, <b>Tissot's ellipse</b>, <b>Tissot ellipse</b>, <b>ellipse of distortion</b>) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician <a href="/wiki/Nicolas_Auguste_Tissot" title="Nicolas Auguste Tissot">Nicolas Auguste Tissot</a> in 1859 and 1871 in order to characterize local distortions due to <a href="/wiki/Map_projection" title="Map projection">map projection</a>. It is the geometry that results from <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projecting</a> a <a href="/wiki/Circle" title="Circle">circle</a> of <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> whose axes indicate the two <a href="/wiki/Principal_curvature" title="Principal curvature">principal directions</a> along which scale is maximal and minimal at that point on the map. </p><p>A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians and parallels. These schematics are important in the study of map projections, both to illustrate distortion and to provide the basis for the calculations that represent the magnitude of distortion precisely at each point. Because the infinitesimal circles represented by the ellipses on the map all have the same area on the underlying curved geometric model, the distortion imposed by the map projection is evident. </p><p>There is a one-to-one correspondence between the Tissot indicatrix and the <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> of the map projection coordinate conversion.<sup id="cite_ref-Goldberg-Gott_1-0" class="reference"><a href="#cite_note-Goldberg-Gott-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tissot's theory was developed in the context of <a href="/wiki/Cartography" title="Cartography">cartographic analysis</a>. Generally the geometric model represents the Earth, and comes in the form of a <a href="/wiki/Sphere" title="Sphere">sphere</a> or <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a>. </p><p>Tissot's indicatrices illustrate linear, angular, and areal distortions of maps: </p> <ul><li>A map distorts distances (linear distortion) wherever the quotient between the lengths of an infinitesimally short line as projected onto the projection surface, and as it originally is on the Earth model, deviates from 1. The quotient is called the <i>scale factor</i>. Unless the projection is <a href="/wiki/Conformal_map" title="Conformal map">conformal</a> at the point being considered, the scale factor varies by direction around the point.</li> <li>A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion which is not a circle.</li> <li>A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection. This is expressed by ellipses of distortion whose areas vary across the map.</li></ul> <p>In conformal maps, where each point preserves angles projected from the geometric model, the Tissot's indicatrices are all circles of size varying by location, possibly also with varying orientation (given the four circle <a href="/wiki/Circular_sector" title="Circular sector">quadrants</a> split by <a href="/wiki/Meridian_(geography)" title="Meridian (geography)">meridians</a> and <a href="/wiki/Parallel_(latitude)" class="mw-redirect" title="Parallel (latitude)">parallels</a>). In <a href="/wiki/Equal-area_projection" title="Equal-area projection">equal-area projections</a>, where area proportions between objects are conserved, the Tissot's indicatrices all have the same area, though their shapes and orientations vary with location. In arbitrary projections, both area and shape vary across the map. </p> <table class="wikitable collapsible collapsed"> <tbody><tr> <th>World maps comparing Tissot's indicatrices on some common projections </th></tr> <tr> <th><ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_equirectangular_proj.svg" class="mw-file-description" title="Equirectangular projection"><img alt="Equirectangular projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tissot_indicatrix_world_map_equirectangular_proj.svg/200px-Tissot_indicatrix_world_map_equirectangular_proj.svg.png" decoding="async" width="200" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tissot_indicatrix_world_map_equirectangular_proj.svg/300px-Tissot_indicatrix_world_map_equirectangular_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Tissot_indicatrix_world_map_equirectangular_proj.svg/400px-Tissot_indicatrix_world_map_equirectangular_proj.svg.png 2x" data-file-width="3000" data-file-height="1500" /></a></span></div> <div class="gallerytext"><a href="/wiki/Equirectangular_projection" title="Equirectangular projection">Equirectangular projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_Mercator_proj.svg" class="mw-file-description" title="Mercator projection"><img alt="Mercator projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Tissot_indicatrix_world_map_Mercator_proj.svg/151px-Tissot_indicatrix_world_map_Mercator_proj.svg.png" decoding="async" width="151" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Tissot_indicatrix_world_map_Mercator_proj.svg/226px-Tissot_indicatrix_world_map_Mercator_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Tissot_indicatrix_world_map_Mercator_proj.svg/302px-Tissot_indicatrix_world_map_Mercator_proj.svg.png 2x" data-file-width="3000" data-file-height="2984" /></a></span></div> <div class="gallerytext"><a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg" class="mw-file-description" title="Gall–Peters projection"><img alt="Gall–Peters projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg/200px-Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg.png" decoding="async" width="200" height="127" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg/300px-Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg/400px-Tissot_indicatrix_world_map_Gall-Peters_equal-area_proj.svg.png 2x" data-file-width="3000" data-file-height="1906" /></a></span></div> <div class="gallerytext"><a href="/wiki/Gall%E2%80%93Peters_projection" title="Gall–Peters projection">Gall–Peters projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_Mollweide_proj.svg" class="mw-file-description" title="Mollweide projection"><img alt="Mollweide projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Tissot_indicatrix_world_map_Mollweide_proj.svg/200px-Tissot_indicatrix_world_map_Mollweide_proj.svg.png" decoding="async" width="200" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Tissot_indicatrix_world_map_Mollweide_proj.svg/300px-Tissot_indicatrix_world_map_Mollweide_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Tissot_indicatrix_world_map_Mollweide_proj.svg/400px-Tissot_indicatrix_world_map_Mollweide_proj.svg.png 2x" data-file-width="3000" data-file-height="1500" /></a></span></div> <div class="gallerytext"><a href="/wiki/Mollweide_projection" title="Mollweide projection">Mollweide projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg" class="mw-file-description" title="Winkel tripel projection"><img alt="Winkel tripel projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg/200px-Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg.png" decoding="async" width="200" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg/300px-Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg/400px-Tissot_indicatrix_world_map_Winkel_Tripel_proj.svg.png 2x" data-file-width="512" data-file-height="313" /></a></span></div> <div class="gallerytext"><a href="/wiki/Winkel_tripel_projection" title="Winkel tripel projection">Winkel tripel projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="Azimuthal equidistant projection"><img alt="Azimuthal equidistant projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png/150px-Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png/225px-Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png/300px-Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="680" data-file-height="680" /></a></span></div> <div class="gallerytext"><a href="/wiki/Azimuthal_equidistant_projection" title="Azimuthal equidistant projection">Azimuthal equidistant projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png" class="mw-file-description" title="Fuller projection"><img alt="Fuller projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png/200px-Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png" decoding="async" width="200" height="95" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png/300px-Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png/400px-Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png 2x" data-file-width="3638" data-file-height="1734" /></a></span></div> <div class="gallerytext"><a href="/wiki/Fuller_projection" class="mw-redirect" title="Fuller projection">Fuller projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png" class="mw-file-description" title="Transverse Mercator projection"><img alt="Transverse Mercator projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png/150px-Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png/225px-Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png/300px-Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png 2x" data-file-width="1240" data-file-height="1240" /></a></span></div> <div class="gallerytext"><a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg" class="mw-file-description" title="Lambert cylindrical equal-area projection"><img alt="Lambert cylindrical equal-area projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg/200px-Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg.png" decoding="async" width="200" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg/300px-Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg/400px-Tissot_indicatrix_world_map_Lambert_cyl_equal-area_proj.svg.png 2x" data-file-width="3000" data-file-height="953" /></a></span></div> <div class="gallerytext"><a href="/wiki/Lambert_cylindrical_equal-area_projection" title="Lambert cylindrical equal-area projection">Lambert cylindrical equal-area projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_sinusoidal_proj.svg" class="mw-file-description" title="Sinusoidal projection"><img alt="Sinusoidal projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Tissot_indicatrix_world_map_sinusoidal_proj.svg/200px-Tissot_indicatrix_world_map_sinusoidal_proj.svg.png" decoding="async" width="200" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Tissot_indicatrix_world_map_sinusoidal_proj.svg/300px-Tissot_indicatrix_world_map_sinusoidal_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Tissot_indicatrix_world_map_sinusoidal_proj.svg/400px-Tissot_indicatrix_world_map_sinusoidal_proj.svg.png 2x" data-file-width="3000" data-file-height="1500" /></a></span></div> <div class="gallerytext"><a href="/wiki/Sinusoidal_projection" title="Sinusoidal projection">Sinusoidal projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Tissot_indicatrix_world_map_Robinson_proj.svg" class="mw-file-description" title="Robinson projection"><img alt="Robinson projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Tissot_indicatrix_world_map_Robinson_proj.svg/200px-Tissot_indicatrix_world_map_Robinson_proj.svg.png" decoding="async" width="200" height="101" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Tissot_indicatrix_world_map_Robinson_proj.svg/300px-Tissot_indicatrix_world_map_Robinson_proj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Tissot_indicatrix_world_map_Robinson_proj.svg/400px-Tissot_indicatrix_world_map_Robinson_proj.svg.png 2x" data-file-width="3000" data-file-height="1522" /></a></span></div> <div class="gallerytext"><a href="/wiki/Robinson_projection" title="Robinson projection">Robinson projection</a></div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 180px;"><span typeof="mw:File"><a href="/wiki/File:Stereographic_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="Stereographic projection"><img alt="Stereographic projection" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_projection_with_Tissot%27s_indicatrix.png/150px-Stereographic_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_projection_with_Tissot%27s_indicatrix.png/225px-Stereographic_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_projection_with_Tissot%27s_indicatrix.png/300px-Stereographic_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="840" data-file-height="840" /></a></span></div> <div class="gallerytext"><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic projection</a></div> </li> </ul> </th></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Mathematics">Mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=2" title="Edit section: Mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the diagram below, the circle <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 ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span> has unit area as defined on the surface of a sphere. The circle <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'B'C'D'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> <msup> <mi>D</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {A'B'C'D'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7328375478a929f961706770ae798e585e5c0d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.968ex; height:2.509ex;" alt="{\displaystyle {A'B'C'D'}}"></span> is the Tissot's indicatrix that results from some projection 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 ABCD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ABCD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412b7d8df4db6ca8093d971320c405598c49c339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.198ex; height:2.176ex;" alt="{\displaystyle ABCD}"></span> onto a plane. Linear scale has not been preserved in this projection, as <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 {OA'\ncong OA}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>≆</mo> <mi>O</mi> <mi>A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {OA'\ncong OA}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db8af26b16d156c7ad33580524f873881e4ec1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.816ex; height:2.843ex;" alt="{\displaystyle {OA'\ncong OA}}"></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 OB'\ncong OB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>≆</mo> <mi>O</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle OB'\ncong OB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f48c65ed48951012c86184e80a0ad098e513ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.858ex; height:2.843ex;" alt="{\displaystyle OB'\ncong OB}"></span>. Because <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 {\angle M'OA'\ncong \angle MOA}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∠</mi> <msup> <mi>M</mi> <mo>′</mo> </msup> <mi>O</mi> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo>≆</mo> <mi mathvariant="normal">∠</mi> <mi>M</mi> <mi>O</mi> <mi>A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\angle M'OA'\ncong \angle MOA}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59cb101b1e53e70aa6f994373da4a63e70ebde5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.798ex; height:2.843ex;" alt="{\displaystyle {\angle M'OA'\ncong \angle MOA}}"></span>, we know that there is an angular distortion. Because <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 \operatorname {Area} (A'B'C'D')\neq \operatorname {Area} (ABCD)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Area</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <msup> <mi>C</mi> <mo>′</mo> </msup> <msup> <mi>D</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>Area</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mi>C</mi> <mi>D</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Area} (A'B'C'D')\neq \operatorname {Area} (ABCD)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8924d8356b99eef5d4e2240dc8c1e8238a8560c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.582ex; height:3.009ex;" alt="{\displaystyle \operatorname {Area} (A'B'C'D')\neq \operatorname {Area} (ABCD)}"></span>, we know there is an areal distortion. </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Indicatrix.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/8/8f/Indicatrix.png" decoding="async" width="308" height="325" class="mw-file-element" data-file-width="308" data-file-height="325" /></a></span> </p><p>The original circle in the above example had a radius of 1, but when dealing with a Tissot indicatrix, one deals with ellipses of infinitesimal radius. Even though the radii of the original circle and its distortion ellipse will all be infinitesimal, by employing <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> the ratios between them can still be meaningfully calculated. For example, if the ratio between the radius of the input circle and a projected circle is equal to 1, then the indicatrix is drawn with as a circle with an area of 1. The size that the indicatrix gets drawn on the map is arbitrary: they are all scaled by the same factor so that their sizes are proportional to one another. Like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> in the diagram, the axes from <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 O}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}"></span> along the parallel and along the meridian may undergo a change of length and a rotation during projection. For a given point, it is common in the literature to represent the scale along the meridian as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> and the scale along the parallel as <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>. Unless the projection is conformal, all angles except the one subtended by the <a href="/wiki/Semi-major_axis" class="mw-redirect" title="Semi-major axis">semi-major axis</a> and <a href="/wiki/Semi-minor_axis" class="mw-redirect" title="Semi-minor axis">semi-minor axis</a> of the ellipse may have changed as well. A particular angle will have changed the most, and the value of that maximum change is known as the angular deformation, denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>. In general, which angle that is and how it is oriented do not figure prominently into distortion analysis; it is the magnitude of the change that is significant. The values 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 h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span>, <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>, 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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> can be computed as follows:<sup id="cite_ref-SnyderManual_2-0" class="reference"><a href="#cite_note-SnyderManual-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 24">: 24 </span></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}h&={\frac {1}{R}}{\sqrt {{{\left({\frac {\partial x}{\partial \varphi }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \varphi }}\right)}^{2}}}}\\[4pt]k&={\frac {1}{R\cos \varphi }}{\sqrt {{{\left({\frac {\partial x}{\partial \lambda }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \lambda }}\right)}^{2}}}}\\[4pt]\sin \theta '&={\frac {1}{R^{2}hk\cos \varphi }}\left({{\frac {\partial y}{\partial \varphi }}{\frac {\partial x}{\partial \lambda }}-{\frac {\partial x}{\partial \varphi }}{\frac {\partial y}{\partial \lambda }}}\right)\\[4pt]a'&={\sqrt {{h^{2}}+{k^{2}}+2hk\sin \theta '}},\quad b'={\sqrt {{h^{2}}+{k^{2}}-2hk\sin \theta '}}\\[4pt]a&={\frac {a'+b'}{2}},\quad b={\frac {a'-b'}{2}}\\[4pt]s&=hk\sin \theta '\\[4pt]\omega &=2\arcsin {\frac {b'}{a'}}\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="0.7em 0.7em 0.7em 0.7em 0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>h</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>R</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>h</mi> <mi>k</mi> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>+</mo> <mn>2</mn> <mi>h</mi> <mi>k</mi> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> </msqrt> </mrow> <mo>,</mo> <mspace width="1em" /> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>h</mi> <mi>k</mi> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>s</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>h</mi> <mi>k</mi> <mi>sin</mi> <mo>⁡</mo> <msup> <mi>θ</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>b</mi> <mo>′</mo> </msup> <msup> <mi>a</mi> <mo>′</mo> </msup> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}h&={\frac {1}{R}}{\sqrt {{{\left({\frac {\partial x}{\partial \varphi }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \varphi }}\right)}^{2}}}}\\[4pt]k&={\frac {1}{R\cos \varphi }}{\sqrt {{{\left({\frac {\partial x}{\partial \lambda }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \lambda }}\right)}^{2}}}}\\[4pt]\sin \theta '&={\frac {1}{R^{2}hk\cos \varphi }}\left({{\frac {\partial y}{\partial \varphi }}{\frac {\partial x}{\partial \lambda }}-{\frac {\partial x}{\partial \varphi }}{\frac {\partial y}{\partial \lambda }}}\right)\\[4pt]a'&={\sqrt {{h^{2}}+{k^{2}}+2hk\sin \theta '}},\quad b'={\sqrt {{h^{2}}+{k^{2}}-2hk\sin \theta '}}\\[4pt]a&={\frac {a'+b'}{2}},\quad b={\frac {a'-b'}{2}}\\[4pt]s&=hk\sin \theta '\\[4pt]\omega &=2\arcsin {\frac {b'}{a'}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20c20bbf5504f7e1f4b1dc8194ad02a4971d63a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.171ex; width:60.567ex; height:45.509ex;" alt="{\displaystyle {\begin{aligned}h&={\frac {1}{R}}{\sqrt {{{\left({\frac {\partial x}{\partial \varphi }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \varphi }}\right)}^{2}}}}\\[4pt]k&={\frac {1}{R\cos \varphi }}{\sqrt {{{\left({\frac {\partial x}{\partial \lambda }}\right)}^{2}}+{{\left({\frac {\partial y}{\partial \lambda }}\right)}^{2}}}}\\[4pt]\sin \theta '&={\frac {1}{R^{2}hk\cos \varphi }}\left({{\frac {\partial y}{\partial \varphi }}{\frac {\partial x}{\partial \lambda }}-{\frac {\partial x}{\partial \varphi }}{\frac {\partial y}{\partial \lambda }}}\right)\\[4pt]a'&={\sqrt {{h^{2}}+{k^{2}}+2hk\sin \theta '}},\quad b'={\sqrt {{h^{2}}+{k^{2}}-2hk\sin \theta '}}\\[4pt]a&={\frac {a'+b'}{2}},\quad b={\frac {a'-b'}{2}}\\[4pt]s&=hk\sin \theta '\\[4pt]\omega &=2\arcsin {\frac {b'}{a'}}\end{aligned}}}"></span> </p><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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> are the latitude and longitude coordinates of a point, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> is the radius of the globe, 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> 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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> are the point's resulting coordinates after projection. </p><p>In the result for any given point, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> 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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> are the maximum and minimum scale factors, analogous to the semimajor and semiminor axes in the diagram; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> represents the amount of inflation or deflation in area, 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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> represents the maximum angular distortion. </p><p>For <a href="/wiki/Map_projection#Conformal" title="Map projection">conformal</a> projections such as the <a href="/wiki/Mercator_projection" title="Mercator projection">Mercator projection</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3dd95ec422d9fe5b53c7858b94a0314e7f0d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.649ex; height:2.176ex;" alt="{\displaystyle h=k}"></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 \theta ={\pi \over 2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ={\pi \over 2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69faf49463da098cc123954e739f58631b837293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.357ex; height:4.676ex;" alt="{\displaystyle \theta ={\pi \over 2}}"></span>, such that at each point the ellipse degenerates into a circle, with the radius being equal to the scale factor. </p><p>For <a href="/wiki/Map_projection#Equal-area" title="Map projection">equal-area</a> such as the <a href="/wiki/Sinusoidal_projection" title="Sinusoidal projection">sinusoidal projection</a>, the semi-major axis of the ellipse is the reciprocal of the semi-minor axis, such that every ellipse has equal area even as their <a href="/wiki/Eccentricity_(mathematics)" title="Eccentricity (mathematics)">eccentricities</a> vary. </p><p>For arbitrary projections, the shape and the area of the ellipses at each point are largely independent from one another.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="An_alternative_derivation_for_numerical_computation">An alternative derivation for numerical computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=3" title="Edit section: An alternative derivation for numerical computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another way to understand and derive Tissot's indicatrix is through the differential geometry of surfaces.<sup id="cite_ref-moderntissot_4-0" class="reference"><a href="#cite_note-moderntissot-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This approach lends itself well to modern numerical methods, as the parameters of Tissot's indicatrix can be computed using <a href="/wiki/Singular_Value_Decomposition" class="mw-redirect" title="Singular Value Decomposition">singular value decomposition</a> (SVD) and <a href="/wiki/Finite_difference" title="Finite difference">central difference approximation</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_distance_on_the_ellipsoid">Differential distance on the ellipsoid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=4" title="Edit section: Differential distance on the ellipsoid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let a 3D point, <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 {\hat {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc59ad6d9a06d55b96b65beb0fdfc89acc1e40e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.843ex;" alt="{\displaystyle {\hat {X}}}"></span>, on an ellipsoid be parameterized as: </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 {\hat {X}}(\lambda ,\phi )=\left[{\begin{matrix}N\cos {\lambda }\cos {\phi }\\-N(1-e^{2})\sin {\phi }\\N\sin {\lambda }\cos {\phi }\end{matrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}(\lambda ,\phi )=\left[{\begin{matrix}N\cos {\lambda }\cos {\phi }\\-N(1-e^{2})\sin {\phi }\\N\sin {\lambda }\cos {\phi }\end{matrix}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7bc8b2688a0fccbadec6fc365b948cfe086e137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:31.351ex; height:9.509ex;" alt="{\displaystyle {\hat {X}}(\lambda ,\phi )=\left[{\begin{matrix}N\cos {\lambda }\cos {\phi }\\-N(1-e^{2})\sin {\phi }\\N\sin {\lambda }\cos {\phi }\end{matrix}}\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 (\lambda ,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda ,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/964f9a2ba8886f77feafe7589b6f17a4702042ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.584ex; height:2.843ex;" alt="{\displaystyle (\lambda ,\phi )}"></span> are longitude and latitude, respectively, 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 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 a function of the equatorial radius, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, and eccentricity, <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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\frac {R}{\sqrt {1-e^{2}\sin ^{2}(\phi )}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\frac {R}{\sqrt {1-e^{2}\sin ^{2}(\phi )}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9342d7992810ad3af1d76f3d0c47ba0f362dee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:21.954ex; height:8.009ex;" alt="{\displaystyle N={\frac {R}{\sqrt {1-e^{2}\sin ^{2}(\phi )}}}}"></span></dd></dl> <p>The element of distance on the sphere, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0fb36e4308227d3e4a1f809c2571ec02527100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle ds}"></span> is defined by the <a href="/wiki/First_fundamental_form" title="First fundamental form">first fundamental form</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}={\begin{bmatrix}d\lambda &d\phi \end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>λ</mi> </mtd> <mtd> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>E</mi> </mtd> <mtd> <mi>F</mi> </mtd> </mtr> <mtr> <mtd> <mi>F</mi> </mtd> <mtd> <mi>G</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}={\begin{bmatrix}d\lambda &d\phi \end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca7e28c4cbac24c16cee63e7a3e5f96b9fdf5fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.938ex; height:6.176ex;" alt="{\displaystyle ds^{2}={\begin{bmatrix}d\lambda &d\phi \end{bmatrix}}{\begin{bmatrix}E&F\\F&G\end{bmatrix}}{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}}"></span></dd></dl> <p>whose coefficients are defined as: </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}&E={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \lambda }}\\&F={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\&G={\frac {\partial {\hat {X}}}{\partial \phi }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">⋅</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">⋅</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">⋅</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&E={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \lambda }}\\&F={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\&G={\frac {\partial {\hat {X}}}{\partial \phi }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b398ec3dada9e68fe68ca35ed2849bc7b4105ab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:14.687ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}&E={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \lambda }}\\&F={\frac {\partial {\hat {X}}}{\partial \lambda }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\&G={\frac {\partial {\hat {X}}}{\partial \phi }}{\boldsymbol {\cdot }}{\frac {\partial {\hat {X}}}{\partial \phi }}\\\end{aligned}}}"></span></dd></dl> <p>Computing the necessary derivatives gives: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {\hat {X}}}{\partial \lambda }}=\left[{\begin{matrix}-N\sin {\lambda }\cos {\phi }\\0\\N\cos {\lambda }\cos {\phi }\end{matrix}}\right]\qquad \qquad {\frac {\partial {\hat {X}}}{\partial \phi }}=\left[{\begin{matrix}-M\cos {\lambda }\sin {\phi }\\-M\cos {\phi }\\M\sin {\lambda }\sin {\phi }\end{matrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>N</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>M</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>M</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {\hat {X}}}{\partial \lambda }}=\left[{\begin{matrix}-N\sin {\lambda }\cos {\phi }\\0\\N\cos {\lambda }\cos {\phi }\end{matrix}}\right]\qquad \qquad {\frac {\partial {\hat {X}}}{\partial \phi }}=\left[{\begin{matrix}-M\cos {\lambda }\sin {\phi }\\-M\cos {\phi }\\M\sin {\lambda }\sin {\phi }\end{matrix}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4371cb1385e98562f6ba8e02c5715ea7025cb01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:60.481ex; height:9.509ex;" alt="{\displaystyle {\frac {\partial {\hat {X}}}{\partial \lambda }}=\left[{\begin{matrix}-N\sin {\lambda }\cos {\phi }\\0\\N\cos {\lambda }\cos {\phi }\end{matrix}}\right]\qquad \qquad {\frac {\partial {\hat {X}}}{\partial \phi }}=\left[{\begin{matrix}-M\cos {\lambda }\sin {\phi }\\-M\cos {\phi }\\M\sin {\lambda }\sin {\phi }\end{matrix}}\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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is a function of the equatorial radius, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, and the ellipsoid eccentricity, <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/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M={\frac {R(1-e^{2})}{(1-e^{2}\sin ^{2}(\phi ))^{\frac {3}{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\frac {R(1-e^{2})}{(1-e^{2}\sin ^{2}(\phi ))^{\frac {3}{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a51f65af58998c22cab3d11d20563289d0c4d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:23.554ex; height:7.843ex;" alt="{\displaystyle M={\frac {R(1-e^{2})}{(1-e^{2}\sin ^{2}(\phi ))^{\frac {3}{2}}}}}"></span></dd></dl> <p>Substituting these values into the first fundamental form gives the formula for elemental distance on the ellipsoid: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=\left(N\cos {\phi }\right)^{2}d\lambda ^{2}+M^{2}d\phi ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=\left(N\cos {\phi }\right)^{2}d\lambda ^{2}+M^{2}d\phi ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66134876d4aac0a4a2edf035511511e92432fe2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.331ex; height:3.343ex;" alt="{\displaystyle ds^{2}=\left(N\cos {\phi }\right)^{2}d\lambda ^{2}+M^{2}d\phi ^{2}}"></span></dd></dl> <p>This result relates the measure of distance on the ellipsoid surface as a function of the spherical coordinate system. </p> <div class="mw-heading mw-heading3"><h3 id="Transforming_the_element_of_distance">Transforming the element of distance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=5" title="Edit section: Transforming the element of distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Recall that the purpose of Tissot's indicatrix is to relate how distances on the sphere change when mapped to a planar surface. Specifically, the desired relation is the transform <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> that relates differential distance along the bases of the spherical coordinate system to differential distance along the bases of the Cartesian coordinate system on the planar map. This can be expressed by the relation: </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{bmatrix}dx\\dy\end{bmatrix}}={\mathcal {T}}{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}dx\\dy\end{bmatrix}}={\mathcal {T}}{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13750d20ea5c8e4c6bb96031e28eaf1734d986ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.69ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}dx\\dy\end{bmatrix}}={\mathcal {T}}{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}}"></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 ds(\lambda ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds(\lambda ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5625d3f5f0951cb4b14e4b3e0f8c3af2edfcb721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.667ex; height:2.843ex;" alt="{\displaystyle ds(\lambda ,0)}"></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 ds(0,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds(0,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88dc8c618e4fb9f459a63b4c0cd9253a22418cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.697ex; height:2.843ex;" alt="{\displaystyle ds(0,\phi )}"></span> represent the computation 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 ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0fb36e4308227d3e4a1f809c2571ec02527100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.306ex; height:2.176ex;" alt="{\displaystyle ds}"></span> along the longitudinal and latitudinal axes, respectively. Computation 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 ds(\lambda ,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds(\lambda ,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5625d3f5f0951cb4b14e4b3e0f8c3af2edfcb721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.667ex; height:2.843ex;" alt="{\displaystyle ds(\lambda ,0)}"></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 ds(0,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds(0,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88dc8c618e4fb9f459a63b4c0cd9253a22418cf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.697ex; height:2.843ex;" alt="{\displaystyle ds(0,\phi )}"></span> can be performed directly from the equation above, yielding: </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}&ds(\lambda ,0)=N\cos(\phi )d\lambda \\&ds(0,\phi )=Md\phi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>M</mi> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&ds(\lambda ,0)=N\cos(\phi )d\lambda \\&ds(0,\phi )=Md\phi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea807bd6922807d8bb109e05bb2cb43e2b991764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.845ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&ds(\lambda ,0)=N\cos(\phi )d\lambda \\&ds(0,\phi )=Md\phi \end{aligned}}}"></span></dd></dl> <p>For the purposes of this computation, it is useful to express this relationship as a matrix operation: </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{bmatrix}d\lambda \\d\phi \end{bmatrix}}=K{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}},\qquad K={\begin{bmatrix}{\frac {1}{N\cos {\phi }}}&0\\0&{\frac {1}{M}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>N</mi> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ϕ</mi> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>M</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}=K{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}},\qquad K={\begin{bmatrix}{\frac {1}{N\cos {\phi }}}&0\\0&{\frac {1}{M}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c915a27f81c92c4cad23e07a95476ccff5d56914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.188ex; margin-bottom: -0.317ex; width:47.316ex; height:8.176ex;" alt="{\displaystyle {\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}=K{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}},\qquad K={\begin{bmatrix}{\frac {1}{N\cos {\phi }}}&0\\0&{\frac {1}{M}}\end{bmatrix}}}"></span></dd></dl> <p>Now, in order to relate the distances on the ellipsoid surface to those on the plane, we need to relate the coordinate systems. From the chain rule, we can write: </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{bmatrix}dx\\dy\end{bmatrix}}=J{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>λ</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>ϕ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}dx\\dy\end{bmatrix}}=J{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e326c95b1c2790a51a92e82d4e13f3a270a1e9ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.13ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}dx\\dy\end{bmatrix}}=J{\begin{bmatrix}d\lambda \\d\phi \end{bmatrix}}}"></span></dd></dl> <p>where J is the <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J={\begin{bmatrix}{\frac {\partial x}{\partial \lambda }}&{\frac {\partial x}{\partial \phi }}\\{\frac {\partial y}{\partial \lambda }}&{\frac {\partial y}{\partial \phi }}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J={\begin{bmatrix}{\frac {\partial x}{\partial \lambda }}&{\frac {\partial x}{\partial \phi }}\\{\frac {\partial y}{\partial \lambda }}&{\frac {\partial y}{\partial \phi }}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/611073d24c5ae26e782f588d1461d1aac4a2846a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:16.219ex; height:8.843ex;" alt="{\displaystyle J={\begin{bmatrix}{\frac {\partial x}{\partial \lambda }}&{\frac {\partial x}{\partial \phi }}\\{\frac {\partial y}{\partial \lambda }}&{\frac {\partial y}{\partial \phi }}\end{bmatrix}}}"></span></dd></dl> <p>Plugging in the matrix expression for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2911481cb08253d29482348954893a519c16cbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.571ex; height:2.176ex;" alt="{\displaystyle d\lambda }"></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 d\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd273cfec7dc419d85c50763eac05d3d9a086c88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.601ex; height:2.509ex;" alt="{\displaystyle d\phi }"></span> yields the definition of the transform <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> represented by the indicatrix: </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{bmatrix}dx\\dy\end{bmatrix}}=JK{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>J</mi> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}dx\\dy\end{bmatrix}}=JK{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134c2ceae607455544219d5f18d194f95b3e36e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.292ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}dx\\dy\end{bmatrix}}=JK{\begin{bmatrix}ds(\lambda ,0)\\ds(0,\phi )\end{bmatrix}}}"></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 {\mathcal {T}}=JK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo>=</mo> <mi>J</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}=JK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b020a6dd326be5f49b66c03f63b85d9c9d29837d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.572ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}=JK}"></span></dd></dl> <p>This transform <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span> encapsulates the mapping from the ellipsoid surface to the plane. Expressed in this form, <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">SVD</a> can be used to parcel out the important components of the local transformation. </p> <div class="mw-heading mw-heading3"><h3 id="Numerical_computation_and_SVD">Numerical computation and SVD</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=6" title="Edit section: Numerical computation and SVD"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to extract the desired distortion information, at any given location in the spherical coordinate system, the values 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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> can be computed directly. The Jacobian, <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 J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359e4f407b49910e02c27c2f52e87a36cd74c053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}"></span>, can be computed analytically from the mapping function itself, but it is often simpler to numerically approximate the values at any location on the map using <a href="/wiki/Finite_difference" title="Finite difference">central differences</a>. Once these values are computed, SVD can be applied to each transformation matrix to extract the local distortion information. Remember that, because distortion is local, every location on the map will have its own transformation. </p><p>Recall the definition of SVD: </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 \mathrm {SVD} ({\mathcal {T}})=U\Lambda V^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">D</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mi mathvariant="normal">Λ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SVD} ({\mathcal {T}})=U\Lambda V^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ae327ad3e1278d038e1c24f82640c1ece3ba516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.357ex; height:3.176ex;" alt="{\displaystyle \mathrm {SVD} ({\mathcal {T}})=U\Lambda V^{T}}"></span></dd></dl> <p>It is the decomposition of the transformation, <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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>, into a rotation in the source domain (i.e. the ellipsoid surface), <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 V^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c64d10f395f64420d45f4779b35a7f89a1e0b38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.306ex; height:2.676ex;" alt="{\displaystyle V^{T}}"></span>, a scaling along the basis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">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/0ac0a4a98a414e3480335f9ba652d12571ec6733" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }"></span>, and a subsequent second rotation, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>. For understanding distortion, the first rotation is irrelevant, as it rotates the axes of the circle but has no bearing on the final orientation of the ellipse. The next operation, represented by the diagonal singular value matrix, scales the circle along its axes, deforming it to an ellipse. Thus, the singular values represent the scale factors along axes of the ellipse. The first singular value provides the semi-major axis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>, and the second provides the semi-minor axis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, which are the directional scaling factors of distortion. Scale distortion can be computed as the area of the ellipse, <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 ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"></span>, or equivalently by the determinant 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 {\mathcal {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8236d074e42310f5dc24d1d2b5b8f5981c3e87ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.343ex;" alt="{\displaystyle {\mathcal {T}}}"></span>. Finally, the orientation of the ellipse, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>, can be extracted from the first column 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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> as: </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 \theta =\arctan \left({\frac {u_{1,0}}{u_{0,0}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan \left({\frac {u_{1,0}}{u_{0,0}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8070ec4a74fc614f212cb64b2f27858d47cd2d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.576ex; height:6.176ex;" alt="{\displaystyle \theta =\arctan \left({\frac {u_{1,0}}{u_{0,0}}}\right)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Gallery">Gallery</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tissot%27s_indicatrix&action=edit&section=7" title="Edit section: Gallery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1248256098">@media all and (max-width:720px){.mw-parser-output .mod-gallery{width:100%!important}}.mw-parser-output .mod-gallery{display:table}.mw-parser-output .mod-gallery-default{background:transparent;margin-top:4px}.mw-parser-output .mod-gallery-center{margin-left:auto;margin-right:auto}.mw-parser-output .mod-gallery-left{float:left}.mw-parser-output .mod-gallery-right{float:right}.mw-parser-output .mod-gallery-none{float:none}.mw-parser-output .mod-gallery-collapsible{width:100%}.mw-parser-output .mod-gallery .title,.mw-parser-output .mod-gallery .main,.mw-parser-output .mod-gallery .footer{display:table-row}.mw-parser-output .mod-gallery .title>div{display:table-cell;padding:0 4px 4px;text-align:center;font-weight:bold}.mw-parser-output .mod-gallery .main>div{display:table-cell}.mw-parser-output .mod-gallery .gallery{line-height:1.35em}.mw-parser-output .mod-gallery .footer>div{display:table-cell;padding:4px;text-align:right;font-size:85%;line-height:1em}.mw-parser-output .mod-gallery .title>div *,.mw-parser-output .mod-gallery .footer>div *{overflow:visible}.mw-parser-output .mod-gallery .gallerybox img{background:none!important}.mw-parser-output .mod-gallery .bordered-images .thumb img{border:solid var(--background-color-neutral,#eaecf0)1px}.mw-parser-output .mod-gallery .whitebg .thumb{background:var(--background-color-base,#fff)!important}</style><div class="mod-gallery mod-gallery-default mod-gallery-center"><div class="main"><div><ul class="gallery mw-gallery-traditional nochecker bordered-images whitebg"> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png" class="mw-file-description" title="The transverse Mercator projection with Tissot's indicatrices"><img alt="The transverse Mercator projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png/160px-Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png/240px-Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png/320px-Transverse_Mercator_projection_of_Standard_meridian_135E-45W_withTissot%27s_indicatrix.png 2x" data-file-width="1240" data-file-height="1240" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">transverse Mercator projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Stereographic_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="The stereographic projection with Tissot's indicatrices"><img alt="The stereographic projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_projection_with_Tissot%27s_indicatrix.png/160px-Stereographic_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_projection_with_Tissot%27s_indicatrix.png/240px-Stereographic_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Stereographic_projection_with_Tissot%27s_indicatrix.png/320px-Stereographic_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="840" data-file-height="840" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Stereographic_projection" title="Stereographic projection">stereographic projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Sinusoidal_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="The sinusoidal projection with Tissot's indicatrices"><img alt="The sinusoidal projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Sinusoidal_projection_with_Tissot%27s_indicatrix.png/160px-Sinusoidal_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="160" height="83" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Sinusoidal_projection_with_Tissot%27s_indicatrix.png/240px-Sinusoidal_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Sinusoidal_projection_with_Tissot%27s_indicatrix.png/320px-Sinusoidal_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="1240" data-file-height="640" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Sinusoidal_projection" title="Sinusoidal projection">sinusoidal projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg" class="mw-file-description" title="The Peirce quincuncial projection with Tissot's indicatrices"><img alt="The Peirce quincuncial projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg/160px-Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg/240px-Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg/320px-Peirce_Quincuncial_with_Tissot%27s_Indicatrices_of_Distortion.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Peirce_quincuncial_projection" title="Peirce quincuncial projection">Peirce quincuncial projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Miller_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="The Miller cylindrical projection with Tissot's indicatrices"><img alt="The Miller cylindrical projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Miller_projection_with_Tissot%27s_indicatrix.png/160px-Miller_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="160" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Miller_projection_with_Tissot%27s_indicatrix.png/240px-Miller_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Miller_projection_with_Tissot%27s_indicatrix.png/320px-Miller_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="840" data-file-height="640" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Miller_cylindrical_projection" title="Miller cylindrical projection">Miller cylindrical projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Hammer_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="The Hammer projection with Tissot's indicatrices"><img alt="The Hammer projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Hammer_projection_with_Tissot%27s_indicatrix.png/160px-Hammer_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="160" height="77" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Hammer_projection_with_Tissot%27s_indicatrix.png/240px-Hammer_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Hammer_projection_with_Tissot%27s_indicatrix.png/320px-Hammer_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="1240" data-file-height="600" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Hammer_projection" title="Hammer projection">Hammer projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png" class="mw-file-description" title="The azimuthal equidistant projection with Tissot's indicatrices"><img alt="The azimuthal equidistant projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png/160px-Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png" decoding="async" width="160" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png/240px-Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png/320px-Azimuthal_equidistant_projection_with_Tissot%27s_indicatrix.png 2x" data-file-width="680" data-file-height="680" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Azimuthal_equidistant_projection" title="Azimuthal equidistant projection">azimuthal equidistant projection</a> with Tissot's indicatrices</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 190px;"><span typeof="mw:File"><a href="/wiki/File:Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png" class="mw-file-description" title="The Fuller projection with Tissot's indicatrices"><img alt="The Fuller projection with Tissot's indicatrices" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png/160px-Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png" decoding="async" width="160" height="76" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png/240px-Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/61/Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png/320px-Fuller_projection_with_Tissot%27s_indicatrix_of_deformation.png 2x" data-file-width="3638" data-file-height="1734" /></a></span></div> <div class="gallerytext">The <a href="/wiki/Fuller_projection" class="mw-redirect" title="Fuller projection">Fuller projection</a> with Tissot's indicatrices</div> </li> </ul></div></div></div> <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=Tissot%27s_indicatrix&action=edit&section=8" 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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Goldberg-Gott-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Goldberg-Gott_1-0">^</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 id="CITEREFGoldbergGott_III2007" class="citation journal cs1">Goldberg, David M.; Gott III, J. Richard (2007). <a rel="nofollow" class="external text" href="http://www.physics.drexel.edu/~goldberg/projections/goldberg_gott.pdf">"Flexion and Skewness in Map Projections of the Earth"</a> <span class="cs1-format">(PDF)</span>. <i>Cartographica</i>. <b>42</b> (4): 297–318. <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/astro-ph/0608501">astro-ph/0608501</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3138%2Fcarto.42.4.297">10.3138/carto.42.4.297</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:11359702">11359702</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2011-11-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cartographica&rft.atitle=Flexion+and+Skewness+in+Map+Projections+of+the+Earth&rft.volume=42&rft.issue=4&rft.pages=297-318&rft.date=2007&rft_id=info%3Aarxiv%2Fastro-ph%2F0608501&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11359702%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.3138%2Fcarto.42.4.297&rft.aulast=Goldberg&rft.aufirst=David+M.&rft.au=Gott+III%2C+J.+Richard&rft_id=http%3A%2F%2Fwww.physics.drexel.edu%2F~goldberg%2Fprojections%2Fgoldberg_gott.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATissot%27s+indicatrix" class="Z3988"></span></span> </li> <li id="cite_note-SnyderManual-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-SnyderManual_2-0">^</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"><a href="/wiki/John_P._Snyder" title="John P. Snyder">Snyder, John P.</a> (1987). <a rel="nofollow" class="external text" href="https://pubs.er.usgs.gov/publication/pp1395"><i>Map projections—A working manual</i></a>. Professional Paper 1395. Denver: <a href="/wiki/United_States_Geological_Survey" title="United States Geological Survey">USGS</a>. p. 383. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1782662228" title="Special:BookSources/978-1782662228"><bdi>978-1782662228</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2015-11-26</span></span>.</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%94A+working+manual&rft.place=Denver&rft.series=Professional+Paper+1395&rft.pages=383&rft.pub=USGS&rft.date=1987&rft.isbn=978-1782662228&rft.aulast=Snyder&rft.aufirst=John+P.&rft_id=https%3A%2F%2Fpubs.er.usgs.gov%2Fpublication%2Fpp1395&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATissot%27s+indicatrix" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">More general example of Tissot's indicatrix: the <a href="/wiki/Scale_(map)#Visualisation_of_point_scale:_the_Tissot_indicatrix" title="Scale (map)">Winkel tripel</a> projection.</span> </li> <li id="cite_note-moderntissot-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-moderntissot_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaskowski1989" class="citation journal cs1">Laskowski, Piotr (1989). "The Traditional and Modern Look at Tissot's Indicatrix". <i>The American Cartographer</i>. <b>16</b> (2): 123–133. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1559%2F152304089783875497">10.1559/152304089783875497</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Cartographer&rft.atitle=The+Traditional+and+Modern+Look+at+Tissot%27s+Indicatrix&rft.volume=16&rft.issue=2&rft.pages=123-133&rft.date=1989&rft_id=info%3Adoi%2F10.1559%2F152304089783875497&rft.aulast=Laskowski&rft.aufirst=Piotr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATissot%27s+indicatrix" class="Z3988"></span></span> </li> </ol></div></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=Tissot%27s_indicatrix&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Map_projections_with_Tissot%27s_indicatrix" class="extiw" title="commons:Category:Map projections with Tissot's indicatrix">Map projections with Tissot's indicatrix</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www.se16.info/js/tissot.htm">Java applet with interactive projections showing Tissot's indicatrix</a></li></ul> <div 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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:Map_projections" title="Template:Map projections"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Map_projections" title="Template talk:Map projections"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Map_projections" title="Special:EditPage/Template:Map projections"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Map_projection" style="font-size:114%;margin:0 4em"><a href="/wiki/Map_projection" title="Map projection">Map projection</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_cartography" title="History of cartography">History</a></li> <li><a href="/wiki/List_of_map_projections" title="List of map projections">List</a></li> <li><a href="/wiki/Portal:Maps" title="Portal:Maps">Portal</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="By_surface" style="font-size:114%;margin:0 4em"><a href="/wiki/Map_projection#Projections_by_surface" title="Map projection">By surface</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" 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/Map_projection#Cylindrical" title="Map projection">Cylindrical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" 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%;font-weight: normal;"><a href="/wiki/Mercator_projection" title="Mercator projection">Mercator</a>-conformal</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/Gauss%E2%80%93Kr%C3%BCger_coordinate_system" class="mw-redirect" title="Gauss–Krüger coordinate system">Gauss–Krüger</a></li> <li><a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator</a></li> <li><a href="/wiki/Oblique_Mercator_projection" title="Oblique Mercator projection">Oblique Mercator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight: normal;"><a href="/wiki/Cylindrical_equal-area_projection" title="Cylindrical equal-area projection">Equal-area</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/Cylindrical_equal-area_projection#Discussion" title="Cylindrical equal-area projection">Balthasart</a></li> <li><a href="/wiki/Behrmann_projection" title="Behrmann projection">Behrmann</a></li> <li><a href="/wiki/Gall%E2%80%93Peters_projection" title="Gall–Peters projection">Gall–Peters</a></li> <li><a href="/wiki/Hobo%E2%80%93Dyer_projection" title="Hobo–Dyer projection">Hobo–Dyer</a></li> <li><a href="/wiki/Lambert_cylindrical_equal-area_projection" title="Lambert cylindrical equal-area projection">Lambert</a></li> <li><a href="/wiki/Cylindrical_equal-area_projection#Discussion" title="Cylindrical equal-area projection">Smyth equal-surface</a></li> <li><a href="/wiki/Cylindrical_equal-area_projection#Discussion" title="Cylindrical equal-area projection">Trystan Edwards</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cassini_projection" title="Cassini projection">Cassini</a></li> <li><a href="/wiki/Central_cylindrical_projection" title="Central cylindrical projection">Central</a></li> <li><a href="/wiki/Equirectangular_projection" title="Equirectangular projection">Equirectangular</a></li> <li><a href="/wiki/Gall_stereographic_projection" title="Gall stereographic projection">Gall stereographic</a></li> <li><a href="/wiki/Gall_isographic_projection" title="Gall isographic projection">Gall isographic</a></li> <li><a href="/wiki/Miller_cylindrical_projection" title="Miller cylindrical projection">Miller</a></li> <li><a href="/wiki/Space-oblique_Mercator_projection" title="Space-oblique Mercator projection">Space-oblique Mercator</a></li> <li><a href="/wiki/Web_Mercator_projection" title="Web Mercator projection">Web Mercator</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Pseudocylindrical" title="Map projection">Pseudocylindrical</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Equal-area" scope="row" class="navbox-group" style="width:1%">Equal-area</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/Collignon_projection" title="Collignon projection">Collignon</a></li> <li><a href="/wiki/Eckert_II_projection" title="Eckert II projection">Eckert II</a></li> <li><a href="/wiki/Eckert_IV_projection" title="Eckert IV projection">Eckert IV</a></li> <li><a href="/wiki/Eckert_VI_projection" title="Eckert VI projection">Eckert VI</a></li> <li><a href="/wiki/Equal_Earth_projection" title="Equal Earth projection">Equal Earth</a></li> <li><a href="/wiki/Goode_homolosine_projection" title="Goode homolosine projection">Goode homolosine</a></li> <li><a href="/wiki/Mollweide_projection" title="Mollweide projection">Mollweide</a></li> <li><a href="/wiki/Sinusoidal_projection" title="Sinusoidal projection">Sinusoidal</a></li> <li><a href="/wiki/Tobler_hyperelliptical_projection" title="Tobler hyperelliptical projection">Tobler hyperelliptical</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Kavrayskiy_VII_projection" title="Kavrayskiy VII projection">Kavrayskiy VII</a></li> <li><a href="/wiki/Wagner_VI_projection" title="Wagner VI projection">Wagner VI</a></li> <li><a href="/wiki/Winkel_projection" title="Winkel projection">Winkel I and II</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Conical" title="Map projection">Conical</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/Albers_projection" title="Albers projection">Albers</a></li> <li><a href="/wiki/Equidistant_conic_projection" title="Equidistant conic projection">Equidistant</a></li> <li><a href="/wiki/Lambert_conformal_conic_projection" title="Lambert conformal conic projection">Lambert conformal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Pseudoconical" title="Map projection">Pseudoconical</a></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/Bonne_projection" title="Bonne projection">Bonne</a></li> <li><a href="/wiki/Bottomley_projection" title="Bottomley projection">Bottomley</a></li> <li><a href="/wiki/Polyconic_projection_class" title="Polyconic projection class">Polyconic</a> <ul><li><a href="/wiki/American_polyconic_projection" title="American polyconic projection">American</a></li> <li><a href="/wiki/Latitudinally_equal-differential_polyconic_projection" title="Latitudinally equal-differential polyconic projection">Chinese</a></li></ul></li> <li><a href="/wiki/Werner_projection" title="Werner projection">Werner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Azimuthal" title="Map projection">Azimuthal<br /><span class="nobold">(planar)</span></a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="General_perspective" scope="row" class="navbox-group" style="width:1%;font-weight: normal;"><a href="/wiki/General_Perspective_projection" title="General Perspective projection">General perspective</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/Gnomonic_projection" title="Gnomonic projection">Gnomonic</a></li> <li><a href="/wiki/Orthographic_map_projection" title="Orthographic map projection">Orthographic</a></li> <li><a href="/wiki/Stereographic_map_projection" title="Stereographic map projection">Stereographic</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Azimuthal_equidistant_projection" title="Azimuthal equidistant projection">Equidistant</a></li> <li><a href="/wiki/Lambert_azimuthal_equal-area_projection" title="Lambert azimuthal equal-area projection">Lambert equal-area</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Pseudoazimuthal" title="Map projection">Pseudoazimuthal</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/Aitoff_projection" title="Aitoff projection">Aitoff</a></li> <li><a href="/wiki/Hammer_projection" title="Hammer projection">Hammer</a></li> <li><a href="/wiki/Wiechel_projection" title="Wiechel projection">Wiechel</a></li> <li><a href="/wiki/Winkel_tripel_projection" title="Winkel tripel projection">Winkel tripel</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="By_metric" style="font-size:114%;margin:0 4em"><a href="/wiki/Map_projection#Projections_by_preservation_of_a_metric_property" title="Map projection">By metric</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" 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/Conformal_map_projection" title="Conformal map projection">Conformal</a></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/Adams_hemisphere-in-a-square_projection" title="Adams hemisphere-in-a-square projection">Adams hemisphere-in-a-square</a></li> <li><a href="/wiki/Gauss%E2%80%93Kr%C3%BCger_coordinate_system" class="mw-redirect" title="Gauss–Krüger coordinate system">Gauss–Krüger</a></li> <li><a href="/wiki/Guyou_hemisphere-in-a-square_projection" title="Guyou hemisphere-in-a-square projection">Guyou hemisphere-in-a-square</a></li> <li><a href="/wiki/Lambert_conformal_conic_projection" title="Lambert conformal conic projection">Lambert conformal conic</a></li> <li><a href="/wiki/Mercator_projection" title="Mercator projection">Mercator</a></li> <li><a href="/wiki/Peirce_quincuncial_projection" title="Peirce quincuncial projection">Peirce quincuncial</a></li> <li><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic</a></li> <li><a href="/wiki/Transverse_Mercator_projection" title="Transverse Mercator projection">Transverse Mercator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Equal-area_projection" title="Equal-area projection">Equal-area</a></th><td class="navbox-list-with-group navbox-list navbox-odd" 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%;font-weight: normal;"><a href="/wiki/Bonne_projection" title="Bonne projection">Bonne</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/Sinusoidal_projection" title="Sinusoidal projection">Sinusoidal</a></li> <li><a href="/wiki/Werner_projection" title="Werner projection">Werner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight: normal;"><a href="/wiki/Bottomley_projection" title="Bottomley projection">Bottomley</a></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/Sinusoidal_projection" title="Sinusoidal projection">Sinusoidal</a></li> <li><a href="/wiki/Werner_projection" title="Werner projection">Werner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight: normal;"><a href="/wiki/Cylindrical_equal-area_projection" title="Cylindrical equal-area projection">Cylindrical</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/Cylindrical_equal-area_projection#Discussion" title="Cylindrical equal-area projection">Balthasart</a></li> <li><a href="/wiki/Behrmann_projection" title="Behrmann projection">Behrmann</a></li> <li><a href="/wiki/Gall%E2%80%93Peters_projection" title="Gall–Peters projection">Gall–Peters</a></li> <li><a href="/wiki/Hobo%E2%80%93Dyer_projection" title="Hobo–Dyer projection">Hobo–Dyer</a></li> <li><a href="/wiki/Lambert_cylindrical_equal-area_projection" title="Lambert cylindrical equal-area projection">Lambert cylindrical equal-area</a></li> <li><a href="/wiki/Cylindrical_equal-area_projection#Discussion" title="Cylindrical equal-area projection">Smyth equal-surface</a></li> <li><a href="/wiki/Cylindrical_equal-area_projection#Discussion" title="Cylindrical equal-area projection">Trystan Edwards</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight: normal;"><a href="/wiki/Tobler_hyperelliptical_projection" title="Tobler hyperelliptical projection">Tobler hyperelliptical</a></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/Collignon_projection" title="Collignon projection">Collignon</a></li> <li><a href="/wiki/Mollweide_projection" title="Mollweide projection">Mollweide</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albers_projection" title="Albers projection">Albers</a></li> <li><a href="/wiki/Hammer_projection#Briesemeister" title="Hammer projection">Briesemeister</a></li> <li><a href="/wiki/Eckert_II_projection" title="Eckert II projection">Eckert II</a></li> <li><a href="/wiki/Eckert_IV_projection" title="Eckert IV projection">Eckert IV</a></li> <li><a href="/wiki/Eckert_VI_projection" title="Eckert VI projection">Eckert VI</a></li> <li><a href="/wiki/Equal_Earth_projection" title="Equal Earth projection">Equal Earth</a></li> <li><a href="/wiki/Goode_homolosine_projection" title="Goode homolosine projection">Goode homolosine</a></li> <li><a href="/wiki/Hammer_projection" title="Hammer projection">Hammer</a></li> <li><a href="/wiki/Lambert_azimuthal_equal-area_projection" title="Lambert azimuthal equal-area projection">Lambert azimuthal equal-area</a></li> <li><a href="/wiki/Quadrilateralized_spherical_cube" title="Quadrilateralized spherical cube">Quadrilateralized spherical cube</a></li> <li><a href="/wiki/Strebe_1995_projection" title="Strebe 1995 projection">Strebe 1995</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Equidistant" title="Map projection">Equidistant in<br />some aspect</a></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/Equidistant_conic_projection" title="Equidistant conic projection">Conic</a></li> <li><a href="/wiki/Equirectangular_projection" title="Equirectangular projection">Equirectangular</a></li> <li><a href="/wiki/Sinusoidal_projection" title="Sinusoidal projection">Sinusoidal</a></li> <li><a href="/wiki/Two-point_equidistant_projection" title="Two-point equidistant projection">Two-point</a></li> <li><a href="/wiki/Werner_projection" title="Werner projection">Werner</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Gnomonic" title="Map projection">Gnomonic</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/Gnomonic_projection" title="Gnomonic projection">Gnomonic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Rhumb_line" title="Map projection">Loxodromic</a></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/Loximuthal_projection" title="Loximuthal projection">Loximuthal</a></li> <li><a href="/wiki/Mercator_projection" title="Mercator projection">Mercator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Retroazimuthal" title="Map projection">Retroazimuthal</a><br /><span class="nobold">(Mecca or Qibla)</span></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/Craig_retroazimuthal_projection" title="Craig retroazimuthal projection">Craig</a></li> <li><a href="/wiki/Hammer_retroazimuthal_projection" title="Hammer retroazimuthal projection">Hammer</a></li> <li><a href="/wiki/Littrow_projection" title="Littrow projection">Littrow</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="By_construction" style="font-size:114%;margin:0 4em"><a href="/wiki/Map_projection#Classification" title="Map projection">By construction</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" 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/Map_projection#Compromise_projections" title="Map projection">Compromise</a></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/Chamberlin_trimetric_projection" title="Chamberlin trimetric projection">Chamberlin trimetric</a></li> <li><a href="/wiki/Kavrayskiy_VII_projection" title="Kavrayskiy VII projection">Kavrayskiy VII</a></li> <li><a href="/wiki/Miller_cylindrical_projection" title="Miller cylindrical projection">Miller cylindrical</a></li> <li><a href="/wiki/Natural_Earth_projection" title="Natural Earth projection">Natural Earth</a></li> <li><a href="/wiki/Robinson_projection" title="Robinson projection">Robinson</a></li> <li><a href="/wiki/Van_der_Grinten_projection" title="Van der Grinten projection">Van der Grinten</a></li> <li><a href="/wiki/Wagner_VI_projection" title="Wagner VI projection">Wagner VI</a></li> <li><a href="/wiki/Winkel_tripel_projection" title="Winkel tripel projection">Winkel tripel</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Hybrid" title="Map projection">Hybrid</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/Goode_homolosine_projection" title="Goode homolosine projection">Goode homolosine</a></li> <li><a href="/wiki/HEALPix" title="HEALPix">HEALPix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Perspective_projections" title="Map projection">Perspective</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Planar" scope="row" class="navbox-group" style="width:1%;font-weight: normal;"><a href="/wiki/General_Perspective_projection" title="General Perspective projection">Planar</a></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/Gnomonic_projection" title="Gnomonic projection">Gnomonic</a></li> <li><a href="/wiki/Orthographic_map_projection" title="Orthographic map projection">Orthographic</a></li> <li><a href="/wiki/Stereographic_projection" title="Stereographic projection">Stereographic</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Central_cylindrical_projection" title="Central cylindrical projection">Central cylindrical</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Map_projection#Polyhedral" title="Map projection">Polyhedral</a></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/AuthaGraph_projection" title="AuthaGraph projection">AuthaGraph</a></li> <li><a href="/wiki/Bernard_J._S._Cahill" title="Bernard J. S. Cahill">Cahill Butterfly</a></li> <li><a href="/wiki/Cahill%E2%80%93Keyes_projection" title="Cahill–Keyes projection">Cahill–Keyes M-shape</a></li> <li><a href="/wiki/Dymaxion_map" title="Dymaxion map">Dymaxion</a></li> <li><a href="/wiki/Snyder_equal-area_projection" title="Snyder equal-area projection">ISEA</a></li> <li><a href="/wiki/Quadrilateralized_spherical_cube" title="Quadrilateralized spherical cube">Quadrilateralized spherical cube</a></li> <li><a href="/wiki/Waterman_butterfly_projection" title="Waterman butterfly projection">Waterman butterfly</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="See_also" style="font-size:114%;margin:0 4em">See also</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Interruption_(map_projection)" title="Interruption (map projection)">Interruption (map projection)</a></li> <li><a href="/wiki/Latitude" title="Latitude">Latitude</a></li> <li><a href="/wiki/Longitude" title="Longitude">Longitude</a></li> <li><a class="mw-selflink selflink">Tissot's indicatrix</a></li> <li><a href="/wiki/Map_projection_of_the_tri-axial_ellipsoid" class="mw-redirect" title="Map projection of the tri-axial ellipsoid">Map projection of the tri-axial ellipsoid</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Tissot%27s_indicatrix&oldid=1251739901">https://en.wikipedia.org/w/index.php?title=Tissot%27s_indicatrix&oldid=1251739901</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Map_projections" title="Category:Map projections">Map projections</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 17 October 2024, at 20:28<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Tissot's indicatrix - Wikipedia