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mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">ابزارهای شخصی</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="منوی کاربری" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_fa.wikipedia.org&uselang=fa"><span>کمک مالی</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D9%88%DB%8C%DA%98%D9%87:%D8%A7%DB%8C%D8%AC%D8%A7%D8%AF_%D8%AD%D8%B3%D8%A7%D8%A8_%DA%A9%D8%A7%D8%B1%D8%A8%D8%B1%DB%8C&returnto=%D9%81%D8%A7%D8%B5%D9%84%D9%87" title="از شما دعوت می‌شود که یک حساب ایجاد کنید و وارد شوید؛ هرچند که این کار اختیاری است."><span class="vector-icon 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class="mw-list-item"><a href="/wiki/%D9%88%DB%8C%DA%98%D9%87:%D9%85%D8%B4%D8%A7%D8%B1%DA%A9%D8%AA%E2%80%8C%D9%87%D8%A7%DB%8C_%D9%85%D9%86" title="فهرست ویرایش‌ها انجام شده از این نشانی آی‌پی [y]" accesskey="y"><span>مشارکت‌ها</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D9%88%DB%8C%DA%98%D9%87:%D8%A8%D8%AD%D8%AB_%D9%85%D9%86" title="بحث پیرامون ویرایش‌های این نشانی آی‌پی [n]" accesskey="n"><span>بحث</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="وبگاه"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="فهرست" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">فهرست</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">انتقال به نوار کناری</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">نهفتن</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">بخش آغازین</div> </a> </li> <li id="toc-در_هندسه" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#در_هندسه"> <div class="vector-toc-text"> <span class="vector-toc-numb">۱</span> <span>در هندسه</span> </div> </a> <button aria-controls="toc-در_هندسه-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>تغییر وضعیت زیربخش‌های در هندسه</span> </button> <ul id="toc-در_هندسه-sublist" class="vector-toc-list"> <li id="toc-فاصلهٔ_اقلیدسی" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#فاصلهٔ_اقلیدسی"> <div class="vector-toc-text"> <span class="vector-toc-numb">۱.۱</span> <span>فاصلهٔ اقلیدسی</span> </div> </a> <ul id="toc-فاصلهٔ_اقلیدسی-sublist" class="vector-toc-list"> <li id="toc-فاصله_میان_نقطه_و_خط" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#فاصله_میان_نقطه_و_خط"> <div class="vector-toc-text"> <span class="vector-toc-numb">۱.۱.۱</span> <span>فاصله میان نقطه و خط</span> </div> </a> <ul id="toc-فاصله_میان_نقطه_و_خط-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-فاصلهٔ_منهتنی" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#فاصلهٔ_منهتنی"> <div class="vector-toc-text"> <span class="vector-toc-numb">۱.۲</span> <span>فاصلهٔ منهتنی</span> </div> </a> <ul id="toc-فاصلهٔ_منهتنی-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-فاصلهٔ_شطرنجی_(فاصلهٔ_چبیشف)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#فاصلهٔ_شطرنجی_(فاصلهٔ_چبیشف)"> <div class="vector-toc-text"> <span class="vector-toc-numb">۱.۳</span> <span>فاصلهٔ شطرنجی (<b>فاصلهٔ چبیشف)</b></span> </div> </a> <ul id="toc-فاصلهٔ_شطرنجی_(فاصلهٔ_چبیشف)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-تعمیم_و_فاصلهٔ_مینکوفسکی" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#تعمیم_و_فاصلهٔ_مینکوفسکی"> <div class="vector-toc-text"> <span class="vector-toc-numb">۱.۴</span> <span>تعمیم و فاصلهٔ مینکوفسکی</span> </div> </a> <ul id="toc-تعمیم_و_فاصلهٔ_مینکوفسکی-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-در_گراف" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#در_گراف"> <div class="vector-toc-text"> <span class="vector-toc-numb">۲</span> <span>در گراف</span> </div> </a> <ul id="toc-در_گراف-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-منابع" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#منابع"> <div class="vector-toc-text"> <span class="vector-toc-numb">۳</span> <span>منابع</span> </div> </a> <ul id="toc-منابع-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="فهرست" 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="تغییر وضعیت فهرست محتویات" > <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">تغییر وضعیت فهرست محتویات</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">فاصله</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="به مقاله‌ای به یک زبان دیگر بروید. به ۹۸ زبان در دسترس است." > <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">۹۸ زبان</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–آفریکانس" lang="af" hreflang="af" data-title="Afstand" data-language-autonym="Afrikaans" data-language-local-name="آفریکانس" 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–امهری" lang="am" hreflang="am" data-title="ርቀት" data-language-autonym="አማርኛ" data-language-local-name="امهری" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Distancia" title="Distancia–آراگونی" lang="an" hreflang="an" data-title="Distancia" data-language-autonym="Aragonés" data-language-local-name="آراگونی" class="interlanguage-link-target"><span>Aragonés</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–عربی" lang="ar" hreflang="ar" data-title="مسافة" data-language-autonym="العربية" data-language-local-name="عربی" class="interlanguage-link-target"><span>العربية</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–عربی مراکشی" lang="ary" hreflang="ary" data-title="بعودية" data-language-autonym="الدارجة" data-language-local-name="عربی مراکشی" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distancia" title="Distancia–آستوری" lang="ast" hreflang="ast" data-title="Distancia" data-language-autonym="Asturianu" data-language-local-name="آستوری" 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–ترکی آذربایجانی" lang="az" hreflang="az" data-title="Məsafə" data-language-autonym="Azərbaycanca" data-language-local-name="ترکی آذربایجانی" 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-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–باشقیری" lang="ba" hreflang="ba" data-title="Алыҫлыҡ" data-language-autonym="Башҡортса" data-language-local-name="باشقیری" class="interlanguage-link-target"><span>Башҡортса</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-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–بلاروسی" lang="be" hreflang="be" data-title="Адлегласць" data-language-autonym="Беларуская" data-language-local-name="بلاروسی" 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–بلاروسی (دستورخط کلاسیک)" lang="be-tarask" hreflang="be-tarask" data-title="Адлегласьць" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="بلاروسی (دستورخط کلاسیک)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</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–بلغاری" lang="bg" hreflang="bg" data-title="Разстояние" data-language-autonym="Български" data-language-local-name="بلغاری" class="interlanguage-link-target"><span>Български</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-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–بنگالی" lang="bn" hreflang="bn" data-title="দূরত্ব" data-language-autonym="বাংলা" data-language-local-name="بنگالی" class="interlanguage-link-target"><span>বাংলা</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–کاتالان" lang="ca" hreflang="ca" data-title="Distància" data-language-autonym="Català" data-language-local-name="کاتالان" class="interlanguage-link-target"><span>Català</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–کردی مرکزی" lang="ckb" hreflang="ckb" data-title="دووری" data-language-autonym="کوردی" data-language-local-name="کردی مرکزی" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vzd%C3%A1lenost" title="Vzdálenost–چکی" lang="cs" hreflang="cs" data-title="Vzdálenost" data-language-autonym="Čeština" data-language-local-name="چکی" class="interlanguage-link-target"><span>Čeština</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–چوواشی" lang="cv" hreflang="cv" data-title="Инçĕш" data-language-autonym="Чӑвашла" data-language-local-name="چوواشی" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Afstandsformlen" title="Afstandsformlen–دانمارکی" lang="da" hreflang="da" data-title="Afstandsformlen" data-language-autonym="Dansk" data-language-local-name="دانمارکی" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abstand" title="Abstand–آلمانی" lang="de" hreflang="de" data-title="Abstand" data-language-autonym="Deutsch" data-language-local-name="آلمانی" class="interlanguage-link-target"><span>Deutsch</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)–یونانی" lang="el" hreflang="el" data-title="Απόσταση (γεωμετρία)" data-language-autonym="Ελληνικά" data-language-local-name="یونانی" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Distance" title="Distance–انگلیسی" lang="en" hreflang="en" data-title="Distance" data-language-autonym="English" data-language-local-name="انگلیسی" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Distanco" title="Distanco–اسپرانتو" lang="eo" hreflang="eo" data-title="Distanco" data-language-autonym="Esperanto" data-language-local-name="اسپرانتو" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Distancia" title="Distancia–اسپانیایی" lang="es" hreflang="es" data-title="Distancia" data-language-autonym="Español" data-language-local-name="اسپانیایی" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kaugus" title="Kaugus–استونیایی" lang="et" hreflang="et" data-title="Kaugus" data-language-autonym="Eesti" data-language-local-name="استونیایی" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Distantzia" title="Distantzia–باسکی" lang="eu" hreflang="eu" data-title="Distantzia" data-language-autonym="Euskara" data-language-local-name="باسکی" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/V%C3%A4limatka" title="Välimatka–فنلاندی" lang="fi" hreflang="fi" data-title="Välimatka" data-language-autonym="Suomi" data-language-local-name="فنلاندی" class="interlanguage-link-target"><span>Suomi</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)–فرانسوی" lang="fr" hreflang="fr" data-title="Distance (mathématiques)" data-language-autonym="Français" data-language-local-name="فرانسوی" 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–گالیسیایی" lang="gl" hreflang="gl" data-title="Distancia" data-language-autonym="Galego" data-language-local-name="گالیسیایی" class="interlanguage-link-target"><span>Galego</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–عبری" lang="he" hreflang="he" data-title="מרחק" data-language-autonym="עברית" data-language-local-name="عبری" class="interlanguage-link-target"><span>עברית</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–هندی" lang="hi" hreflang="hi" data-title="दूरी" data-language-autonym="हिन्दी" data-language-local-name="هندی" 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–کروات" lang="hr" hreflang="hr" data-title="Daljina" data-language-autonym="Hrvatski" data-language-local-name="کروات" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Distans" title="Distans–هائیتیایی" lang="ht" hreflang="ht" data-title="Distans" data-language-autonym="Kreyòl ayisyen" data-language-local-name="هائیتیایی" class="interlanguage-link-target"><span>Kreyòl ayisyen</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–مجاری" lang="hu" hreflang="hu" data-title="Távolság" data-language-autonym="Magyar" data-language-local-name="مجاری" class="interlanguage-link-target"><span>Magyar</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)–ارمنی" lang="hy" hreflang="hy" data-title="Հեռավորություն (երկրաչափություն)" data-language-autonym="Հայերեն" data-language-local-name="ارمنی" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Distantia" title="Distantia–اینترلینگوا" lang="ia" hreflang="ia" data-title="Distantia" data-language-autonym="Interlingua" data-language-local-name="اینترلینگوا" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Penyauh" title="Penyauh–ایبانی" lang="iba" hreflang="iba" data-title="Penyauh" data-language-autonym="Jaku Iban" data-language-local-name="ایبانی" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Jarak" title="Jarak–اندونزیایی" lang="id" hreflang="id" data-title="Jarak" data-language-autonym="Bahasa Indonesia" data-language-local-name="اندونزیایی" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Disto" title="Disto–ایدو" lang="io" hreflang="io" data-title="Disto" data-language-autonym="Ido" data-language-local-name="ایدو" class="interlanguage-link-target"><span>Ido</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–ایسلندی" lang="is" hreflang="is" data-title="Fjarlægð" data-language-autonym="Íslenska" data-language-local-name="ایسلندی" 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)–ایتالیایی" lang="it" hreflang="it" data-title="Distanza (matematica)" data-language-autonym="Italiano" data-language-local-name="ایتالیایی" class="interlanguage-link-target"><span>Italiano</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–ژاپنی" lang="ja" hreflang="ja" data-title="距離" data-language-autonym="日本語" data-language-local-name="ژاپنی" class="interlanguage-link-target"><span>日本語</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–گرجی" lang="ka" hreflang="ka" data-title="მანძილი" data-language-autonym="ქართული" data-language-local-name="گرجی" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Ameccaq" title="Ameccaq–قبایلی" lang="kab" hreflang="kab" data-title="Ameccaq" data-language-autonym="Taqbaylit" data-language-local-name="قبایلی" class="interlanguage-link-target"><span>Taqbaylit</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–قزاقی" lang="kk" hreflang="kk" data-title="Арақашықтық" data-language-autonym="Қазақша" data-language-local-name="قزاقی" class="interlanguage-link-target"><span>Қазақша</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–کانارا" lang="kn" hreflang="kn" data-title="ದೂರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="کانارا" class="interlanguage-link-target"><span>ಕನ್ನಡ</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–کره‌ای" lang="ko" hreflang="ko" data-title="거리" data-language-autonym="한국어" data-language-local-name="کره‌ای" class="interlanguage-link-target"><span>한국어</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–قره‌چایی‐بالکاری" lang="krc" hreflang="krc" data-title="Узакълыкъ" data-language-autonym="Къарачай-малкъар" data-language-local-name="قره‌چایی‐بالکاری" class="interlanguage-link-target"><span>Къарачай-малкъар</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–کردی" lang="ku" hreflang="ku" data-title="Dûrahî" data-language-autonym="Kurdî" data-language-local-name="کردی" 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–قرقیزی" lang="ky" hreflang="ky" data-title="Аралык" data-language-autonym="Кыргызча" data-language-local-name="قرقیزی" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Ofstand" title="Ofstand–لوگزامبورگی" lang="lb" hreflang="lb" data-title="Ofstand" data-language-autonym="Lëtzebuergesch" data-language-local-name="لوگزامبورگی" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Aafstandj" title="Aafstandj–لیمبورگی" lang="li" hreflang="li" data-title="Aafstandj" data-language-autonym="Limburgs" data-language-local-name="لیمبورگی" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Atstumas" title="Atstumas–لیتوانیایی" lang="lt" hreflang="lt" data-title="Atstumas" data-language-autonym="Lietuvių" data-language-local-name="لیتوانیایی" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Att%C4%81lums" title="Attālums–لتونیایی" lang="lv" hreflang="lv" data-title="Attālums" data-language-autonym="Latviešu" data-language-local-name="لتونیایی" class="interlanguage-link-target"><span>Latviešu</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–مقدونی" lang="mk" hreflang="mk" data-title="Растојание" data-language-autonym="Македонски" data-language-local-name="مقدونی" 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–مراتی" lang="mr" hreflang="mr" data-title="अंतर" data-language-autonym="मराठी" data-language-local-name="مراتی" 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–مالایی" lang="ms" hreflang="ms" data-title="Jarak" data-language-autonym="Bahasa Melayu" data-language-local-name="مالایی" 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–برمه‌ای" lang="my" hreflang="my" data-title="အကွာအဝေး" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="برمه‌ای" 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–هلندی" lang="nl" hreflang="nl" data-title="Afstand" data-language-autonym="Nederlands" data-language-local-name="هلندی" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Avstand" title="Avstand–نروژی نی‌نُشک" lang="nn" hreflang="nn" data-title="Avstand" data-language-autonym="Norsk nynorsk" data-language-local-name="نروژی نی‌نُشک" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Avstand" title="Avstand–نروژی بوک‌مُل" lang="nb" hreflang="nb" data-title="Avstand" data-language-autonym="Norsk bokmål" data-language-local-name="نروژی بوک‌مُل" class="interlanguage-link-target"><span>Norsk bokmål</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–پنجابی" lang="pa" hreflang="pa" data-title="ਪੈਂਡਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="پنجابی" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</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–لهستانی" lang="pl" hreflang="pl" data-title="Odległość" data-language-autonym="Polski" data-language-local-name="لهستانی" class="interlanguage-link-target"><span>Polski</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–پشتو" lang="ps" hreflang="ps" data-title="واټن" data-language-autonym="پښتو" data-language-local-name="پشتو" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dist%C3%A2ncia" title="Distância–پرتغالی" lang="pt" hreflang="pt" data-title="Distância" data-language-autonym="Português" data-language-local-name="پرتغالی" 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–روسی" lang="ru" hreflang="ru" data-title="Расстояние" data-language-autonym="Русский" data-language-local-name="روسی" 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–اسکاتلندی" lang="sco" hreflang="sco" data-title="Distance" data-language-autonym="Scots" data-language-local-name="اسکاتلندی" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Rastojanje" title="Rastojanje–صرب و کرواتی" lang="sh" hreflang="sh" data-title="Rastojanje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="صرب و کرواتی" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Distance" title="Distance–انگلیسی ساده" lang="en-simple" hreflang="en-simple" data-title="Distance" data-language-autonym="Simple English" data-language-local-name="انگلیسی ساده" 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–اسلواکی" lang="sk" hreflang="sk" data-title="Vzdialenosť" data-language-autonym="Slovenčina" data-language-local-name="اسلواکی" 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–اسلوونیایی" lang="sl" hreflang="sl" data-title="Razdalja" data-language-autonym="Slovenščina" data-language-local-name="اسلوونیایی" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Nhambwe" title="Nhambwe–شونایی" lang="sn" hreflang="sn" data-title="Nhambwe" data-language-autonym="ChiShona" data-language-local-name="شونایی" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Distanca" title="Distanca–آلبانیایی" lang="sq" hreflang="sq" data-title="Distanca" data-language-autonym="Shqip" data-language-local-name="آلبانیایی" class="interlanguage-link-target"><span>Shqip</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–صربی" lang="sr" hreflang="sr" data-title="Растојање" data-language-autonym="Српски / srpski" data-language-local-name="صربی" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Avst%C3%A5nd" title="Avstånd–سوئدی" lang="sv" hreflang="sv" data-title="Avstånd" data-language-autonym="Svenska" data-language-local-name="سوئدی" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Umbali" title="Umbali–سواحیلی" lang="sw" hreflang="sw" data-title="Umbali" data-language-autonym="Kiswahili" data-language-local-name="سواحیلی" class="interlanguage-link-target"><span>Kiswahili</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–تامیلی" lang="ta" hreflang="ta" data-title="தூரம்" data-language-autonym="தமிழ்" data-language-local-name="تامیلی" class="interlanguage-link-target"><span>தமிழ்</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–تلوگویی" lang="te" hreflang="te" data-title="దూరం" data-language-autonym="తెలుగు" data-language-local-name="تلوگویی" class="interlanguage-link-target"><span>తెలుగు</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–تاجیکی" lang="tg" hreflang="tg" data-title="Масофа" data-language-autonym="Тоҷикӣ" data-language-local-name="تاجیکی" class="interlanguage-link-target"><span>Тоҷикӣ</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–تایلندی" lang="th" hreflang="th" data-title="ระยะทาง" data-language-autonym="ไทย" data-language-local-name="تایلندی" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Distansiya" title="Distansiya–تاگالوگی" lang="tl" hreflang="tl" data-title="Distansiya" data-language-autonym="Tagalog" data-language-local-name="تاگالوگی" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Mesafe" title="Mesafe–ترکی استانبولی" lang="tr" hreflang="tr" data-title="Mesafe" data-language-autonym="Türkçe" data-language-local-name="ترکی استانبولی" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-ts mw-list-item"><a href="https://ts.wikipedia.org/wiki/Mpfhuka" title="Mpfhuka–تسونگایی" lang="ts" hreflang="ts" data-title="Mpfhuka" data-language-autonym="Xitsonga" data-language-local-name="تسونگایی" class="interlanguage-link-target"><span>Xitsonga</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–اوکراینی" lang="uk" hreflang="uk" data-title="Відстань" data-language-autonym="Українська" data-language-local-name="اوکراینی" 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–اردو" lang="ur" hreflang="ur" data-title="فاصلہ" data-language-autonym="اردو" data-language-local-name="اردو" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Masofa" title="Masofa–ازبکی" lang="uz" hreflang="uz" data-title="Masofa" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="ازبکی" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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–ویتنامی" lang="vi" hreflang="vi" data-title="Khoảng cách" data-language-autonym="Tiếng Việt" data-language-local-name="ویتنامی" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Distansiya" title="Distansiya–وارایی" lang="war" hreflang="war" data-title="Distansiya" data-language-autonym="Winaray" data-language-local-name="وارایی" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%B7%9D%E7%A6%BB" title="距离–وو" lang="wuu" hreflang="wuu" data-title="距离" data-language-autonym="吴语" data-language-local-name="وو" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/Umgama" title="Umgama–خوسایی" lang="xh" hreflang="xh" data-title="Umgama" data-language-autonym="IsiXhosa" data-language-local-name="خوسایی" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%95%D7%95%D7%99%D7%99%D7%98%D7%A7%D7%99%D7%99%D7%98" title="ווייטקייט–یدی" lang="yi" hreflang="yi" data-title="ווייטקייט" data-language-autonym="ייִדיש" data-language-local-name="یدی" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B7%9D%E7%A6%BB" title="距离–چینی" lang="zh" hreflang="zh" data-title="距离" data-language-autonym="中文" data-language-local-name="چینی" class="interlanguage-link-target"><span>中文</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–چینی ادبی" lang="lzh" hreflang="lzh" data-title="距" data-language-autonym="文言" data-language-local-name="چینی ادبی" class="interlanguage-link-target"><span>文言</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–مین نان" lang="nan" hreflang="nan" data-title="Kū-lī" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="مین نان" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%B7%9D%E9%9B%A2" title="距離–کانتونی" lang="yue" hreflang="yue" data-title="距離" data-language-autonym="粵語" data-language-local-name="کانتونی" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q126017#sitelinks-wikipedia" title="ویرایش پیوندهای بین‌زبانی" class="wbc-editpage">ویرایش پیوندها</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="فضاهای نام"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87" title="دیدن صفحهٔ محتویات [c]" accesskey="c"><span>مقاله</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D8%A8%D8%AD%D8%AB:%D9%81%D8%A7%D8%B5%D9%84%D9%87" rel="discussion" title="گفتگو پیرامون محتوای صفحه [t]" accesskey="t"><span>بحث</span></a></li> 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mw-list-item"><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=history"><span>نمایش تاریخچه</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> عمومی </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D9%88%DB%8C%DA%98%D9%87:%D9%BE%DB%8C%D9%88%D9%86%D8%AF_%D8%A8%D9%87_%D8%A7%DB%8C%D9%86_%D8%B5%D9%81%D8%AD%D9%87/%D9%81%D8%A7%D8%B5%D9%84%D9%87" title="فهرست همهٔ صفحه‌هایی که به این صفحه پیوند می‌دهند [j]" accesskey="j"><span>پیوندها به این صفحه</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D9%88%DB%8C%DA%98%D9%87:%D8%AA%D8%BA%DB%8C%DB%8C%D8%B1%D8%A7%D8%AA_%D9%85%D8%B1%D8%AA%D8%A8%D8%B7/%D9%81%D8%A7%D8%B5%D9%84%D9%87" rel="nofollow" title="تغییرات اخیر صفحه‌هایی که این صفحه به آن‌ها پیوند دارد [k]" accesskey="k"><span>تغییرات مرتبط</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/ویکی‌پدیا:بارگذاری" title="بارگذاری تصاویر و پرونده‌های دیگر [u]" accesskey="u"><span>بارگذاری پرونده</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D9%88%DB%8C%DA%98%D9%87:%D8%B5%D9%81%D8%AD%D9%87%E2%80%8C%D9%87%D8%A7%DB%8C_%D9%88%DB%8C%DA%98%D9%87" title="فهرستی از همهٔ صفحه‌های ویژه [q]" accesskey="q"><span>صفحه‌های ویژه</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&oldid=40624652" title="پیوند پایدار به این نسخه از این صفحه"><span>پیوند پایدار</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=info" title="اطلاعات بیشتر دربارهٔ این صفحه"><span>اطلاعات صفحه</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D9%88%DB%8C%DA%98%D9%87:%DB%8C%D8%A7%D8%AF%DA%A9%D8%B1%D8%AF&page=%D9%81%D8%A7%D8%B5%D9%84%D9%87&id=40624652&wpFormIdentifier=titleform" title="اطلاعات در خصوص چگونگی یادکرد این صفحه"><span>یادکرد این صفحه</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D9%88%DB%8C%DA%98%D9%87:UrlShortener&url=https%3A%2F%2Ffa.wikipedia.org%2Fwiki%2F%25D9%2581%25D8%25A7%25D8%25B5%25D9%2584%25D9%2587"><span>دریافت نشانی کوتاه‌شده</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D9%88%DB%8C%DA%98%D9%87:QrCode&url=https%3A%2F%2Ffa.wikipedia.org%2Fwiki%2F%25D9%2581%25D8%25A7%25D8%25B5%25D9%2584%25D9%2587"><span>بارگیری کد QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> نسخه‌برداری </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D9%88%DB%8C%DA%98%D9%87:DownloadAsPdf&page=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=show-download-screen"><span>بارگیری به‌صورت PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&printable=yes" title="نسخهٔ قابل چاپ این صفحه [p]" accesskey="p"><span>نسخهٔ قابل چاپ</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> در پروژه‌های دیگر </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Distance" hreflang="en"><span>ویکی‌انبار</span></a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://fa.wikiquote.org/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87" hreflang="fa"><span>ویکی‌گفتاورد</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q126017" title="پیوند به آیتم متصل‌شدۀ مخزن داده‌ها [g]" accesskey="g"><span>آیتم ویکی‌داده</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="ابزارهای صفحه"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="ظاهر"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">ظاهر</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">انتقال به نوار کناری</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">نهفتن</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">از ویکی‌پدیا، دانشنامهٔ آزاد</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-rtl mw-parser-output" lang="fa" dir="rtl"><style data-mw-deduplicate="TemplateStyles:r35841839">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-right:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">برای دیگر کاربردها، <a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_(%D8%A7%D8%A8%D9%87%D8%A7%D9%85%E2%80%8C%D8%B2%D8%AF%D8%A7%DB%8C%DB%8C)" class="mw-disambig" title="فاصله (ابهام‌زدایی)">فاصله (ابهام‌زدایی)</a> را ببینید.</div> <p><b>فاصِله</b> یک <a href="/wiki/%DA%A9%D9%85%DB%8C%D8%AA_%D9%86%D8%B1%D8%AF%D9%87%E2%80%8C%D8%A7%DB%8C" title="کمیت نرده‌ای">کمیت عددی</a> برای تعیین مقدار دوری یا نزدیکی دو چیز است. </p><p>«فاصله» در زمینه‌های مختلفی تعریف می‌شود. به عنوان مثال، فاصلهٔ دو جسم از هم یا فاصلهٔ دو روز از سال. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="در_هندسه"><span id=".D8.AF.D8.B1_.D9.87.D9.86.D8.AF.D8.B3.D9.87"></span>در هندسه</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=1" title="ویرایش بخش: در هندسه"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>در <a href="/wiki/%D9%87%D9%86%D8%AF%D8%B3%D9%87_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="هندسه اقلیدسی">هندسهٔ اقلیدسی</a>، انواع مختلفی از فاصله تعریف می‌شود؛ امّا به طور کلّی منظور از «فاصله» همان <a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="فاصله اقلیدسی">فاصلهٔ اقلیدسی</a> است. </p> <div class="mw-heading mw-heading3"><h3 id="فاصلهٔ_اقلیدسی"><span id=".D9.81.D8.A7.D8.B5.D9.84.D9.87.D9.94_.D8.A7.D9.82.D9.84.DB.8C.D8.AF.D8.B3.DB.8C"></span>فاصلهٔ اقلیدسی</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=2" title="ویرایش بخش: فاصلهٔ اقلیدسی"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r35841839"><div role="note" class="hatnote navigation-not-searchable">مقالهٔ اصلی: <a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="فاصله اقلیدسی">فاصله اقلیدسی</a></div> <p>به طول کوتاه‌ترین خط بین دو نقطه‌ی <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> <a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_%D8%A7%D9%82%D9%84%DB%8C%D8%AF%D8%B3%DB%8C" title="فاصله اقلیدسی"><b>فاصلهٔ اقلیدسی</b></a> (با نماد <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 {\overline {AB}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f91c39c977790f3cc4e768d5aad89bb1696110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.622ex; height:3.009ex;" alt="{\displaystyle {\overline {AB}}}"></span><sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>۱<span class="cite-bracket">]</span></a></sup>) گفته می‌شود. </p><p>در <a href="/wiki/%D9%87%D9%86%D8%AF%D8%B3%D9%87_%D8%AA%D8%AD%D9%84%DB%8C%D9%84%DB%8C" title="هندسه تحلیلی">هندسهٔ تحلیلی</a> فاصلهٔ بین دو نقطه‌ی <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(A_{x},A_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{x},A_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f1e9883e8a3a71f8aa0f96c4542d80d9bee79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.393ex; height:3.009ex;" alt="{\displaystyle A=(A_{x},A_{y})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(B_{x},B_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(B_{x},B_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d25b4ffaea44c7630b4b34685b4463e40ae45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.456ex; height:3.009ex;" alt="{\displaystyle B=(B_{x},B_{y})}"></span> را می‌توان به کمک <a href="/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D9%81%DB%8C%D8%AB%D8%A7%D8%BA%D9%88%D8%B1%D8%B3" title="قضیه فیثاغورس">قضیهٔ فیثاغورث</a> پیدا کرد که به فرمول زیر می‌رسد<sup id="cite_ref-:02_2-0" class="reference"><a href="#cite_note-:02-2"><span class="cite-bracket">[</span>۲<span class="cite-bracket">]</span></a></sup>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AB}}={\sqrt {\Delta x^{2}+\Delta y^{2}}}={\sqrt {(A_{x}-B_{x})^{2}+(A_{y}-B_{y})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>y</mi> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}={\sqrt {\Delta x^{2}+\Delta y^{2}}}={\sqrt {(A_{x}-B_{x})^{2}+(A_{y}-B_{y})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8ead1c7b1a388f8662ab594186ab5b751ea327" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:51.483ex; height:4.843ex;" alt="{\displaystyle {\overline {AB}}={\sqrt {\Delta x^{2}+\Delta y^{2}}}={\sqrt {(A_{x}-B_{x})^{2}+(A_{y}-B_{y})^{2}}}}"></span> </p><p>این فرمول را می‌توان به ابعاد بالاتر تعمیم داد؛ یعنی اگر <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(A_{1},A_{2},\dots ,A_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{1},A_{2},\dots ,A_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d21e4bededfa2a4a29941be639869439cbbc783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.419ex; height:2.843ex;" alt="{\displaystyle A=(A_{1},A_{2},\dots ,A_{n})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(B_{1},B_{2},\dots ,B_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(B_{1},B_{2},\dots ,B_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d983861e86b6d9efa0c9915fba2938008a5310c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.503ex; height:2.843ex;" alt="{\displaystyle B=(B_{1},B_{2},\dots ,B_{n})}"></span>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {AB}}={\sqrt {\sum _{i=1}^{n}{(\Delta A_{i})^{2}}}}={\sqrt {(A_{1}-B_{1})^{2}+(A_{2}-B_{2})^{2}+\dots +(A_{n}-B_{n})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>A</mi> <mi>B</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {AB}}={\sqrt {\sum _{i=1}^{n}{(\Delta A_{i})^{2}}}}={\sqrt {(A_{1}-B_{1})^{2}+(A_{2}-B_{2})^{2}+\dots +(A_{n}-B_{n})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d83a3367fe03a49e89c7e3fdfeed4bca934a6a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:71.082ex; height:7.509ex;" alt="{\displaystyle {\overline {AB}}={\sqrt {\sum _{i=1}^{n}{(\Delta A_{i})^{2}}}}={\sqrt {(A_{1}-B_{1})^{2}+(A_{2}-B_{2})^{2}+\dots +(A_{n}-B_{n})^{2}}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="فاصله_میان_نقطه_و_خط"><span id=".D9.81.D8.A7.D8.B5.D9.84.D9.87_.D9.85.DB.8C.D8.A7.D9.86_.D9.86.D9.82.D8.B7.D9.87_.D9.88_.D8.AE.D8.B7"></span>فاصله میان نقطه و خط</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=3" title="ویرایش بخش: فاصله میان نقطه و خط"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>فاصله میان نقطه (<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 P=(x_{p},y_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x_{p},y_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce6e9dc3c087f1a73f2336d9125ed77b2ed940c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.37ex; height:3.009ex;" alt="{\displaystyle P=(x_{p},y_{p}}"></span> و خط <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span>، از طریق نقاط <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></span> برابر است با: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(P,l)={\sqrt {(x_{p}-x_{0}-\lambda _{q}(x_{1}-x_{0}))^{2}+(y_{p}-y_{0}-\lambda _{q}(y_{1}-y_{0}))^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <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"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(P,l)={\sqrt {(x_{p}-x_{0}-\lambda _{q}(x_{1}-x_{0}))^{2}+(y_{p}-y_{0}-\lambda _{q}(y_{1}-y_{0}))^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7429dd47f8070ed3bd43c3d55507decaef87870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:64.155ex; height:4.843ex;" alt="{\displaystyle d(P,l)={\sqrt {(x_{p}-x_{0}-\lambda _{q}(x_{1}-x_{0}))^{2}+(y_{p}-y_{0}-\lambda _{q}(y_{1}-y_{0}))^{2}}}}"></span></dd></dl> <p>که در آن: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{q}={\frac {(x_{1}-x_{0})(x_{p}-x_{0})+(y_{1}-y_{0})(y_{p}-y_{0})}{(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</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>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{q}={\frac {(x_{1}-x_{0})(x_{p}-x_{0})+(y_{1}-y_{0})(y_{p}-y_{0})}{(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa94b2620ee0acc8c27e523c2b7b92de03b82de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.037ex; height:6.509ex;" alt="{\displaystyle \lambda _{q}={\frac {(x_{1}-x_{0})(x_{p}-x_{0})+(y_{1}-y_{0})(y_{p}-y_{0})}{(x_{1}-x_{0})^{2}+(y_{1}-y_{0})^{2}}}}"></span></dd></dl> <p>اگر مقدار <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ce0d3452435efa2eba679b287568618d059dfd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.344ex; height:2.843ex;" alt="{\displaystyle \lambda _{q}}"></span> میان ۰ و ۱ باشد نقطهٔ تقاطع <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> و خط گذرنده از <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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> و عمود بر <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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> بین نقاط <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{0},y_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{0},y_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c296094af9a1c665425debeac5eaab99a37a04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{0},y_{0})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},y_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc74086e56542bd28b46a84faaee3cebdd4a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.421ex; height:2.843ex;" alt="{\displaystyle (x_{1},y_{1})}"></span> جای می‌گیرد. </p> <div class="mw-heading mw-heading3"><h3 id="فاصلهٔ_منهتنی"><span id=".D9.81.D8.A7.D8.B5.D9.84.D9.87.D9.94_.D9.85.D9.86.D9.87.D8.AA.D9.86.DB.8C"></span>فاصلهٔ منهتنی</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=4" title="ویرایش بخش: فاصلهٔ منهتنی"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r35841839"><div role="note" class="hatnote navigation-not-searchable">مقالهٔ اصلی: <a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_%D9%85%D9%86%D9%87%D8%AA%D9%86" title="فاصله منهتن">فاصله منهتن</a></div><figure typeof="mw:File/Thumb"><a href="/wiki/%D9%BE%D8%B1%D9%88%D9%86%D8%AF%D9%87:Manhattan_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/246px-Manhattan_distance.svg.png" decoding="async" width="246" height="246" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/369px-Manhattan_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/492px-Manhattan_distance.svg.png 2x" data-file-width="283" data-file-height="283" /></a><figcaption>در منهتن انتخاب بین مسیر‌های زرد و آبی و قرمز فرقی در مسافت طی‌شدهٔ نهایی ایجاد نمی‌کند<div style="direction: ltr;"><div class="thumb tright" style="clear:none;text-align:center;max-width:222px"><div class="thumbinner" style="width:214px"><table cellpadding="0" cellspacing="0" style="background:white;font-size:88%;border:1px #b0b0b0 solid;padding:0;margin:auto"><tbody><tr style="vertical-align:middle"><td style="vertical-align:inherit;padding:0"></td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">a</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">b</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">c</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">d</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">e</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">f</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">g</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">h</td><td style="vertical-align:inherit;padding:0"></td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;width:18px;height:22px">8</td><td colspan="8" rowspan="8" style="padding:0;vertical-align:inherit"><div class="chess-board" style="position:relative"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Chessboard480.svg/176px-Chessboard480.svg.png" decoding="async" width="176" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Chessboard480.svg/264px-Chessboard480.svg.png 1.5x, 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src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Chess_x5t45.svg/22px-Chess_x5t45.svg.png" decoding="async" width="22" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Chess_x5t45.svg/33px-Chess_x5t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Chess_x5t45.svg/44px-Chess_x5t45.svg.png 2x" data-file-width="45" data-file-height="45" /></a></span></div></div></td><td style="padding:0;vertical-align:inherit;width:18px;height:22px">8</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">7</td><td style="padding:0;vertical-align:inherit;height:22px">7</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">6</td><td style="padding:0;vertical-align:inherit;height:22px">6</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">5</td><td style="padding:0;vertical-align:inherit;height:22px">5</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">4</td><td style="padding:0;vertical-align:inherit;height:22px">4</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">3</td><td style="padding:0;vertical-align:inherit;height:22px">3</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">2</td><td style="padding:0;vertical-align:inherit;height:22px">2</td></tr><tr style="vertical-align:middle"><td style="padding:0;vertical-align:inherit;height:22px">1</td><td style="padding:0;vertical-align:inherit;height:22px">1</td></tr><tr style="vertical-align:middle"><td style="vertical-align:inherit;padding:0"></td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">a</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">b</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">c</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">d</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">e</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">f</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">g</td><td style="padding:0;vartical-align:inherit;height:18px;width:22px">h</td><td style="vertical-align:inherit;padding:0"></td></tr></tbody></table></div></div></div></figcaption></figure> <p>نام این نوع فاصله از <a href="/wiki/%D9%85%D9%86%D9%87%D8%AA%D9%86" title="منهتن">منهتن در نیویورک آمریکا</a> الهام گرفته شده؛ به این دلیل که نقشهٔ جادّه‌های آنجا بلوک‌بندی شده. فاصلهٔ منهتنی دو خانه برابر مسافتی ست که یک تاکسی باید طی کند تا به مقصد برسد. </p><p>فاصلهٔ منهتنی دو نقطه‌ی <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(A_{x},A_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{x},A_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f1e9883e8a3a71f8aa0f96c4542d80d9bee79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.393ex; height:3.009ex;" alt="{\displaystyle A=(A_{x},A_{y})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(B_{x},B_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(B_{x},B_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d25b4ffaea44c7630b4b34685b4463e40ae45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.456ex; height:3.009ex;" alt="{\displaystyle B=(B_{x},B_{y})}"></span> (با نماد <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span>) با فرمول زیر تعریف می‌شود: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d_{1}} (A,B)=\left\vert \Delta x\right\vert +\left\vert \Delta y\right\vert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d_{1}} (A,B)=\left\vert \Delta x\right\vert +\left\vert \Delta y\right\vert }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1373d0b600b879d6c12fa1f743c9f7335c53e4a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.58ex; height:2.843ex;" alt="{\displaystyle \mathrm {d_{1}} (A,B)=\left\vert \Delta x\right\vert +\left\vert \Delta y\right\vert }"></span> </p><p>این فرمول را نیز می‌توان به ابعاد بالاتر تعمیم داد. </p> <div class="mw-heading mw-heading3"><h3 id="فاصلهٔ_شطرنجی_(فاصلهٔ_چبیشف)"><span id=".D9.81.D8.A7.D8.B5.D9.84.D9.87.D9.94_.D8.B4.D8.B7.D8.B1.D9.86.D8.AC.DB.8C_.28.D9.81.D8.A7.D8.B5.D9.84.D9.87.D9.94_.DA.86.D8.A8.DB.8C.D8.B4.D9.81.29"></span>فاصلهٔ شطرنجی (<b>فاصلهٔ چبیشف)</b></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=5" title="ویرایش بخش: فاصلهٔ شطرنجی (فاصلهٔ چبیشف)"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>این نوع فاصله به این دلیل <i>شطرنجی</i> نامیده شده که در شطرنج برابر تعداد نوبت‌های مورد نیاز <i><a href="/wiki/%D8%B4%D8%A7%D9%87_(%D8%B4%D8%B7%D8%B1%D9%86%D8%AC)" title="شاه (شطرنج)">شاه</a></i> برای رسیدن به مقصدش می‌باشد. </p><p>فاصلهٔ شطرنجی دو نقطه‌ی <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(A_{x},A_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{x},A_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f1e9883e8a3a71f8aa0f96c4542d80d9bee79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.393ex; height:3.009ex;" alt="{\displaystyle A=(A_{x},A_{y})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(B_{x},B_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(B_{x},B_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d25b4ffaea44c7630b4b34685b4463e40ae45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.456ex; height:3.009ex;" alt="{\displaystyle B=(B_{x},B_{y})}"></span> (با نماد <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"></span>) با فرمول زیر تعریف می‌شود: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d_{\infty }} (A,B)=\max\{\Delta x,\Delta y\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">Δ</mi> <mi>x</mi> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mi>y</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d_{\infty }} (A,B)=\max\{\Delta x,\Delta y\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b9a410cb0190d1a61d4fe202d8f632edc10801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.658ex; height:2.843ex;" alt="{\displaystyle \mathrm {d_{\infty }} (A,B)=\max\{\Delta x,\Delta y\}}"></span> </p><p>این فرمول را نیز می‌توان به ابعاد بالاتر تعمیم داد. </p> <div class="mw-heading mw-heading3"><h3 id="تعمیم_و_فاصلهٔ_مینکوفسکی"><span id=".D8.AA.D8.B9.D9.85.DB.8C.D9.85_.D9.88_.D9.81.D8.A7.D8.B5.D9.84.D9.87.D9.94_.D9.85.DB.8C.D9.86.DA.A9.D9.88.D9.81.D8.B3.DA.A9.DB.8C"></span>تعمیم و فاصلهٔ مینکوفسکی</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=6" title="ویرایش بخش: تعمیم و فاصلهٔ مینکوفسکی"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r35841839"><div role="note" class="hatnote navigation-not-searchable">مقالهٔ اصلی: <a href="/wiki/%D9%86%D8%B1%D9%85_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="نرم (ریاضیات)">نرم (ریاضیات)</a></div> <p>در فضای <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 \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> فاصلهٔ مینکوفسکی از مرتبهٔ <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> (یا <a href="/wiki/%D9%86%D8%B1%D9%85_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="نرم (ریاضیات)">p-نُرم</a> با نماد <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb49196cdddd8fb8b68c9f0a85951e064922e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.352ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} _{p}}"></span>) بین دو نقطه‌ی <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=(A_{1},A_{2},\dots ,A_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{1},A_{2},\dots ,A_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d21e4bededfa2a4a29941be639869439cbbc783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.419ex; height:2.843ex;" alt="{\displaystyle A=(A_{1},A_{2},\dots ,A_{n})}"></span> و <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=(B_{1},B_{2},\dots ,B_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=(B_{1},B_{2},\dots ,B_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d983861e86b6d9efa0c9915fba2938008a5310c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.503ex; height:2.843ex;" alt="{\displaystyle B=(B_{1},B_{2},\dots ,B_{n})}"></span> به صورت زیر تعریف می‌شود: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\sum _{i=1}^{n}\left|A_{i}-B_{i}\right|^{p}\right)^{1/p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\sum _{i=1}^{n}\left|A_{i}-B_{i}\right|^{p}\right)^{1/p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/820994d4798707ae41af8d191afaf1becc1fd0c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.426ex; height:8.009ex;" alt="{\displaystyle \left(\sum _{i=1}^{n}\left|A_{i}-B_{i}\right|^{p}\right)^{1/p}}"></span> </p><p>فاصلهٔ منهتنی معادل ۱-نرم، فاصلهٔ شطرنجی معادل ∞-نرم و فاصلهٔ اقلیدسی معادل ۲-نرم (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df459e56cd108403aaa8be0866fd0d552e64c0af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.347ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} _{2}}"></span>) هستند. </p> <div class="mw-heading mw-heading2"><h2 id="در_گراف"><span id=".D8.AF.D8.B1_.DA.AF.D8.B1.D8.A7.D9.81"></span>در گراف</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=7" title="ویرایش بخش: در گراف"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r35841839"><div role="note" class="hatnote navigation-not-searchable">مقالهٔ اصلی: <a href="/wiki/%D9%81%D8%A7%D8%B5%D9%84%D9%87_%D8%AF%D8%B1_%DA%AF%D8%B1%D8%A7%D9%81" title="فاصله در گراف">فاصله در گراف</a></div> <p>در <a href="/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D8%A7%D9%81" title="نظریه گراف">نظریّهٔ گراف‌ها</a>، فاصلهٔ دو <a href="/wiki/%D8%B1%D8%A3%D8%B3_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D8%A7%D9%81)" title="رأس (نظریه گراف)">رأس</a> (با نماد <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>) برابر طول کوتاه‌ترین <a href="/wiki/%D9%85%D8%B3%DB%8C%D8%B1_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D8%A7%D9%81)" title="مسیر (نظریه گراف)">مسیر</a> بین دو آن دو تعریف می‌شود<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>۳<span class="cite-bracket">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="منابع"><span id=".D9.85.D9.86.D8.A7.D8.A8.D8.B9"></span>منابع</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D9%81%D8%A7%D8%B5%D9%84%D9%87&action=edit&section=8" title="ویرایش بخش: منابع"><span>ویرایش</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r37199623">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.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-:0-1"><span class="mw-cite-backlink"><a href="#cite_ref-:0_1-0">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r39868036">.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 a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),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:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFسازمان_بین‌المللی_استانداردسازی" class="citation web cs1 cs1-prop-year-range-abbreviated cs1-prop-foreign-lang-source"><a href="/wiki/%D8%B3%D8%A7%D8%B2%D9%85%D8%A7%D9%86_%D8%A8%DB%8C%D9%86%E2%80%8C%D8%A7%D9%84%D9%85%D9%84%D9%84%DB%8C_%D8%A7%D8%B3%D8%AA%D8%A7%D9%86%D8%AF%D8%A7%D8%B1%D8%AF%D8%B3%D8%A7%D8%B2%DB%8C" title="سازمان بین‌المللی استانداردسازی">سازمان بین‌المللی استانداردسازی</a> (2019-08). <a rel="nofollow" class="external text" href="https://www.iso.org/standard/64973.html">"ISO 80000-2:2019 Quantities and units — Part 2: Mathematics"</a> (به انگلیسی).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ISO+80000-2%3A2019+Quantities+and+units+%E2%80%94+Part+2%3A+Mathematics&rft.au=%D8%B3%D8%A7%D8%B2%D9%85%D8%A7%D9%86+%D8%A8%DB%8C%D9%86%E2%80%8C%D8%A7%D9%84%D9%85%D9%84%D9%84%DB%8C+%D8%A7%D8%B3%D8%AA%D8%A7%D9%86%D8%AF%D8%A7%D8%B1%D8%AF%D8%B3%D8%A7%D8%B2%DB%8C&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F64973.html&rfr_id=info%3Asid%2Ffa.wikipedia.org%3A%D9%81%D8%A7%D8%B5%D9%84%D9%87" class="Z3988"></span> <span class="cs1-visible-error citation-comment"><code class="cs1-code">{{<a href="/wiki/%D8%A7%D9%84%DA%AF%D9%88:Cite_web" title="الگو:Cite web">cite web</a>}}</code>: </span><span class="cs1-visible-error citation-comment">Check date values in: <code class="cs1-code">|تاریخ انتشار=</code> (<a href="/wiki/%D8%B1%D8%A7%D9%87%D9%86%D9%85%D8%A7:CS1_errors#bad_date" class="mw-redirect" title="راهنما:CS1 errors">help</a>)</span>; <span class="cs1-visible-error citation-comment">line feed character in <code class="cs1-code">|عنوان=</code> at position 17 (<a href="/wiki/%D8%B1%D8%A7%D9%87%D9%86%D9%85%D8%A7:CS1_errors#invisible_char" class="mw-redirect" title="راهنما:CS1 errors">help</a>)</span></span> </li> <li id="cite_note-:02-2"><span class="mw-cite-backlink"><a href="#cite_ref-:02_2-0">↑</a></span> <span class="reference-text"><cite id="978-0134438986" class="citation book">«۱۲٫۱». <i>Thomas' Calculus (14th Edition)</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=12.1&rft.btitle=Thomas%27+Calculus+%2814th+Edition%29&rfr_id=info%3Asid%2Ffa.wikipedia.org%3A%D9%81%D8%A7%D8%B5%D9%84%D9%87" class="Z3988"></span><style data-mw-deduplicate="TemplateStyles:r28290294">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"\"""\"""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center;padding-right:1em;padding-left:0}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center;padding-right:1em;padding-left:0}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center;padding-right:1em;padding-left:0}.mw-parser-output div[dir=ltr] .cs1-lock-free a,.mw-parser-output div[dir=ltr] .cs1-lock-subscription a,.mw-parser-output div[dir=ltr] .cs1-lock-limited a,.mw-parser-output div[dir=ltr] .cs1-lock-registration a{background-position:left .1em center;padding-left:1em;padding-right:0}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output 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West.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.1&rft.btitle=Introduction+to+Graph+Theory+%282nd+Edition%29&rfr_id=info%3Asid%2Ffa.wikipedia.org%3A%D9%81%D8%A7%D8%B5%D9%84%D9%87" class="Z3988"></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r28290294"></span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r39940593">.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 0.9em 2px 0;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0 2px 0.9em;text-align:center}@media(min-width:500px){.mw-parser-output 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الگو">ب</abbr></a></li><li class="nv-ویرایش"><a href="/wiki/%D9%88%DB%8C%DA%98%D9%87:%D9%88%DB%8C%D8%B1%D8%A7%DB%8C%D8%B4_%D8%B5%D9%81%D8%AD%D9%87/%D8%A7%D9%84%DA%AF%D9%88:%D8%B3%DB%8C%D9%86%D9%85%D8%A7%D8%AA%DB%8C%DA%A9" title="ویژه:ویرایش صفحه/الگو:سینماتیک"><abbr title="ویرایش کردن این الگو">و</abbr></a></li></ul></div><div id="سینماتیک" style="font-size:114%;margin:0 4em"><a href="/wiki/%D8%B3%DB%8C%D9%86%D9%85%D8%A7%D8%AA%DB%8C%DA%A9_(%D9%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9)" class="mw-redirect" title="سینماتیک (مکانیک)">سینماتیک</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>→ <a href="/wiki/%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84" title="انتگرال">انتگرال</a> … <a href="/wiki/%D9%85%D8%B4%D8%AA%D9%82" title="مشتق">مشتق</a> ←</li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a 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