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<span>Toggle Distances in physics and geometry subsection</span> </button> <ul id="toc-Distances_in_physics_and_geometry-sublist" class="vector-toc-list"> <li id="toc-Straight-line_or_Euclidean_distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Straight-line_or_Euclidean_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Straight-line or Euclidean distance</span> </div> </a> <ul id="toc-Straight-line_or_Euclidean_distance-sublist" class="vector-toc-list"> <li id="toc-Measurement" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1.1</span> <span>Measurement</span> </div> </a> <ul id="toc-Measurement-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Shortest-path_distance_on_a_curved_surface" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Shortest-path_distance_on_a_curved_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Shortest-path distance on a curved surface</span> </div> </a> <ul id="toc-Shortest-path_distance_on_a_curved_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Effects_of_relativity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Effects_of_relativity"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Effects of relativity</span> </div> </a> <ul id="toc-Effects_of_relativity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_spatial_distances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_spatial_distances"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Other spatial distances</span> </div> </a> <ul id="toc-Other_spatial_distances-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Metaphorical_distances" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Metaphorical_distances"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Metaphorical distances</span> </div> </a> <button aria-controls="toc-Metaphorical_distances-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Metaphorical distances subsection</span> </button> <ul id="toc-Metaphorical_distances-sublist" class="vector-toc-list"> <li id="toc-Statistical_distances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Statistical_distances"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Statistical distances</span> </div> </a> <ul id="toc-Statistical_distances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Edit_distances" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Edit_distances"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Edit distances</span> </div> </a> <ul id="toc-Edit_distances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distance_in_graph_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distance_in_graph_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Distance in graph theory</span> </div> </a> <ul id="toc-Distance_in_graph_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_the_social_sciences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_the_social_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>In the social sciences</span> </div> </a> <ul id="toc-In_the_social_sciences-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematical_formalization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematical_formalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematical formalization</span> </div> </a> <ul id="toc-Mathematical_formalization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distance_between_sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Distance_between_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Distance between sets</span> </div> </a> <ul id="toc-Distance_between_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_ideas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_ideas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Related ideas</span> </div> </a> <button aria-controls="toc-Related_ideas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Related ideas subsection</span> </button> <ul id="toc-Related_ideas-sublist" class="vector-toc-list"> <li id="toc-Distance_travelled" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distance_travelled"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Distance travelled</span> </div> </a> <ul id="toc-Distance_travelled-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Displacement_and_directed_distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Displacement_and_directed_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Displacement and directed distance</span> </div> </a> <ul id="toc-Displacement_and_directed_distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Signed_distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Signed_distance"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Signed distance</span> </div> </a> <ul id="toc-Signed_distance-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Library_support" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Library_support"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Library support</span> </div> </a> <ul id="toc-Library_support-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Distance</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 98 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-98" 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">98 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Afstand" title="Afstand – Afrikaans" lang="af" hreflang="af" data-title="Afstand" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%AD%E1%89%80%E1%89%B5" title="ርቀት – Amharic" lang="am" hreflang="am" data-title="ርቀት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D9%81%D8%A9" title="مسافة – Arabic" lang="ar" hreflang="ar" data-title="مسافة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Distancia" title="Distancia – Aragonese" lang="an" hreflang="an" data-title="Distancia" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distancia" title="Distancia – Asturian" lang="ast" hreflang="ast" data-title="Distancia" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/M%C9%99saf%C9%99" title="Məsafə – Azerbaijani" lang="az" hreflang="az" data-title="Məsafə" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D8%B3%D8%A7%D9%81%D8%AA" title="مسافت – South Azerbaijani" lang="azb" hreflang="azb" data-title="مسافت" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A6%E0%A7%82%E0%A6%B0%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="দূরত্ব – Bangla" lang="bn" hreflang="bn" data-title="দূরত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C5%AB-l%C4%AB" title="Kū-lī – Minnan" lang="nan" hreflang="nan" data-title="Kū-lī" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D1%8B%D2%AB%D0%BB%D1%8B%D2%A1" title="Алыҫлыҡ – Bashkir" lang="ba" hreflang="ba" data-title="Алыҫлыҡ" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B4%D0%BB%D0%B5%D0%B3%D0%BB%D0%B0%D1%81%D1%86%D1%8C" title="Адлегласць – Belarusian" lang="be" hreflang="be" data-title="Адлегласць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%B4%D0%BB%D0%B5%D0%B3%D0%BB%D0%B0%D1%81%D1%8C%D1%86%D1%8C" title="Адлегласьць – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Адлегласьць" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%A6%E0%A5%81%E0%A4%B0%E0%A4%A4%E0%A5%8D%E0%A4%B5" title="दुरत्व – Bhojpuri" lang="bh" hreflang="bh" data-title="दुरत्व" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bcl mw-list-item"><a href="https://bcl.wikipedia.org/wiki/Distansya" title="Distansya – Central Bikol" lang="bcl" hreflang="bcl" data-title="Distansya" data-language-autonym="Bikol Central" data-language-local-name="Central Bikol" class="interlanguage-link-target"><span>Bikol Central</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Разстояние – Bulgarian" lang="bg" hreflang="bg" data-title="Разстояние" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%97%D0%B0%D0%B9" title="Зай – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Зай" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target"><span>Буряад</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dist%C3%A0ncia" title="Distància – Catalan" lang="ca" hreflang="ca" data-title="Distància" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BD%C3%A7%C4%95%D1%88" title="Инçĕш – Chuvash" lang="cv" hreflang="cv" data-title="Инçĕш" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vzd%C3%A1lenost" title="Vzdálenost – Czech" lang="cs" hreflang="cs" data-title="Vzdálenost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhambwe" title="Nhambwe – Shona" lang="sn" hreflang="sn" data-title="Nhambwe" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Afstandsformlen" title="Afstandsformlen – Danish" lang="da" hreflang="da" data-title="Afstandsformlen" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D8%A8%D8%B9%D9%88%D8%AF%D9%8A%D8%A9" title="بعودية – Moroccan Arabic" lang="ary" hreflang="ary" data-title="بعودية" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abstand" title="Abstand – German" lang="de" hreflang="de" data-title="Abstand" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kaugus" title="Kaugus – Estonian" lang="et" hreflang="et" data-title="Kaugus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%80%CF%8C%CF%83%CF%84%CE%B1%CF%83%CE%B7_(%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1)" title="Απόσταση (γεωμετρία) – Greek" lang="el" hreflang="el" data-title="Απόσταση (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distancia" title="Distancia – Spanish" lang="es" hreflang="es" data-title="Distancia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Distanco" title="Distanco – Esperanto" lang="eo" hreflang="eo" data-title="Distanco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Distantzia" title="Distantzia – Basque" lang="eu" hreflang="eu" data-title="Distantzia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87" title="فاصله – Persian" lang="fa" hreflang="fa" data-title="فاصله" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Distance_(math%C3%A9matiques)" title="Distance (mathématiques) – French" lang="fr" hreflang="fr" data-title="Distance (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Distancia" title="Distancia – Galician" lang="gl" hreflang="gl" data-title="Distancia" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B1%B0%EB%A6%AC" title="거리 – Korean" lang="ko" hreflang="ko" data-title="거리" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A5%D5%BC%D5%A1%D5%BE%D5%B8%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6)" title="Հեռավորություն (երկրաչափություն) – Armenian" lang="hy" hreflang="hy" data-title="Հեռավորություն (երկրաչափություն)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A6%E0%A5%82%E0%A4%B0%E0%A5%80" title="दूरी – Hindi" lang="hi" hreflang="hi" data-title="दूरी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Daljina" title="Daljina – Croatian" lang="hr" hreflang="hr" data-title="Daljina" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Disto" title="Disto – Ido" lang="io" hreflang="io" data-title="Disto" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jarak" title="Jarak – Indonesian" lang="id" hreflang="id" data-title="Jarak" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Distantia" title="Distantia – Interlingua" lang="ia" hreflang="ia" data-title="Distantia" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Umgama" title="Umgama – Xhosa" lang="xh" hreflang="xh" data-title="Umgama" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fjarl%C3%A6g%C3%B0" title="Fjarlægð – Icelandic" lang="is" hreflang="is" data-title="Fjarlægð" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Distanza_(matematica)" title="Distanza (matematica) – Italian" lang="it" hreflang="it" data-title="Distanza (matematica)" 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%A8%D7%97%D7%A7" title="מרחק – Hebrew" lang="he" hreflang="he" data-title="מרחק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A6%E0%B3%82%E0%B2%B0" title="ದೂರ – Kannada" lang="kn" hreflang="kn" data-title="ದೂರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-krc mw-list-item"><a href="https://krc.wikipedia.org/wiki/%D0%A3%D0%B7%D0%B0%D0%BA%D1%8A%D0%BB%D1%8B%D0%BA%D1%8A" title="Узакълыкъ – Karachay-Balkar" lang="krc" hreflang="krc" data-title="Узакълыкъ" data-language-autonym="Къарачай-малкъар" data-language-local-name="Karachay-Balkar" class="interlanguage-link-target"><span>Къарачай-малкъар</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%9C%E1%83%AB%E1%83%98%E1%83%9A%E1%83%98" title="მანძილი – Georgian" lang="ka" hreflang="ka" data-title="მანძილი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D2%9B%D0%B0%D1%88%D1%8B%D2%9B%D1%82%D1%8B%D2%9B" title="Арақашықтық – Kazakh" lang="kk" hreflang="kk" data-title="Арақашықтық" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Umbali" title="Umbali – Swahili" lang="sw" hreflang="sw" data-title="Umbali" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Distans" title="Distans – Haitian Creole" lang="ht" hreflang="ht" data-title="Distans" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/D%C3%BBrah%C3%AE" title="Dûrahî – Kurdish" lang="ku" hreflang="ku" data-title="Dûrahî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D1%80%D0%B0%D0%BB%D1%8B%D0%BA" title="Аралык – Kyrgyz" lang="ky" hreflang="ky" data-title="Аралык" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Att%C4%81lums" title="Attālums – Latvian" lang="lv" hreflang="lv" data-title="Attālums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ofstand" title="Ofstand – Luxembourgish" lang="lb" hreflang="lb" data-title="Ofstand" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atstumas" title="Atstumas – Lithuanian" lang="lt" hreflang="lt" data-title="Atstumas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Aafstandj" title="Aafstandj – Limburgish" lang="li" hreflang="li" data-title="Aafstandj" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%A1vols%C3%A1g" title="Távolság – Hungarian" lang="hu" hreflang="hu" data-title="Távolság" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%82%D0%BE%D1%98%D0%B0%D0%BD%D0%B8%D0%B5" title="Растојание – Macedonian" lang="mk" hreflang="mk" data-title="Растојание" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%82%E0%A4%A4%E0%A4%B0" title="अंतर – Marathi" lang="mr" hreflang="mr" data-title="अंतर" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jarak" title="Jarak – Malay" lang="ms" hreflang="ms" data-title="Jarak" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%80%E1%80%BD%E1%80%AC%E1%80%A1%E1%80%9D%E1%80%B1%E1%80%B8" title="အကွာအဝေး – Burmese" lang="my" hreflang="my" data-title="အကွာအဝေး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Afstand" title="Afstand – Dutch" lang="nl" hreflang="nl" data-title="Afstand" 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/%E8%B7%9D%E9%9B%A2" title="距離 – Japanese" lang="ja" hreflang="ja" data-title="距離" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Avstand" title="Avstand – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Avstand" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Avstand" title="Avstand – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Avstand" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Masofa" title="Masofa – Uzbek" lang="uz" hreflang="uz" data-title="Masofa" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%88%E0%A8%82%E0%A8%A1%E0%A8%BE" title="ਪੈਂਡਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਪੈਂਡਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%88%D8%A7%D9%BC%D9%86" title="واټن – Pashto" lang="ps" hreflang="ps" data-title="واټن" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Odleg%C5%82o%C5%9B%C4%87" title="Odległość – Polish" lang="pl" hreflang="pl" data-title="Odległość" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dist%C3%A2ncia" title="Distância – Portuguese" lang="pt" hreflang="pt" data-title="Distância" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D0%B5" title="Расстояние – Russian" lang="ru" hreflang="ru" data-title="Расстояние" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Distance" title="Distance – Scots" lang="sco" hreflang="sco" data-title="Distance" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Distanca" title="Distanca – Albanian" lang="sq" hreflang="sq" data-title="Distanca" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Distance" title="Distance – Simple English" lang="en-simple" hreflang="en-simple" data-title="Distance" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Vzdialenos%C5%A5" title="Vzdialenosť – Slovak" lang="sk" hreflang="sk" data-title="Vzdialenosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Razdalja" title="Razdalja – Slovenian" lang="sl" hreflang="sl" data-title="Razdalja" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%D9%88%D9%88%D8%B1%DB%8C" title="دووری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="دووری" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D1%82%D0%BE%D1%98%D0%B0%D1%9A%D0%B5" title="Растојање – Serbian" lang="sr" hreflang="sr" data-title="Растојање" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rastojanje" title="Rastojanje – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Rastojanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4limatka" title="Välimatka – Finnish" lang="fi" hreflang="fi" data-title="Välimatka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Avst%C3%A5nd" title="Avstånd – Swedish" lang="sv" hreflang="sv" data-title="Avstånd" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distansiya" title="Distansiya – Tagalog" lang="tl" hreflang="tl" data-title="Distansiya" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%82%E0%AE%B0%E0%AE%AE%E0%AF%8D" title="தூரம் – Tamil" lang="ta" hreflang="ta" data-title="தூரம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Ameccaq" title="Ameccaq – Kabyle" lang="kab" hreflang="kab" data-title="Ameccaq" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A6%E0%B1%82%E0%B0%B0%E0%B0%82" title="దూరం – Telugu" lang="te" hreflang="te" data-title="దూరం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%A2%E0%B8%B0%E0%B8%97%E0%B8%B2%E0%B8%87" title="ระยะทาง – Thai" lang="th" hreflang="th" data-title="ระยะทาง" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%81%D0%BE%D1%84%D0%B0" title="Масофа – Tajik" lang="tg" hreflang="tg" data-title="Масофа" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mesafe" title="Mesafe – Turkish" lang="tr" hreflang="tr" data-title="Mesafe" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D1%96%D0%B4%D1%81%D1%82%D0%B0%D0%BD%D1%8C" title="Відстань – Ukrainian" lang="uk" hreflang="uk" data-title="Відстань" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%DB%81" title="فاصلہ – Urdu" lang="ur" hreflang="ur" data-title="فاصلہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Distansa" title="Distansa – Venetian" lang="vec" hreflang="vec" data-title="Distansa" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kho%E1%BA%A3ng_c%C3%A1ch" title="Khoảng cách – Vietnamese" lang="vi" hreflang="vi" data-title="Khoảng cách" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E8%B7%9D" title="距 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="距" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Distansiya" title="Distansiya – Waray" lang="war" hreflang="war" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Separation between two points</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Distance_(disambiguation)" class="mw-disambig" title="Distance (disambiguation)">Distance (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Distance" title="Special:EditPage/Distance">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Distance%22">"Distance"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Distance%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Distance%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Distance%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Distance%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Distance%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Distance_board_in_Vizag_cropped.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Distance_board_in_Vizag_cropped.jpg/275px-Distance_board_in_Vizag_cropped.jpg" decoding="async" width="275" height="425" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Distance_board_in_Vizag_cropped.jpg/413px-Distance_board_in_Vizag_cropped.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Distance_board_in_Vizag_cropped.jpg/550px-Distance_board_in_Vizag_cropped.jpg 2x" data-file-width="913" data-file-height="1412" /></a><figcaption>A board showing distances near <a href="/wiki/Visakhapatnam" title="Visakhapatnam">Visakhapatnam</a>, India</figcaption></figure> <p><b>Distance</b> is a numerical or occasionally qualitative <a href="/wiki/Measurement" title="Measurement">measurement</a> of how far apart objects, points, people, or ideas are. In <a href="/wiki/Physics" title="Physics">physics</a> or everyday usage, distance may refer to a physical <a href="/wiki/Length" title="Length">length</a> or an estimation based on other criteria (e.g. "two counties over"). The term is also frequently used metaphorically<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> to mean a measurement of the amount of difference between two similar objects (such as <a href="/wiki/Statistical_distance" title="Statistical distance">statistical distance</a> between <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a> or <a href="/wiki/Edit_distance" title="Edit distance">edit distance</a> between <a href="/wiki/String_(computer_science)" title="String (computer science)">strings of text</a>) or a degree of separation (as exemplified by <a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">distance</a> between people in a <a href="/wiki/Social_network" title="Social network">social network</a>). Most such notions of distance, both physical and metaphorical, are formalized in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> using the notion of a <a href="/wiki/Metric_space" title="Metric space">metric space</a>. </p><p>In the <a href="/wiki/Social_science" title="Social science">social sciences</a>, <b>distance</b> can refer to a qualitative measurement of separation, such as <a href="/wiki/Social_distance" title="Social distance">social distance</a> or <a href="/wiki/Psychological_distance" title="Psychological distance">psychological distance</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Distances_in_physics_and_geometry">Distances in physics and geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=1" title="Edit section: Distances in physics and geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The distance between physical locations can be defined in different ways in different contexts. </p> <div class="mw-heading mw-heading3"><h3 id="Straight-line_or_Euclidean_distance">Straight-line or Euclidean distance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=2" title="Edit section: Straight-line or Euclidean distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a></div> <p>The distance between two points in physical <a href="/wiki/Space" title="Space">space</a> is the <a href="/wiki/Length" title="Length">length</a> of a <a href="/wiki/Line_segment" title="Line segment">straight line</a> between them, which is the shortest possible path. This is the usual meaning of distance in <a href="/wiki/Classical_physics" title="Classical physics">classical physics</a>, including <a href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</a>. </p><p>Straight-line distance is formalized mathematically as the <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> in <a href="/wiki/Two-dimensional_Euclidean_space" class="mw-redirect" title="Two-dimensional Euclidean space">two-</a> and <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional space</a>. In <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, the distance between two points <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> is often denoted <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"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/fa0bf6dba77a8b8734eb4356cac20b49e9a80c3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.801ex; height:2.843ex;" alt="{\displaystyle |AB|}"></span>. In <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">coordinate geometry</a>, Euclidean distance is computed using the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. The distance between points <span class="texhtml">(<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>)</span> and <span class="texhtml">(<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>)</span> in the plane is given by:<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><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> <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 d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/617b88d273f6cec8288acc4a071c855ce441e49b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:51.035ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"></span> Similarly, given points (<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>, <i>z</i><sub>1</sub>) and (<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>, <i>z</i><sub>2</sub>) in three-dimensional space, the distance between them is:<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <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 d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8518a4bd2334b23780a251a54df9c43fd67a26c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:72.578ex; height:4.843ex;" alt="{\displaystyle d={\sqrt {(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}}}={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}.}"></span> This idea generalizes to higher-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Measurement">Measurement</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=3" title="Edit section: Measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distance_measurement" class="mw-redirect" title="Distance measurement">Distance measurement</a></div> <p>There are many ways of <a href="/wiki/Measuring" class="mw-redirect" title="Measuring">measuring</a> straight-line distances. For example, it can be done directly using a <a href="/wiki/Ruler" title="Ruler">ruler</a>, or indirectly with a <a href="/wiki/Radar" title="Radar">radar</a> (for long distances) or <a href="/wiki/Interferometry" title="Interferometry">interferometry</a> (for very short distances). The <a href="/wiki/Cosmic_distance_ladder" title="Cosmic distance ladder">cosmic distance ladder</a> is a set of ways of measuring extremely long distances. </p> <div class="mw-heading mw-heading3"><h3 id="Shortest-path_distance_on_a_curved_surface">Shortest-path distance on a curved surface</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=4" title="Edit section: Shortest-path distance on a curved surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Greatcircle_Jetstream_routes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Greatcircle_Jetstream_routes.svg/400px-Greatcircle_Jetstream_routes.svg.png" decoding="async" width="400" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Greatcircle_Jetstream_routes.svg/600px-Greatcircle_Jetstream_routes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Greatcircle_Jetstream_routes.svg/800px-Greatcircle_Jetstream_routes.svg.png 2x" data-file-width="815" data-file-height="374" /></a><figcaption>Airline routes between <a href="/wiki/Los_Angeles" title="Los Angeles">Los Angeles</a> and <a href="/wiki/Tokyo" title="Tokyo">Tokyo</a> approximately follow a <a href="/wiki/Great_circle" title="Great circle">great circle</a> going west (top) but use the <a href="/wiki/Jet_stream" title="Jet stream">jet stream</a> (bottom) when heading eastwards. The shortest route appears as a curve rather than a straight line because the <a href="/wiki/Map_projection" title="Map projection">map projection</a> does not scale all distances equally compared to the real spherical surface of the Earth.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Geographic_distance" class="mw-redirect" title="Geographic distance">Geographic distance</a> and <a href="/wiki/Geodesic" title="Geodesic">geodesic</a></div> <p>The straight-line distance between two points on the surface of the Earth is not very useful for most purposes, since we cannot tunnel straight through the <a href="/wiki/Earth%27s_mantle" title="Earth's mantle">Earth's mantle</a>. Instead, one typically measures the shortest path along the <a href="/wiki/Surface_of_the_Earth" class="mw-redirect" title="Surface of the Earth">surface of the Earth</a>, <a href="/wiki/As_the_crow_flies" title="As the crow flies">as the crow flies</a>. This is approximated mathematically by the <a href="/wiki/Great-circle_distance" title="Great-circle distance">great-circle distance</a> on a sphere. </p><p>More generally, the shortest path between two points along a <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">curved surface</a> is known as a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a>. The <a href="/wiki/Arc_length" title="Arc length">arc length</a> of geodesics gives a way of measuring distance from the perspective of an <a href="/wiki/Ant" title="Ant">ant</a> or other flightless creature living on that surface. </p> <div class="mw-heading mw-heading3"><h3 id="Effects_of_relativity">Effects of relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=5" title="Edit section: Effects of relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distance_measure" title="Distance measure">Distance measure</a></div> <p>In the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>, because of phenomena such as <a href="/wiki/Length_contraction" title="Length contraction">length contraction</a> and the <a href="/wiki/Relativity_of_simultaneity" title="Relativity of simultaneity">relativity of simultaneity</a>, distances between objects depend on a choice of <a href="/wiki/Inertial_frame_of_reference" title="Inertial frame of reference">inertial frame of reference</a>. On galactic and larger scales, the measurement of distance is also affected by the <a href="/wiki/Expansion_of_the_universe" title="Expansion of the universe">expansion of the universe</a>. In practice, a number of <a href="/wiki/Distance_measure" title="Distance measure">distance measures</a> are used in <a href="/wiki/Cosmology" title="Cosmology">cosmology</a> to quantify such distances. </p> <div class="mw-heading mw-heading3"><h3 id="Other_spatial_distances">Other spatial distances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=6" title="Edit section: Other spatial distances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Manhattan_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/200px-Manhattan_distance.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/300px-Manhattan_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/400px-Manhattan_distance.svg.png 2x" data-file-width="283" data-file-height="283" /></a><figcaption><a href="/wiki/Manhattan_distance" class="mw-redirect" title="Manhattan distance">Manhattan distance</a> on a grid</figcaption></figure> <p>Unusual definitions of distance can be helpful to model certain physical situations, but are also used in theoretical mathematics: </p> <ul><li>In practice, one is often interested in the travel distance between two points along roads, rather than as the crow flies. In a <a href="/wiki/Grid_plan" title="Grid plan">grid plan</a>, the travel distance between street corners is given by the <a href="/wiki/Manhattan_distance" class="mw-redirect" title="Manhattan distance">Manhattan distance</a>: the number of east–west and north–south blocks one must traverse to get between those two points.</li> <li>Chessboard distance, formalized as <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a>, is the minimum number of moves a <a href="/wiki/King_(chess)" title="King (chess)">king</a> must make on a <a href="/wiki/Chessboard" title="Chessboard">chessboard</a> in order to travel between two squares.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Metaphorical_distances">Metaphorical distances</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=7" title="Edit section: Metaphorical distances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many abstract notions of distance used in mathematics, science and engineering represent a degree of difference or separation between similar objects. This page gives a few examples. </p> <div class="mw-heading mw-heading3"><h3 id="Statistical_distances">Statistical distances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=8" title="Edit section: Statistical distances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Statistical_distance" title="Statistical distance">Statistical distance</a></div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Information_geometry" title="Information geometry">information geometry</a>, <a href="/wiki/Statistical_distance" title="Statistical distance">statistical distances</a> measure the degree of difference between two <a href="/wiki/Probability_distribution" title="Probability distribution">probability distributions</a>. There are many kinds of statistical distances, typically formalized as <a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergences</a>; these allow a set of probability distributions to be understood as a <a href="/wiki/Space_(mathematics)" title="Space (mathematics)">geometrical object</a> called a <a href="/wiki/Statistical_manifold" title="Statistical manifold">statistical manifold</a>. The most elementary is the <a href="/wiki/Squared_Euclidean_distance" class="mw-redirect" title="Squared Euclidean distance">squared Euclidean distance</a>, which is minimized by the <a href="/wiki/Least_squares" title="Least squares">least squares</a> method; this is the most basic <a href="/wiki/Bregman_divergence" title="Bregman divergence">Bregman divergence</a>. The most important in <a href="/wiki/Information_theory" title="Information theory">information theory</a> is the <a href="/wiki/Relative_entropy" class="mw-redirect" title="Relative entropy">relative entropy</a> (<a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="Kullback–Leibler divergence">Kullback–Leibler divergence</a>), which allows one to analogously study <a href="/wiki/Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimation</a> geometrically; this is an example of both an <a href="/wiki/F-divergence" title="F-divergence"><i>f</i>-divergence</a> and a Bregman divergence (and in fact the only example which is both). Statistical manifolds corresponding to Bregman divergences are <a href="/wiki/Flat_manifold" title="Flat manifold">flat manifolds</a> in the corresponding geometry, allowing an analog of the <a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a> (which holds for squared Euclidean distance) to be used for <a href="/wiki/Linear_inverse_problem" class="mw-redirect" title="Linear inverse problem">linear inverse problems</a> in inference by <a href="/wiki/Optimization_theory" class="mw-redirect" title="Optimization theory">optimization theory</a>. </p><p>Other important statistical distances include the <a href="/wiki/Mahalanobis_distance" title="Mahalanobis distance">Mahalanobis distance</a> and the <a href="/wiki/Energy_distance" title="Energy distance">energy distance</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Edit_distances">Edit distances</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=9" title="Edit section: Edit distances"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, an <a href="/wiki/Edit_distance" title="Edit distance">edit distance</a> or <a href="/wiki/String_metric" title="String metric">string metric</a> between two <a href="/wiki/String_(computer_science)" title="String (computer science)">strings</a> measures how different they are. For example, the words "dog" and "dot", which differ by just one letter, are closer than "dog" and "cat", which have no letters in common. This idea is used in <a href="/wiki/Spell_checker" title="Spell checker">spell checkers</a> and in <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>, and is mathematically formalized in a number of different ways, including <a href="/wiki/Levenshtein_distance" title="Levenshtein distance">Levenshtein distance</a>, <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a>, <a href="/wiki/Lee_distance" title="Lee distance">Lee distance</a>, and <a href="/wiki/Jaro%E2%80%93Winkler_distance" title="Jaro–Winkler distance">Jaro–Winkler distance</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Distance_in_graph_theory">Distance in graph theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=10" title="Edit section: Distance in graph theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">Distance (graph theory)</a></div> <p>In a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>, the <a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">distance</a> between two vertices is measured by the length of the shortest <a href="/wiki/Path_(graph_theory)" title="Path (graph theory)">edge path</a> between them. For example, if the graph represents a <a href="/wiki/Social_network" title="Social network">social network</a>, then the idea of <a href="/wiki/Six_degrees_of_separation" title="Six degrees of separation">six degrees of separation</a> can be interpreted mathematically as saying that the distance between any two vertices is at most six. Similarly, the <a href="/wiki/Erd%C5%91s_number" title="Erdős number">Erdős number</a> and the <a href="/wiki/Bacon_number" class="mw-redirect" title="Bacon number">Bacon number</a>—the number of collaborative relationships away a person is from prolific mathematician <a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> and actor <a href="/wiki/Kevin_Bacon" title="Kevin Bacon">Kevin Bacon</a>, respectively—are distances in the graphs whose edges represent mathematical or artistic collaborations. </p> <div class="mw-heading mw-heading3"><h3 id="In_the_social_sciences">In the social sciences</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=11" title="Edit section: In the social sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Psychology" title="Psychology">psychology</a>, <a href="/wiki/Human_geography" title="Human geography">human geography</a>, and the <a href="/wiki/Social_science" title="Social science">social sciences</a>, distance is often theorized not as an objective numerical measurement, but as a qualitative description of a subjective experience.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Psychological_distance" title="Psychological distance">psychological distance</a> is "the different ways in which an object might be removed from" the self along dimensions such as "time, space, social distance, and hypotheticality".<sup id="cite_ref-psych_5-0" class="reference"><a href="#cite_note-psych-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In <a href="/wiki/Sociology" title="Sociology">sociology</a>, <a href="/wiki/Social_distance" title="Social distance">social distance</a> describes the separation between individuals or <a href="/wiki/Social_groups" class="mw-redirect" title="Social groups">social groups</a> in <a href="/wiki/Society" title="Society">society</a> along dimensions such as <a href="/wiki/Social_class" title="Social class">social class</a>, <a href="/wiki/Race_(classification_of_human_beings)" class="mw-redirect" title="Race (classification of human beings)">race</a>/<a href="/wiki/Ethnicity" title="Ethnicity">ethnicity</a>, <a href="/wiki/Gender" title="Gender">gender</a> or <a href="/wiki/Human_sexuality" title="Human sexuality">sexuality</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formalization">Mathematical formalization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=12" title="Edit section: Mathematical formalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Metric_space" title="Metric space">Metric space</a></div> <p>Most of the notions of distance between two points or objects described above are examples of the mathematical idea of a <a href="/wiki/Metric_space" title="Metric space">metric</a>. A <i>metric</i> or <i>distance function</i> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <span class="texhtml mvar" style="font-style:italic;">d</span> which takes pairs of points or objects to <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> and satisfies the following rules: </p> <ol><li>The distance between an object and itself is always zero.</li> <li>The distance between distinct objects is always positive.</li> <li>Distance is <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a>: the distance from <span class="texhtml mvar" style="font-style:italic;">x</span> to <span class="texhtml mvar" style="font-style:italic;">y</span> is always the same as the distance from <span class="texhtml mvar" style="font-style:italic;">y</span> to <span class="texhtml mvar" style="font-style:italic;">x</span>.</li> <li>Distance satisfies the <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a>: if <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="texhtml mvar" style="font-style:italic;">z</span> are three objects, then <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 d(x,z)\leq d(x,y)+d(y,z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7072502a79725a25151d5c96b7310e7dbb97a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.91ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}"></span> This condition can be described informally as "intermediate stops can't speed you up."</li></ol> <p>As an exception, many of the <a href="/wiki/Divergence_(statistics)" title="Divergence (statistics)">divergences</a> used in statistics are not metrics. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Distance_function_animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Distance_function_animation.gif/220px-Distance_function_animation.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Distance_function_animation.gif/330px-Distance_function_animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Distance_function_animation.gif/440px-Distance_function_animation.gif 2x" data-file-width="800" data-file-height="800" /></a><figcaption>Animation visualizing the function (abs(x)^r + abs(y)^r)^(1/r) for various values of r.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Distance_between_sets">Distance between sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=13" title="Edit section: Distance between sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Distance_between_sets.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Distance_between_sets.svg/220px-Distance_between_sets.svg.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Distance_between_sets.svg/330px-Distance_between_sets.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Distance_between_sets.svg/440px-Distance_between_sets.svg.png 2x" data-file-width="200" data-file-height="201" /></a><figcaption>The distances between these three sets do not satisfy the triangle inequality:<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 d(A,B)>d(A,C)+d(C,B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)>d(A,C)+d(C,B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c30d1db0082ceb2240ec12e1c1bd88d37154dce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.663ex; height:2.843ex;" alt="{\displaystyle d(A,B)>d(A,C)+d(C,B)}"></span></figcaption></figure> <p>There are multiple ways of measuring the physical distance between objects that <a href="/wiki/Extension_(metaphysics)" title="Extension (metaphysics)">consist of more than one point</a>: </p> <ul><li>One may measure the distance between representative points such as the <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>; this is used for astronomical distances such as the <a href="/wiki/Lunar_distance_(astronomy)" class="mw-redirect" title="Lunar distance (astronomy)">Earth–Moon distance</a>.</li> <li>One may measure the distance between the closest points of the two objects; in this sense, the <a href="/wiki/Altitude" title="Altitude">altitude</a> of an airplane or spacecraft is its distance from the Earth. The same sense of distance is used in Euclidean geometry to define <a href="/wiki/Distance_from_a_point_to_a_line" title="Distance from a point to a line">distance from a point to a line</a>, <a href="/wiki/Distance_from_a_point_to_a_plane" title="Distance from a point to a plane">distance from a point to a plane</a>, or, more generally, <a href="/wiki/Perpendicular_distance" title="Perpendicular distance">perpendicular distance</a> between <a href="/wiki/Affine_subspace" class="mw-redirect" title="Affine subspace">affine subspaces</a>.</li></ul> <dl><dd>Even more generally, this idea can be used to define the distance between two <a href="/wiki/Subset" title="Subset">subsets</a> of a metric space. The distance between sets <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> is the <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> of the distances between any two of their respective points:<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 d(A,B)=\inf _{x\in A,y\in B}d(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)=\inf _{x\in A,y\in B}d(x,y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9209edb97082243dbaf8e4041de388b3aebe097e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.13ex; height:4.343ex;" alt="{\displaystyle d(A,B)=\inf _{x\in A,y\in B}d(x,y).}"></span> This does not define a metric on the set of such subsets: the distance between overlapping sets is zero, and this distance does not satisfy the triangle inequality for any metric space with two or more points (consider the triple of sets consisting of two distinct singletons and their union).</dd></dl> <ul><li>The <a href="/wiki/Hausdorff_distance" title="Hausdorff distance">Hausdorff distance</a> between two subsets of a metric space can be thought of as measuring how far they are from perfectly overlapping. Somewhat more precisely, the Hausdorff distance between <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> is either the distance from <span class="texhtml mvar" style="font-style:italic;">A</span> to the farthest point of <span class="texhtml mvar" style="font-style:italic;">B</span>, or the distance from <span class="texhtml mvar" style="font-style:italic;">B</span> to the farthest point of <span class="texhtml mvar" style="font-style:italic;">A</span>, whichever is larger. (Here "farthest point" must be interpreted as a supremum.) The Hausdorff distance defines a metric on the set of <a href="/wiki/Compact_space" title="Compact space">compact subsets</a> of a metric space.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Related_ideas">Related ideas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=14" title="Edit section: Related ideas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Length" title="Length">Length</a></div> <p>The word <b>distance</b> is also used for related concepts that are not encompassed by the description "a numerical measurement of how far apart points or objects are". </p> <div class="mw-heading mw-heading3"><h3 id="Distance_travelled">Distance travelled</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=15" title="Edit section: Distance travelled"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>distance travelled</b> by an object is the length of a specific path travelled between two points,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> such as the distance walked while navigating a <a href="/wiki/Maze" title="Maze">maze</a>. This can even be a <b>closed distance</b> along a <a href="/wiki/Closed_curve" class="mw-redirect" title="Closed curve">closed curve</a> which starts and ends at the same point, such as a ball thrown straight up, or the Earth when it completes one <a href="/wiki/Orbit" title="Orbit">orbit</a>. This is formalized mathematically as the <a href="/wiki/Arc_length" title="Arc length">arc length</a> of the curve. </p><p>The distance travelled may also be <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">signed</a>: a "forward" distance is positive and a "backward" distance is negative. </p><p><b>Circular distance</b> is the distance traveled by a point on the circumference of a <a href="/wiki/Wheel" title="Wheel">wheel</a>, which can be useful to consider when designing vehicles or mechanical gears (see also <a href="/wiki/Odometry" title="Odometry">odometry</a>). The circumference of the wheel is <span class="texhtml">2π × radius</span>; if the radius is 1, each revolution of the wheel causes a vehicle to travel <span class="texhtml">2π</span> radians. </p> <div class="mw-heading mw-heading3"><h3 id="Displacement_and_directed_distance">Displacement and directed distance<span class="anchor" id="Displacement"></span><span class="anchor" id="Directed_distance"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=16" title="Edit section: Displacement and directed distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Distancedisplacement.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Distancedisplacement.svg/220px-Distancedisplacement.svg.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Distancedisplacement.svg/330px-Distancedisplacement.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Distancedisplacement.svg/440px-Distancedisplacement.svg.png 2x" data-file-width="323" data-file-height="200" /></a><figcaption>Distance along a path compared with displacement. The Euclidean distance is the length of the displacement vector.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">Displacement (vector)</a></div> <p>The <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> in classical physics measures the change in position of an object during an interval of time. While distance is a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> quantity, or a <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>, displacement is a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a> quantity with both magnitude and <a href="/wiki/Direction_(geometry,_geography)" class="mw-redirect" title="Direction (geometry, geography)">direction</a>. In general, the vector measuring the difference between two locations (the <a href="/wiki/Relative_position" class="mw-redirect" title="Relative position">relative position</a>) is sometimes called the <b>directed distance</b>.<sup id="cite_ref-ittcKU_7-0" class="reference"><a href="#cite_note-ittcKU-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> For example, the directed distance from the <a href="/wiki/New_York_Public_Library_Main_Branch" title="New York Public Library Main Branch">New York City Main Library</a> flag pole to the <a href="/wiki/Statue_of_Liberty" title="Statue of Liberty">Statue of Liberty</a> flag pole has: </p> <ul><li>A starting point: library flag pole</li> <li>An ending point: statue flag pole</li> <li>A direction: -38°</li> <li>A distance: 8.72 km</li></ul> <div class="mw-heading mw-heading3"><h3 id="Signed_distance">Signed distance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=17" title="Edit section: Signed distance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="excerpt-block"><style data-mw-deduplicate="TemplateStyles:r1066933788">.mw-parser-output .excerpt-hat .mw-editsection-like{font-style:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="/wiki/Signed_distance_function" title="Signed distance function">Signed distance function</a>.<span class="mw-editsection-like plainlinks"><span class="mw-editsection-bracket">[</span><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Signed_distance_function&action=edit">edit</a><span class="mw-editsection-bracket">]</span></span></div><div class="excerpt"> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and its applications, the <a href="/wiki/Signed_distance_function" title="Signed distance function">signed distance function</a> or signed distance field (SDF) is the <a href="/wiki/Orthogonal_distance" class="mw-redirect" title="Orthogonal distance">orthogonal distance</a> of a given point <i>x</i> to the <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> Ω in a <a href="/wiki/Metric_space" title="Metric space">metric space</a> (such as the surface of a geometric shape), with the <a href="/wiki/Sign_(mathematics)" title="Sign (mathematics)">sign</a> determined by whether or not <i>x</i> is in the <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a> of Ω. The <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> has positive values at points <i>x</i> inside Ω, it decreases in value as <i>x</i> approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside).<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The concept also sometimes goes by the name oriented distance function/field.</div></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=18" title="Edit section: See also"><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 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decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Distance" class="extiw" title="q:Special:Search/Distance">Distance</a></b></i>.</div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Absolute_difference" title="Absolute difference">Absolute difference</a></li> <li><a href="/wiki/Astronomical_system_of_units" title="Astronomical system of units">Astronomical system of units</a></li> <li><a href="/wiki/Color_difference" title="Color difference">Color difference</a></li> <li><a href="/wiki/Closeness_(mathematics)" title="Closeness (mathematics)">Closeness (mathematics)</a></li> <li><a href="/wiki/Distance_geometry_problem" class="mw-redirect" title="Distance geometry problem">Distance geometry problem</a></li> <li><a href="/wiki/Dijkstra%27s_algorithm" title="Dijkstra's algorithm">Dijkstra's algorithm</a></li> <li><a href="/wiki/Distance_matrix" title="Distance matrix">Distance matrix</a></li> <li><a href="/wiki/Distance_set" title="Distance set">Distance set</a></li> <li><a href="/wiki/Engineering_tolerance" title="Engineering tolerance">Engineering tolerance</a></li> <li><a href="/wiki/Multiplicative_distance" title="Multiplicative distance">Multiplicative distance</a></li> <li><a href="/wiki/Optical_path_length" title="Optical path length">Optical path length</a></li> <li><a href="/wiki/Orders_of_magnitude_(length)" title="Orders of magnitude (length)">Orders of magnitude (length)</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proxemics" title="Proxemics">Proxemics</a> – physical distance between people</li> <li><a href="/wiki/Signed_distance_function" title="Signed distance function">Signed distance function</a></li> <li><a href="/wiki/Similarity_measure" title="Similarity measure">Similarity measure</a></li> <li><a href="/wiki/Social_distancing" title="Social distancing">Social distancing</a></li> <li><a href="/wiki/Vertical_distance" class="mw-redirect" title="Vertical distance">Vertical distance</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Library_support">Library support</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=19" title="Edit section: Library support"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python (programming language)</a> <ul><li><a rel="nofollow" class="external text" href="https://docs.scipy.org/doc/scipy/reference/spatial.distance.html">SciPy</a> -Distance computations (<code>scipy.spatial.distance</code>)</li></ul></li> <li><a href="/wiki/Julia_(programming_language)" title="Julia (programming language)">Julia (programming language)</a> <ul><li><a rel="nofollow" class="external text" href="https://github.com/JuliaStats/Distances.jl">Julia Statistics Distance</a> -A Julia package for evaluating distances (metrics) between vectors.</li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=20" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSchnall2014" class="citation book cs1">Schnall, Simone (2014). "Are there basic metaphors?". <i>The power of metaphor: Examining its influence on social life</i>. American Psychological Association. pp. 225–247. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2F14278-010">10.1037/14278-010</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Are+there+basic+metaphors%3F&rft.btitle=The+power+of+metaphor%3A+Examining+its+influence+on+social+life.&rft.pages=225-247&rft.pub=American+Psychological+Association&rft.date=2014&rft_id=info%3Adoi%2F10.1037%2F14278-010&rft.aulast=Schnall&rft.aufirst=Simone&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> <li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Distance.html">"Distance"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Distance&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDistance.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/algebra/distance-2-points.html">"Distance Between 2 Points"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. 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Retrieved <span class="nowrap">2020-07-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.hawaii.edu&rft.atitle=SOCIAL+DISTANCES&rft_id=https%3A%2F%2Fwww.hawaii.edu%2Fpowerkills%2FTCH.CHAP16.HTM&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> <li id="cite_note-psych-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-psych_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTropeLiberman2010" class="citation journal cs1">Trope Y, Liberman N (April 2010). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3152826">"Construal-level theory of psychological distance"</a>. <i>Psychological Review</i>. <b>117</b> (2): 440–63. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1037%2Fa0018963">10.1037/a0018963</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3152826">3152826</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20438233">20438233</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Psychological+Review&rft.atitle=Construal-level+theory+of+psychological+distance&rft.volume=117&rft.issue=2&rft.pages=440-63&rft.date=2010-04&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3152826%23id-name%3DPMC&rft_id=info%3Apmid%2F20438233&rft_id=info%3Adoi%2F10.1037%2Fa0018963&rft.aulast=Trope&rft.aufirst=Y&rft.au=Liberman%2C+N&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC3152826&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.khanacademy.org/science/physics/one-dimensional-motion/displacement-velocity-time/a/what-is-displacement">"What is displacement? (article)"</a>. <i>Khan Academy</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-07-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Khan+Academy&rft.atitle=What+is+displacement%3F+%28article%29&rft_id=https%3A%2F%2Fwww.khanacademy.org%2Fscience%2Fphysics%2Fone-dimensional-motion%2Fdisplacement-velocity-time%2Fa%2Fwhat-is-displacement&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> <li id="cite_note-ittcKU-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-ittcKU_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20161110044317/http://www.ittc.ku.edu/~jstiles/220/handouts/The%20Directed%20Distance.pdf">"The Directed Distance"</a> <span class="cs1-format">(PDF)</span>. <i>Information and Telecommunication Technology Center</i>. University of Kansas. Archived from <a rel="nofollow" class="external text" href="http://www.ittc.ku.edu/~jstiles/220/handouts/The%20Directed%20Distance.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 10 November 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">18 September</span> 2018</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Information+and+Telecommunication+Technology+Center&rft.atitle=The+Directed+Distance&rft_id=http%3A%2F%2Fwww.ittc.ku.edu%2F~jstiles%2F220%2Fhandouts%2FThe%2520Directed%2520Distance.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChan,_T.Zhu,_W.2005" class="citation conference cs1">Chan, T.; Zhu, W. (2005). <i>Level set based shape prior segmentation</i>. IEEE Computer Society Conference on Computer Vision and Pattern Recognition. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FCVPR.2005.212">10.1109/CVPR.2005.212</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Level+set+based+shape+prior+segmentation&rft.date=2005&rft_id=info%3Adoi%2F10.1109%2FCVPR.2005.212&rft.au=Chan%2C+T.&rft.au=Zhu%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalladi,_R.Sethian,_J.A.Vemuri,_B.C.1995" class="citation journal cs1">Malladi, R.; Sethian, J.A.; Vemuri, B.C. (1995). "Shape modeling with front propagation: a level set approach". <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>. <b>17</b> (2): 158–175. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.33.2443">10.1.1.33.2443</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/10.1109%2F34.368173">10.1109/34.368173</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:9505101">9505101</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Pattern+Analysis+and+Machine+Intelligence&rft.atitle=Shape+modeling+with+front+propagation%3A+a+level+set+approach&rft.volume=17&rft.issue=2&rft.pages=158-175&rft.date=1995&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.33.2443%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A9505101%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2F34.368173&rft.au=Malladi%2C+R.&rft.au=Sethian%2C+J.A.&rft.au=Vemuri%2C+B.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Distance&action=edit&section=21" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDezaDeza2006" class="citation book cs1"><a href="/wiki/Elena_Deza" title="Elena Deza">Deza E</a>, <a href="/wiki/Michel_Deza" title="Michel Deza">Deza M</a> (2006). <i>Dictionary of Distances</i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-52087-2" title="Special:BookSources/0-444-52087-2"><bdi>0-444-52087-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dictionary+of+Distances&rft.pub=Elsevier&rft.date=2006&rft.isbn=0-444-52087-2&rft.aulast=Deza&rft.aufirst=E&rft.au=Deza%2C+M&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADistance" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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