Frenet–Serret formulas

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class="vector-toc-numb">2</span> <span>Formulas in <i>n</i> dimensions</span> </div> </a> <ul id="toc-Formulas_in_n_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof_of_the_Frenet-Serret_formulas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof_of_the_Frenet-Serret_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Proof of the Frenet-Serret formulas</span> </div> </a> <ul id="toc-Proof_of_the_Frenet-Serret_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_and_interpretation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_and_interpretation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications and interpretation</span> </div> </a> <button aria-controls="toc-Applications_and_interpretation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications and interpretation subsection</span> </button> <ul id="toc-Applications_and_interpretation-sublist" class="vector-toc-list"> <li id="toc-Kinematics_of_the_frame" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kinematics_of_the_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Kinematics of the frame</span> </div> </a> <ul id="toc-Kinematics_of_the_frame-sublist" class="vector-toc-list"> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graphical_Illustrations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Graphical_Illustrations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Graphical Illustrations</span> </div> </a> <ul id="toc-Graphical_Illustrations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Frenet–Serret_formulas_in_calculus" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frenet–Serret_formulas_in_calculus"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Frenet–Serret formulas in calculus</span> </div> </a> <ul id="toc-Frenet–Serret_formulas_in_calculus-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Taylor_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Taylor_expansion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Taylor expansion</span> </div> </a> <ul id="toc-Taylor_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ribbons_and_tubes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ribbons_and_tubes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Ribbons and tubes</span> </div> </a> <ul id="toc-Ribbons_and_tubes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruence_of_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congruence_of_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Congruence of curves</span> </div> </a> <ul id="toc-Congruence_of_curves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_expressions_of_the_frame" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_expressions_of_the_frame"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Other expressions of the frame</span> </div> </a> <ul id="toc-Other_expressions_of_the_frame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_cases" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_cases"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Special cases</span> </div> </a> <button aria-controls="toc-Special_cases-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Special cases subsection</span> </button> <ul id="toc-Special_cases-sublist" class="vector-toc-list"> <li id="toc-Plane_curves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plane_curves"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Plane curves</span> </div> </a> <ul id="toc-Plane_curves-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button 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Available in 17 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" 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">17 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B5%D9%8A%D8%BA_%D9%81%D8%B1%D9%8A%D9%86%D9%8A-%D8%B3%D9%8A%D8%B1%D9%8A" title="صيغ فريني-سيري – Arabic" lang="ar" hreflang="ar" data-title="صيغ فريني-سيري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/F%C3%B3rmules_Frenet-Serret" title="Fórmules Frenet-Serret – Catalan" lang="ca" hreflang="ca" data-title="Fórmules Frenet-Serret" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Frenetsche_Formeln" title="Frenetsche Formeln – German" lang="de" hreflang="de" data-title="Frenetsche Formeln" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/F%C3%B3rmulas_de_Frenet-Serret" title="Fórmulas de Frenet-Serret – Spanish" lang="es" hreflang="es" data-title="Fórmulas de Frenet-Serret" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Freneten_formulak" title="Freneten formulak – Basque" lang="eu" hreflang="eu" data-title="Freneten formulak" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Rep%C3%A8re_de_Frenet" title="Repère de Frenet – French" lang="fr" hreflang="fr" data-title="Repère de Frenet" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%84%EB%A0%88%EB%84%A4-%EC%84%B8%EB%A0%88_%EA%B3%B5%EC%8B%9D" title="프레네-세레 공식 – Korean" lang="ko" hreflang="ko" data-title="프레네-세레 공식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%A4%D7%A8%D7%A0%D7%94-%D7%A1%D7%A8%D7%94" title="משוואות פרנה-סרה – Hebrew" lang="he" hreflang="he" data-title="משוואות פרנה-סרה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Formules_van_Frenet-Serret" title="Formules van Frenet-Serret – Dutch" lang="nl" hreflang="nl" data-title="Formules van Frenet-Serret" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%83%AC%E3%83%8D%E3%83%BB%E3%82%BB%E3%83%AC%E3%81%AE%E5%85%AC%E5%BC%8F" title="フレネ・セレの公式 – Japanese" lang="ja" hreflang="ja" data-title="フレネ・セレの公式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wzory_Freneta" title="Wzory Freneta – Polish" lang="pl" hreflang="pl" data-title="Wzory Freneta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Triedro_de_Frenet" title="Triedro de Frenet – Portuguese" lang="pt" hreflang="pt" data-title="Triedro de Frenet" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Formulele_lui_Frenet" title="Formulele lui Frenet – Romanian" lang="ro" hreflang="ro" data-title="Formulele lui Frenet" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D1%91%D1%85%D0%B3%D1%80%D0%B0%D0%BD%D0%BD%D0%B8%D0%BA_%D0%A4%D1%80%D0%B5%D0%BD%D0%B5" title="Трёхгранник Френе – Russian" lang="ru" hreflang="ru" data-title="Трёхгранник Френе" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Frenet%E2%80%93Serrets_formler" title="Frenet–Serrets formler – Swedish" lang="sv" hreflang="sv" data-title="Frenet–Serrets formler" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B3%D1%80%D0%B0%D0%BD%D0%BD%D0%B8%D0%BA_%D0%A4%D1%80%D0%B5%D0%BD%D0%B5" title="Тригранник Френе – Ukrainian" lang="uk" hreflang="uk" data-title="Тригранник Френе" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BC%97%E8%8E%B1%E7%BA%B3%E5%85%AC%E5%BC%8F" title="弗莱纳公式 – Chinese" lang="zh" hreflang="zh" data-title="弗莱纳公式" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Formulas in differential geometry</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Binormal" redirects here. For the category-theoretic meaning of this word, see <a href="/wiki/Normal_morphism" title="Normal morphism">normal morphism</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Frenet.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Frenet.svg/300px-Frenet.svg.png" decoding="async" width="300" height="358" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Frenet.svg/450px-Frenet.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Frenet.svg/600px-Frenet.svg.png 2x" data-file-width="628" data-file-height="749" /></a><figcaption>A space curve; the vectors <span class="texhtml"><b>T</b>, <b>N</b>, <b>B</b></span>; and the <a href="/wiki/Osculating_plane" title="Osculating plane">osculating plane</a> spanned by <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>N</b></span></figcaption></figure> <p>In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the <b>Frenet–Serret formulas</b> describe the <a href="/wiki/Kinematic" class="mw-redirect" title="Kinematic">kinematic</a> properties of a particle moving along a differentiable <a href="/wiki/Curve" title="Curve">curve</a> in three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3},}"> <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"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"></span> or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the <a href="/wiki/Derivative" title="Derivative">derivatives</a> of the so-called <b>tangent, normal, and binormal</b> <a href="/wiki/Unit_vector" title="Unit vector">unit vectors</a> in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: <a href="/wiki/Jean_Fr%C3%A9d%C3%A9ric_Frenet" title="Jean Frédéric Frenet">Jean Frédéric Frenet</a>, in his thesis of 1847, and <a href="/wiki/Joseph_Alfred_Serret" class="mw-redirect" title="Joseph Alfred Serret">Joseph Alfred Serret</a>, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery. </p><p>The tangent, normal, and binormal unit vectors, often called <span class="texhtml"><b>T</b></span>, <span class="texhtml"><b>N</b></span>, and <span class="texhtml"><b>B</b></span>, or collectively the <b>Frenet–Serret frame</b> (<b>TNB frame</b> or <b>TNB basis</b>), together form an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> that <a href="/wiki/Linear_span" title="Linear span">spans</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3},}"> <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"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/deb17c1074c77de2cf88d45bcd6d7a795b0f5d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.379ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} ^{3},}"></span> and are defined as follows: </p> <ul><li><span class="texhtml"><b>T</b></span> is the unit vector <a href="/wiki/Tangent_vector" title="Tangent vector">tangent</a> to the curve, pointing in the direction of motion.</li> <li><span class="texhtml"><b>N</b></span> is the <a href="/wiki/Normal_vector" class="mw-redirect" title="Normal vector">normal</a> unit vector, the derivative of <span class="texhtml"><b>T</b></span> with respect to the <a href="/wiki/Rectifiable_path" class="mw-redirect" title="Rectifiable path">arclength parameter</a> of the curve, divided by its length.</li> <li><span class="texhtml"><b>B</b></span> is the binormal unit vector, the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>N</b></span>.</li></ul> <p>The Frenet–Serret formulas are: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>+</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24223bb31c70e4fbbc153723ec6b0a4426c9f61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.361ex; margin-bottom: -0.31ex; width:19.667ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d}{ds}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {d}{ds}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86dfe71ddb0da32602843f8ac868cd1aced16242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.467ex; height:3.843ex;" alt="{\displaystyle {\tfrac {d}{ds}}}"></span> is the derivative with respect to arclength, <span class="texhtml mvar" style="font-style:italic;">κ</span> is the <a href="/wiki/Curvature" title="Curvature">curvature</a>, and <span class="texhtml mvar" style="font-style:italic;">τ</span> is the <a href="/wiki/Torsion_of_curves" class="mw-redirect" title="Torsion of curves">torsion</a> of the space curve. (Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.) The <span class="texhtml"><b>TNB</b></span> basis combined with the two <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>, <span class="texhtml mvar" style="font-style:italic;">κ</span> and <span class="texhtml mvar" style="font-style:italic;">τ</span>, is called collectively the <b>Frenet–Serret apparatus</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:FrenetTN.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/FrenetTN.svg/350px-FrenetTN.svg.png" decoding="async" width="350" height="116" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/FrenetTN.svg/525px-FrenetTN.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/FrenetTN.svg/700px-FrenetTN.svg.png 2x" data-file-width="827" data-file-height="274" /></a><figcaption> <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:solid black 2px;"> </span> The <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>N</b></span> vectors at two points on a plane curve</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:dotted black 2px;"> </span> A translated version of the second frame.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><div class="legend"><span class="legend-line mw-no-invert" style="display: inline-block; vertical-align: middle; width: 1.67em; height: 0; border-style: none; border-top: 2px dotted black;border-top:dashed black 2px;"> </span> The change in <span class="texhtml"><b>T</b>: δ<b>T'</b></span>.</div> <span class="texhtml mvar" style="font-style:italic;">δs</span> is the distance between the points. In the limit <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 {\tfrac {d\mathbf {T} }{ds}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {d\mathbf {T} }{ds}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69184e38fb3f06e0e3d138e49893c566a695fcec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:3.011ex; height:3.843ex;" alt="{\displaystyle {\tfrac {d\mathbf {T} }{ds}}}"></span> will be in the direction <span class="texhtml"><b>N</b></span> and the curvature describes the speed of rotation of the frame.</figcaption></figure> <p>Let <span class="texhtml"><b>r</b>(<i>t</i>)</span> be a <a href="/wiki/Curve" title="Curve">curve</a> in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, representing the <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position vector</a> of the particle as a function of time. The Frenet–Serret formulas apply to curves which are <i>non-degenerate</i>, which roughly means that they have nonzero <a href="/wiki/Curvature" title="Curvature">curvature</a>. More formally, in this situation the <a href="/wiki/Velocity" title="Velocity">velocity</a> vector <span class="texhtml"><b>r</b>′(<i>t</i>)</span> and the <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> vector <span class="texhtml"><b>r</b>′′(<i>t</i>)</span> are required not to be proportional. </p><p>Let <span class="texhtml"><i>s</i>(<i>t</i>)</span> represent the <a href="/wiki/Arc_length" title="Arc length">arc length</a> which the particle has moved along the <a href="/wiki/Curve" title="Curve">curve</a> in time <span class="texhtml mvar" style="font-style:italic;">t</span>. The quantity <span class="texhtml mvar" style="font-style:italic;">s</span> is used to give the curve traced out by the trajectory of the particle a <a href="/wiki/Rectifiable_path" class="mw-redirect" title="Rectifiable path">natural parametrization</a> by arc length (i.e. <a href="/wiki/Differentiable_curve#Length_and_natural_parametrization" title="Differentiable curve">arc-length parametrization</a>), since many different particle paths may trace out the same geometrical curve by traversing it at different rates. In detail, <span class="texhtml mvar" style="font-style:italic;">s</span> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)=\int _{0}^{t}\left\|\mathbf {r} '(\sigma )\right\|d\sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mi>d</mi> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)=\int _{0}^{t}\left\|\mathbf {r} '(\sigma )\right\|d\sigma .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8098edcc9e5f22169a3fec39313e709f11eadd7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.345ex; height:6.176ex;" alt="{\displaystyle s(t)=\int _{0}^{t}\left\|\mathbf {r} '(\sigma )\right\|d\sigma .}"></span> Moreover, since we have assumed that <span class="texhtml"><b>r</b>′ ≠ 0</span>, it follows that <span class="texhtml"><i>s</i>(<i>t</i>)</span> is a strictly monotonically increasing function. Therefore, it is possible to solve for <span class="texhtml mvar" style="font-style:italic;">t</span> as a function of <span class="texhtml mvar" style="font-style:italic;">s</span>, and thus to write <span class="texhtml"><b>r</b>(<i>s</i>) = <b>r</b>(<i>t</i>(<i>s</i>))</span>. The curve is thus parametrized in a preferred manner by its arc length. </p><p>With a non-degenerate curve <span class="texhtml"><b>r</b>(<i>s</i>)</span>, parameterized by its arc length, it is now possible to define the <b>Frenet–Serret frame</b> (or <b><span class="texhtml">TNB</span> frame</b>): </p> <ul> <li> The tangent unit vector <span class="texhtml"><b>T</b></span> is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} :={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} :={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21a775091228fed8f0ce0b1d4b1b278d81ba96c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.482ex; height:5.509ex;" alt="{\displaystyle \mathbf {T} :={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} s}}.}"></span> </li> <li> The normal unit vector <span class="texhtml"><b>N</b></span> is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} :={{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}} \over \left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} :={{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}} \over \left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17719991c8193512bfba3678c77d1770d6f48af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:12.709ex; height:8.176ex;" alt="{\displaystyle \mathbf {N} :={{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}} \over \left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|},}"></span> from which it follows, since <span class="texhtml"><b>T</b></span> always has unit <a href="/wiki/Magnitude_(mathematics)" title="Magnitude (mathematics)">magnitude</a>, that <span class="texhtml"><b>N</b></span> (the change of <span class="texhtml"><b>T</b></span>) is always perpendicular to <span class="texhtml"><b>T</b></span>, since there is no change in length of <span class="texhtml"><b>T</b></span>. Note that by calling curvature <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 \kappa =\left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mo symmetric="true">‖</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =\left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11e218a9ca1cb13904f7dbd044e8933e90ad427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.75ex; height:5.843ex;" alt="{\displaystyle \kappa =\left\|{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}\right\|}"></span> we automatically obtain the first relation. </li> </ul><ul> <li> The binormal unit vector <span class="texhtml"><b>B</b></span> is defined as the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>N</b></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} :=\mathbf {T} \times \mathbf {N} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} :=\mathbf {T} \times \mathbf {N} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8248d2359aec4d83028f9380ac85ce70e6d552bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.084ex; height:2.509ex;" alt="{\displaystyle \mathbf {B} :=\mathbf {T} \times \mathbf {N} ,}"></span> </li> </ul> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Frenetframehelix.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/8/87/Frenetframehelix.gif" decoding="async" width="290" height="205" class="mw-file-element" data-file-width="290" data-file-height="205" /></a><figcaption>The Frenet–Serret frame moving along a <a href="/wiki/Helix" title="Helix">helix</a>. The <span class="texhtml"><b>T</b></span> is represented by the blue arrow, <span class="texhtml"><b>N</b></span> is represented by the red arrow while <span class="texhtml"><b>B</b></span> is represented by the black arrow.</figcaption></figure> <p>from which it follows that <span class="texhtml"><b>B</b></span> is always perpendicular to both <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>N</b></span>. Thus, the three unit vectors <span class="texhtml"><b>T</b>, <b>N</b>, <b>B</b></span> are all perpendicular to each other. </p><p>The <b>Frenet–Serret formulas</b> are: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>+</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d24223bb31c70e4fbbc153723ec6b0a4426c9f61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.361ex; margin-bottom: -0.31ex; width:19.667ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\mathrm {d} \mathbf {T} }{\mathrm {d} s}}&=\kappa \mathbf {N} ,\\[4pt]{\frac {\mathrm {d} \mathbf {N} }{\mathrm {d} s}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\[4pt]{\frac {\mathrm {d} \mathbf {B} }{\mathrm {d} s}}&=-\tau \mathbf {N} ,\end{aligned}}}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">κ</span> is the <a href="/wiki/Curvature" title="Curvature">curvature</a> and <span class="texhtml mvar" style="font-style:italic;">τ</span> is the <a href="/wiki/Torsion_of_curves" class="mw-redirect" title="Torsion of curves">torsion</a>. </p><p>The Frenet–Serret formulas are also known as <i>Frenet–Serret theorem</i>, and can be stated more concisely using matrix notation:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\mathbf {T'} \\\mathbf {N'} \\\mathbf {B'} \end{bmatrix}}={\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">T</mi> <mo>′</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">N</mi> <mo>′</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>κ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>κ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>τ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\mathbf {T'} \\\mathbf {N'} \\\mathbf {B'} \end{bmatrix}}={\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe09655c47b65c82a2535da318ecb0216bf861a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.174ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}\mathbf {T'} \\\mathbf {N'} \\\mathbf {B'} \end{bmatrix}}={\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.}"></span> </p><p>This matrix is <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Formulas_in_n_dimensions">Formulas in <i>n</i> dimensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=2" title="Edit section: Formulas in n dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Frenet–Serret formulas were generalized to higher-dimensional Euclidean spaces by <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a> in 1874. </p><p>Suppose that <span class="texhtml"><b>r</b>(<i>s</i>)</span> is a smooth curve in <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> <mo>,</mo> </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/d7035fcb9fe3ebecc6bc9f372f82d0352202c8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{n},}"></span> and that the first <span class="texhtml mvar" style="font-style:italic;">n</span> derivatives of <span class="texhtml"><b>r</b></span> are linearly independent.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The vectors in the Frenet–Serret frame are an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> constructed by applying the <a href="/wiki/Gram-Schmidt_process" class="mw-redirect" title="Gram-Schmidt process">Gram-Schmidt process</a> to the vectors <span class="texhtml">(<b>r</b>′(<i>s</i>), <b>r</b>′′(<i>s</i>), ..., <b>r</b><sup>(<i>n</i>)</sup>(<i>s</i>))</span>. </p><p>In detail, the unit tangent vector is the first Frenet vector <span class="texhtml"><i>e</i><sub>1</sub>(<i>s</i>)</span> and is defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}(s)={\frac {{\overline {\mathbf {e} _{1}}}(s)}{\|{\overline {\mathbf {e} _{1}}}(s)\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{1}(s)={\frac {{\overline {\mathbf {e} _{1}}}(s)}{\|{\overline {\mathbf {e} _{1}}}(s)\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fd62b4feed03f796709f61e2fd47ee661c9f6a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.733ex; height:6.509ex;" alt="{\displaystyle \mathbf {e} _{1}(s)={\frac {{\overline {\mathbf {e} _{1}}}(s)}{\|{\overline {\mathbf {e} _{1}}}(s)\|}}}"></span> </p><p>where </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {e} _{1}}}(s)=\mathbf {r} '(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {e} _{1}}}(s)=\mathbf {r} '(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2dadfcacfe6c9180a148749f6bfdce3c54be43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.079ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbf {e} _{1}}}(s)=\mathbf {r} '(s)}"></span> </p><p>The <b>normal vector</b>, sometimes called the <b>curvature vector</b>, indicates the deviance of the curve from being a straight line. It is defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b4649c12f1afeb6f68ef39cedd22a34ececb261" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.1ex; height:3.009ex;" alt="{\displaystyle {\overline {\mathbf {e} _{2}}}(s)=\mathbf {r} ''(s)-\langle \mathbf {r} ''(s),\mathbf {e} _{1}(s)\rangle \,\mathbf {e} _{1}(s)}"></span> </p><p>Its normalized form, the <b>unit normal vector</b>, is the second Frenet vector <span class="texhtml"><b>e</b><sub>2</sub>(<i>s</i>)</span> and defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{2}(s)={\frac {{\overline {\mathbf {e} _{2}}}(s)}{\|{\overline {\mathbf {e} _{2}}}(s)\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{2}(s)={\frac {{\overline {\mathbf {e} _{2}}}(s)}{\|{\overline {\mathbf {e} _{2}}}(s)\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232a8802b35b864a11b2256b5aaaf8a0a595dbbc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.733ex; height:6.509ex;" alt="{\displaystyle \mathbf {e} _{2}(s)={\frac {{\overline {\mathbf {e} _{2}}}(s)}{\|{\overline {\mathbf {e} _{2}}}(s)\|}}}"></span> </p><p>The tangent and the normal vector at point <span class="texhtml mvar" style="font-style:italic;">s</span> define the <i><a href="/wiki/Osculating_plane" title="Osculating plane">osculating plane</a></i> at point <span class="texhtml"><b>r</b>(<i>s</i>)</span>. </p><p>The remaining vectors in the frame (the binormal, trinormal, etc.) are defined similarly by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {e} _{j}(s)&={\frac {{\overline {\mathbf {e} _{j}}}(s)}{\|{\overline {\mathbf {e} _{j}}}(s)\|}},\\{\overline {\mathbf {e} _{j}}}(s)&=\mathbf {r} ^{(j)}(s)-\sum _{i=1}^{j-1}\langle \mathbf {r} ^{(j)}(s),\mathbf {e} _{i}(s)\rangle \,\mathbf {e} _{i}(s).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo fence="false" stretchy="false">⟨</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {e} _{j}(s)&={\frac {{\overline {\mathbf {e} _{j}}}(s)}{\|{\overline {\mathbf {e} _{j}}}(s)\|}},\\{\overline {\mathbf {e} _{j}}}(s)&=\mathbf {r} ^{(j)}(s)-\sum _{i=1}^{j-1}\langle \mathbf {r} ^{(j)}(s),\mathbf {e} _{i}(s)\rangle \,\mathbf {e} _{i}(s).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a0cc306bd848c467b634d19a3cfeee7dcdd3350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.363ex; margin-bottom: -0.308ex; width:41.303ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {e} _{j}(s)&={\frac {{\overline {\mathbf {e} _{j}}}(s)}{\|{\overline {\mathbf {e} _{j}}}(s)\|}},\\{\overline {\mathbf {e} _{j}}}(s)&=\mathbf {r} ^{(j)}(s)-\sum _{i=1}^{j-1}\langle \mathbf {r} ^{(j)}(s),\mathbf {e} _{i}(s)\rangle \,\mathbf {e} _{i}(s).\end{aligned}}}"></span></dd></dl> <p>The last vector in the frame is defined by the cross-product of the first <span class="texhtml"><i>n</i> − 1</span> vectors: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{n}(s)=\mathbf {e} _{1}(s)\times \mathbf {e} _{2}(s)\times \dots \times \mathbf {e} _{n-2}(s)\times \mathbf {e} _{n-1}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{n}(s)=\mathbf {e} _{1}(s)\times \mathbf {e} _{2}(s)\times \dots \times \mathbf {e} _{n-2}(s)\times \mathbf {e} _{n-1}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e60a96a77019a145622cf49757a90d6bb50a3e8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.773ex; height:2.843ex;" alt="{\displaystyle \mathbf {e} _{n}(s)=\mathbf {e} _{1}(s)\times \mathbf {e} _{2}(s)\times \dots \times \mathbf {e} _{n-2}(s)\times \mathbf {e} _{n-1}(s)}"></span> </p><p>The real valued functions used below <span class="texhtml"><i>χ<sub>i</sub></i>(<i>s</i>)</span> are called <b>generalized curvature</b> and are defined as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{i}(s)={\frac {\langle \mathbf {e} _{i}'(s),\mathbf {e} _{i+1}(s)\rangle }{\|\mathbf {r} '(s)\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{i}(s)={\frac {\langle \mathbf {e} _{i}'(s),\mathbf {e} _{i+1}(s)\rangle }{\|\mathbf {r} '(s)\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48aadea192189b799e756cf28cf7523f8e2aacf3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.882ex; height:6.676ex;" alt="{\displaystyle \chi _{i}(s)={\frac {\langle \mathbf {e} _{i}'(s),\mathbf {e} _{i+1}(s)\rangle }{\|\mathbf {r} '(s)\|}}}"></span> </p><p>The <b>Frenet–Serret formulas</b>, stated in matrix language, are </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\mathbf {e} _{1}'(s)\\\vdots \\\mathbf {e} _{n}'(s)\\\end{bmatrix}}=\|\mathbf {r} '(s)\|\cdot {\begin{bmatrix}0&\chi _{1}(s)&0&0\\[4pt]-\chi _{1}(s)&\ddots &\ddots &0\\[4pt]0&\ddots &\ddots &\chi _{n-1}(s)\\[4pt]0&0&-\chi _{n-1}(s)&0\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(s)\\\vdots \\\mathbf {e} _{n}(s)\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="0.8em 0.8em 0.8em 0.4em" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\mathbf {e} _{1}'(s)\\\vdots \\\mathbf {e} _{n}'(s)\\\end{bmatrix}}=\|\mathbf {r} '(s)\|\cdot {\begin{bmatrix}0&\chi _{1}(s)&0&0\\[4pt]-\chi _{1}(s)&\ddots &\ddots &0\\[4pt]0&\ddots &\ddots &\chi _{n-1}(s)\\[4pt]0&0&-\chi _{n-1}(s)&0\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(s)\\\vdots \\\mathbf {e} _{n}(s)\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750498c7230d9400f90eeee57566cb543480eb02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:70.782ex; height:19.176ex;" alt="{\displaystyle {\begin{bmatrix}\mathbf {e} _{1}'(s)\\\vdots \\\mathbf {e} _{n}'(s)\\\end{bmatrix}}=\|\mathbf {r} '(s)\|\cdot {\begin{bmatrix}0&\chi _{1}(s)&0&0\\[4pt]-\chi _{1}(s)&\ddots &\ddots &0\\[4pt]0&\ddots &\ddots &\chi _{n-1}(s)\\[4pt]0&0&-\chi _{n-1}(s)&0\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(s)\\\vdots \\\mathbf {e} _{n}(s)\\\end{bmatrix}}}"></span> </p><p>Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources. The top curvature <span class="texhtml"><i>χ</i><sub><i>n</i>-1</sub></span> (also called the torsion, in this context) and the last vector in the frame <span class="texhtml"><b>e</b><sub><i>n</i></sub></span>, differ by a sign </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {or} \left(\mathbf {r} ^{(1)},\dots ,\mathbf {r} ^{(n)}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>or</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {or} \left(\mathbf {r} ^{(1)},\dots ,\mathbf {r} ^{(n)}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c771a3c926f5f9501e7b7340b291dc51a5b5f4b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.063ex; height:4.843ex;" alt="{\displaystyle \operatorname {or} \left(\mathbf {r} ^{(1)},\dots ,\mathbf {r} ^{(n)}\right)}"></span> </p><p>(the orientation of the basis) from the usual torsion. The Frenet–Serret formulas are invariant under flipping the sign of both <span class="texhtml"><i>χ</i><sub><i>n</i>-1</sub></span> and <span class="texhtml"><b>e</b><sub><i>n</i></sub></span>, and this change of sign makes the frame positively oriented. As defined above, the frame inherits its orientation from the jet of <span class="texhtml"><b>r</b></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Proof_of_the_Frenet-Serret_formulas">Proof of the Frenet-Serret formulas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=3" title="Edit section: Proof of the Frenet-Serret formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first Frenet-Serret formula holds by the definition of the normal <span class="texhtml"><b>N</b></span> and the curvature <span class="texhtml mvar" style="font-style:italic;">κ</span>, and the third Frenet-Serret formula holds by the definition of the torsion <span class="texhtml mvar" style="font-style:italic;">τ</span>. Thus what is needed is to show the second Frenet-Serret formula. </p><p>Since <span class="texhtml"><b>T</b>, <b>N</b>, <b>B</b></span> are orthogonal unit vectors with <span class="texhtml"><b>B</b> = <b>T</b> × <b>N</b></span>, one also has <span class="texhtml"><b>T</b> = <b>N</b> × <b>B</b></span> and <span class="texhtml"><b>N</b> = <b>B</b> × <b>T</b></span>. Differentiating the last equation with respect to <span class="texhtml mvar" style="font-style:italic;">s</span> gives </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {N} }{\partial s}}=\left({\frac {\partial \mathbf {B} }{\partial s}}\right)\times \mathbf {T} +\mathbf {B} \times \left({\frac {\partial \mathbf {T} }{\partial s}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {N} }{\partial s}}=\left({\frac {\partial \mathbf {B} }{\partial s}}\right)\times \mathbf {T} +\mathbf {B} \times \left({\frac {\partial \mathbf {T} }{\partial s}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e89d25d0937f8d8136b35a8962b60b05ce0db3b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.537ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \mathbf {N} }{\partial s}}=\left({\frac {\partial \mathbf {B} }{\partial s}}\right)\times \mathbf {T} +\mathbf {B} \times \left({\frac {\partial \mathbf {T} }{\partial s}}\right)}"></span> </p><p>Using that <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 {\tfrac {\partial \mathbf {B} }{\partial s}}=-\tau \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mo>−</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial \mathbf {B} }{\partial s}}=-\tau \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291aba7bc636b7a27204cdbcd8e26e9508a04443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.312ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial \mathbf {B} }{\partial s}}=-\tau \mathbf {N} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\partial \mathbf {T} }{\partial s}}=\kappa \mathbf {N} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\partial \mathbf {T} }{\partial s}}=\kappa \mathbf {N} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/304e68b09c42260226d05f48acb84967fde6078e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.259ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\partial \mathbf {T} }{\partial s}}=\kappa \mathbf {N} ,}"></span> this becomes </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {N} }{\partial s}}&=-\tau (\mathbf {N} \times \mathbf {T} )+\kappa (\mathbf {B} \times \mathbf {N} )\\&=\tau \mathbf {B} -\kappa \mathbf {T} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>κ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>−</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {N} }{\partial s}}&=-\tau (\mathbf {N} \times \mathbf {T} )+\kappa (\mathbf {B} \times \mathbf {N} )\\&=\tau \mathbf {B} -\kappa \mathbf {T} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2efe648fd7958171b1b7b5b8aef4c04152a1705" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:32.528ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial \mathbf {N} }{\partial s}}&=-\tau (\mathbf {N} \times \mathbf {T} )+\kappa (\mathbf {B} \times \mathbf {N} )\\&=\tau \mathbf {B} -\kappa \mathbf {T} \end{aligned}}}"></span> </p><p>This is exactly the second Frenet-Serret formula. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_and_interpretation">Applications and interpretation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=4" title="Edit section: Applications and interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Kinematics_of_the_frame">Kinematics of the frame</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=5" title="Edit section: Kinematics of the frame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Frenet-Serret_moving_frame1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/d1/Frenet-Serret_moving_frame1.png" decoding="async" width="172" height="315" class="mw-file-element" data-file-width="172" data-file-height="315" /></a><figcaption>The Frenet–Serret frame moving along a <a href="/wiki/Helix" title="Helix">helix</a> in space</figcaption></figure> <p>The Frenet–Serret frame consisting of the tangent <span class="texhtml"><b>T</b></span>, normal <span class="texhtml"><b>N</b></span>, and binormal <span class="texhtml"><b>B</b></span> collectively forms an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> of 3-space. At each point of the curve, this <i>attaches</i> a <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a> or <a href="/wiki/Rectilinear_grid" class="mw-redirect" title="Rectilinear grid">rectilinear</a> <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a> (see image). </p><p>The Frenet–Serret formulas admit a <a href="/wiki/Kinematics" title="Kinematics">kinematic</a> interpretation. Imagine that an observer moves along the curve in time, using the attached frame at each point as their coordinate system. The Frenet–Serret formulas mean that this coordinate system is constantly rotating as an observer moves along the curve. Hence, this coordinate system is always <a href="/wiki/Non-inertial_reference_frame" title="Non-inertial reference frame">non-inertial</a>. The <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> of the observer's coordinate system is proportional to the <a href="/wiki/Darboux_vector" title="Darboux vector">Darboux vector</a> of the frame. </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:TNB_frame_momenta.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/TNB_frame_momenta.svg/220px-TNB_frame_momenta.svg.png" decoding="async" width="220" height="288" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/TNB_frame_momenta.svg/330px-TNB_frame_momenta.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/TNB_frame_momenta.svg/440px-TNB_frame_momenta.svg.png 2x" data-file-width="702" data-file-height="920" /></a><figcaption>A top whose axis is situated along the binormal is observed to rotate with angular speed <span class="texhtml mvar" style="font-style:italic;">κ</span>. If the axis is along the tangent, it is observed to rotate with angular speed <span class="texhtml mvar" style="font-style:italic;">τ</span>.</figcaption></figure> <p>Concretely, suppose that the observer carries an (inertial) <a href="/wiki/Spinning_top" title="Spinning top">top</a> (or <a href="/wiki/Gyroscope" title="Gyroscope">gyroscope</a>) with them along the curve. If the axis of the top points along the tangent to the curve, then it will be observed to rotate about its axis with angular velocity -τ relative to the observer's non-inertial coordinate system. If, on the other hand, the axis of the top points in the binormal direction, then it is observed to rotate with angular velocity -κ. This is easily visualized in the case when the curvature is a positive constant and the torsion vanishes. The observer is then in <a href="/wiki/Uniform_circular_motion" class="mw-redirect" title="Uniform circular motion">uniform circular motion</a>. If the top points in the direction of the binormal, then by <a href="/wiki/Conservation_of_angular_momentum" class="mw-redirect" title="Conservation of angular momentum">conservation of angular momentum</a> it must rotate in the <i>opposite</i> direction of the circular motion. In the limiting case when the curvature vanishes, the observer's normal <a href="/wiki/Precess" class="mw-redirect" title="Precess">precesses</a> about the tangent vector, and similarly the top will rotate in the opposite direction of this precession. </p><p>The general case is illustrated <a href="#Illustrations">below</a>. There are further <a href="https://commons.wikimedia.org/wiki/Category:Illustrations_for_curvature_and_torsion_of_curves" class="extiw" title="commons:Category:Illustrations for curvature and torsion of curves">illustrations</a> on Wikimedia. </p> <div class="mw-heading mw-heading4"><h4 id="Applications">Applications</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=6" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The kinematics of the frame have many applications in the sciences. </p> <ul><li>In the <a href="/wiki/Life_sciences" class="mw-redirect" title="Life sciences">life sciences</a>, particularly in models of microbial motion, considerations of the Frenet–Serret frame have been used to explain the mechanism by which a moving organism in a viscous medium changes its direction.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li>In physics, the Frenet–Serret frame is useful when it is impossible or inconvenient to assign a natural coordinate system for a trajectory. Such is often the case, for instance, in <a href="/wiki/Relativity_theory" class="mw-redirect" title="Relativity theory">relativity theory</a>. Within this setting, Frenet–Serret frames have been used to model the precession of a gyroscope in a gravitational well.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading4"><h4 id="Graphical_Illustrations">Graphical Illustrations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=7" title="Edit section: Graphical Illustrations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol><li>Example of a moving Frenet basis (<span class="texhtml"><b>T</b></span> in blue, <span class="texhtml"><b>N</b></span> in green, <span class="texhtml"><b>B</b></span> in purple) along <a href="/wiki/Viviani%27s_curve" title="Viviani's curve">Viviani's curve</a>.</li></ol> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Frenet-Serret-frame_along_Vivani-curve.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/14/Frenet-Serret-frame_along_Vivani-curve.gif" decoding="async" width="500" height="534" class="mw-file-element" data-file-width="500" data-file-height="534" /></a></span> </p> <ol><li class="mw-empty-elt"></li><li value="2"> On the example of a <a href="/wiki/Torus_knot" title="Torus knot">torus knot</a>, the tangent vector <span class="texhtml"><b>T</b></span>, the normal vector <span class="texhtml"><b>N</b></span>, and the binormal vector <span class="texhtml"><b>B</b></span>, along with the curvature <span class="texhtml"><i>κ</i>(<i>s</i>)</span>, and the torsion <span class="texhtml"><i>τ</i>(<i>s</i>)</span> are displayed. <br /> At the peaks of the torsion function the rotation of the Frenet–Serret frame <span class="texhtml">(<b>T</b>,<b>N</b>,<b>B</b>)</span> around the tangent vector is clearly visible.</li></ol> <p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Torus-Knot_nebeneinander_animated.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/70/Torus-Knot_nebeneinander_animated.gif" decoding="async" width="1024" height="435" class="mw-file-element" data-file-width="1024" data-file-height="435" /></a></span> </p> <ol><li class="mw-empty-elt"></li><li value="3"> The kinematic significance of the curvature is best illustrated with plane curves (having constant torsion equal to zero). See the page on <a href="/wiki/Curvature#Curvature_of_plane_curves" title="Curvature">curvature of plane curves</a>.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Frenet–Serret_formulas_in_calculus"><span id="Frenet.E2.80.93Serret_formulas_in_calculus"></span>Frenet–Serret formulas in calculus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=8" title="Edit section: Frenet–Serret formulas in calculus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Frenet–Serret formulas are frequently introduced in courses on <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a> as a companion to the study of space curves such as the <a href="/wiki/Helix" title="Helix">helix</a>. A helix can be characterized by the height <span class="texhtml">2π<i>h</i></span> and radius <span class="texhtml mvar" style="font-style:italic;">r</span> of a single turn. The curvature and torsion of a helix (with constant radius) are given by the formulas <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\kappa &={\frac {r}{r^{2}+h^{2}}}\\[4pt]\tau &=\pm {\frac {h}{r^{2}+h^{2}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.7em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>κ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>τ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\kappa &={\frac {r}{r^{2}+h^{2}}}\\[4pt]\tau &=\pm {\frac {h}{r^{2}+h^{2}}}.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af071782cf53d9b25d77cdac6aa05957bb35b750" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:15.817ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\kappa &={\frac {r}{r^{2}+h^{2}}}\\[4pt]\tau &=\pm {\frac {h}{r^{2}+h^{2}}}.\end{aligned}}}"></span> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Frenet-Serret_helices.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Frenet-Serret_helices.png/220px-Frenet-Serret_helices.png" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Frenet-Serret_helices.png/330px-Frenet-Serret_helices.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Frenet-Serret_helices.png/440px-Frenet-Serret_helices.png 2x" data-file-width="529" data-file-height="406" /></a><figcaption>Two helices (slinkies) in space. (a) A more compact helix with higher curvature and lower torsion. (b) A stretched out helix with slightly higher torsion but lower curvature.</figcaption></figure> <p>The sign of the torsion is determined by the right-handed or left-handed <a href="/wiki/Right-hand_rule" title="Right-hand rule">sense</a> in which the helix twists around its central axis. Explicitly, the parametrization of a single turn of a right-handed helix with height <span class="texhtml">2π<i>h</i></span> and radius <span class="texhtml mvar" style="font-style:italic;">r</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=r\cos t\\y&=r\sin t\\z&=ht\\(0&\leq t\leq 2\pi )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>h</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=r\cos t\\y&=r\sin t\\z&=ht\\(0&\leq t\leq 2\pi )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b0156516fa09375cfeca54eba941202b0d2b8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:13.254ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}x&=r\cos t\\y&=r\sin t\\z&=ht\\(0&\leq t\leq 2\pi )\end{aligned}}}"></span> and, for a left-handed helix, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}x&=r\cos t\\y&=-r\sin t\\z&=ht\\(0&\leq t\leq 2\pi ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>h</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mn>0</mn> </mtd> <mtd> <mi></mi> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=r\cos t\\y&=-r\sin t\\z&=ht\\(0&\leq t\leq 2\pi ).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4122b1bfc9d2940268a208ad2a1920c8fa20b756" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:13.901ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}x&=r\cos t\\y&=-r\sin t\\z&=ht\\(0&\leq t\leq 2\pi ).\end{aligned}}}"></span> Note that these are not the arc length parametrizations (in which case, each of <span class="texhtml"><i>x</i>, <i>y</i>, <i>z</i></span> would need to be divided by <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 {\sqrt {h^{2}+r^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {h^{2}+r^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5b8c97820c60aad39c4618f784dadc1f7ee7fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.66ex; height:3.509ex;" alt="{\displaystyle {\sqrt {h^{2}+r^{2}}}}"></span>.) </p><p>In his expository writings on the geometry of curves, <a href="/wiki/Rudy_Rucker" title="Rudy Rucker">Rudy Rucker</a><sup id="cite_ref-rucker_5-0" class="reference"><a href="#cite_note-rucker-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> employs the model of a <a href="/wiki/Slinky" title="Slinky">slinky</a> to explain the meaning of the torsion and curvature. The slinky, he says, is characterized by the property that the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{2}=h^{2}+r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{2}=h^{2}+r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9d42cddfc55f7dcceba4b2b9713deb1a73a825" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.232ex; height:2.843ex;" alt="{\displaystyle A^{2}=h^{2}+r^{2}}"></span> remains constant if the slinky is vertically stretched out along its central axis. (Here <span class="texhtml">2π<i>h</i></span> is the height of a single twist of the slinky, and <span class="texhtml mvar" style="font-style:italic;">r</span> the radius.) In particular, curvature and torsion are complementary in the sense that the torsion can be increased at the expense of curvature by stretching out the slinky. </p> <div class="mw-heading mw-heading3"><h3 id="Taylor_expansion">Taylor expansion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=9" title="Edit section: Taylor expansion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Repeatedly differentiating the curve and applying the Frenet–Serret formulas gives the following <a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor approximation</a> to the curve near <span class="texhtml"><i>s</i> = 0</span> if the curve is parameterized by arclength:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} (s)=\mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>κ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>κ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>κ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (s)=\mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c4b6813490cbceb016432f568a30b9378779a1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:97.086ex; height:7.509ex;" alt="{\displaystyle \mathbf {r} (s)=\mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3}).}"></span> </p><p>For a generic curve with nonvanishing torsion, the projection of the curve onto various coordinate planes in the <span class="texhtml"><b>T</b>, <b>N</b>, <b>B</b></span> coordinate system at <span class="nowrap"><i>s</i> = 0</span> have the following interpretations: </p> <ul><li>The <i><a href="/wiki/Osculating_plane" title="Osculating plane">osculating plane</a></i> is the plane <a href="/wiki/Linear_span" title="Linear span">containing</a> <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>N</b></span>. The projection of the curve onto this plane has the form:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} (0)+s\mathbf {T} (0)+{\frac {s^{2}\kappa (0)}{2}}\mathbf {N} (0)+o(s^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>κ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (0)+s\mathbf {T} (0)+{\frac {s^{2}\kappa (0)}{2}}\mathbf {N} (0)+o(s^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f4f059ac57dde4145e3fabb1e85dc749f5ca14d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.6ex; height:5.843ex;" alt="{\displaystyle \mathbf {r} (0)+s\mathbf {T} (0)+{\frac {s^{2}\kappa (0)}{2}}\mathbf {N} (0)+o(s^{2}).}"></span>This is a <a href="/wiki/Parabola" title="Parabola">parabola</a> up to terms of order <span class="texhtml"><i>O</i>(<i>s</i><sup>2</sup>)</span>, whose curvature at 0 is equal to <span class="texhtml"><i>κ</i>(0)</span>. The osculating plane has the special property that the distance from the curve to the osculating plane is <span class="texhtml"><i>O</i>(<i>s</i><sup>3</sup>)</span>, while the distance from the curve to any other plane is no better than <span class="texhtml"><i>O</i>(<i>s</i><sup>2</sup>)</span>. This can be seen from the above Taylor expansion. Thus in a sense the osculating plane is the closest plane to the curve at a given point.</li> <li>The <i><a href="/wiki/Normal_plane_(geometry)" title="Normal plane (geometry)">normal plane</a></i> is the plane containing <span class="texhtml"><b>N</b></span> and <span class="texhtml"><b>B</b></span>. The projection of the curve onto this plane has the form:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>κ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>κ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>κ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfca0bbe506f2dd1b48323c901a4369bbfa4607" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.323ex; height:7.509ex;" alt="{\displaystyle \mathbf {r} (0)+\left({\frac {s^{2}\kappa (0)}{2}}+{\frac {s^{3}\kappa '(0)}{6}}\right)\mathbf {N} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}"></span>which is a <a href="/wiki/Cuspidal_cubic" class="mw-redirect" title="Cuspidal cubic">cuspidal cubic</a> to order <span class="texhtml"><i>o</i>(<i>s</i><sup>3</sup>)</span>.</li> <li>The <b>rectifying plane</b> is the plane containing <span class="texhtml"><b>T</b></span> and <span class="texhtml"><b>B</b></span>. The projection of the curve onto this plane is:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>κ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5f40328623d657b01515997755287bc98b4f75c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.259ex; height:7.509ex;" alt="{\displaystyle \mathbf {r} (0)+\left(s-{\frac {s^{3}\kappa ^{2}(0)}{6}}\right)\mathbf {T} (0)+\left({\frac {s^{3}\kappa (0)\tau (0)}{6}}\right)\mathbf {B} (0)+o(s^{3})}"></span>which traces out the graph of a <a href="/wiki/Cubic_polynomial" class="mw-redirect" title="Cubic polynomial">cubic polynomial</a> to order <span class="texhtml"><i>o</i>(<i>s</i><sup>3</sup>)</span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Ribbons_and_tubes">Ribbons and tubes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=10" title="Edit section: Ribbons and tubes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Ribbon-Frenet.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Ribbon-Frenet.png/350px-Ribbon-Frenet.png" decoding="async" width="350" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Ribbon-Frenet.png/525px-Ribbon-Frenet.png 1.5x, //upload.wikimedia.org/wikipedia/commons/d/dd/Ribbon-Frenet.png 2x" data-file-width="552" data-file-height="346" /></a><figcaption>A ribbon defined by a curve of constant torsion and a highly oscillating curvature. The arc length parameterization of the curve was defined via integration of the Frenet–Serret equations.</figcaption></figure> <p>The Frenet–Serret apparatus allows one to define certain optimal <i>ribbons</i> and <i>tubes</i> centered around a curve. These have diverse applications in <a href="/wiki/Materials_science" title="Materials science">materials science</a> and <a href="/wiki/Elasticity_theory" class="mw-redirect" title="Elasticity theory">elasticity theory</a>,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> as well as to <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The <b>Frenet ribbon</b><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> along a curve <span class="texhtml mvar" style="font-style:italic;">C</span> is the surface traced out by sweeping the line segment <span class="texhtml">[−<b>N</b>,<b>N</b>]</span> generated by the unit normal along the curve. This surface is sometimes confused with the <a href="/wiki/Tangent_developable" title="Tangent developable">tangent developable</a>, which is the <a href="/wiki/Envelope_(mathematics)" title="Envelope (mathematics)">envelope</a> <span class="texhtml mvar" style="font-style:italic;">E</span> of the osculating planes of <span class="texhtml mvar" style="font-style:italic;">C</span>. This is perhaps because both the Frenet ribbon and <span class="texhtml mvar" style="font-style:italic;">E</span> exhibit similar properties along <span class="texhtml mvar" style="font-style:italic;">C</span>. Namely, the tangent planes of both sheets of <span class="texhtml mvar" style="font-style:italic;">E</span>, near the singular locus <span class="texhtml mvar" style="font-style:italic;">C</span> where these sheets intersect, approach the osculating planes of <span class="texhtml mvar" style="font-style:italic;">C</span>; the tangent planes of the Frenet ribbon along <span class="texhtml mvar" style="font-style:italic;">C</span> are equal to these osculating planes. The Frenet ribbon is in general not developable. </p> <div class="mw-heading mw-heading3"><h3 id="Congruence_of_curves">Congruence of curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=11" title="Edit section: Congruence of curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In classical <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, one is interested in studying the properties of figures in the plane which are <i>invariant</i> under congruence, so that if two figures are congruent then they must have the same properties. The Frenet–Serret apparatus presents the curvature and torsion as numerical invariants of a space curve. </p><p>Roughly speaking, two curves <span class="texhtml mvar" style="font-style:italic;">C</span> and <span class="texhtml mvar" style="font-style:italic;">C'</span> in space are <i>congruent</i> if one can be rigidly moved to the other. A rigid motion consists of a combination of a translation and a rotation. A translation moves one point of <span class="texhtml mvar" style="font-style:italic;">C</span> to a point of <span class="texhtml mvar" style="font-style:italic;">C'</span>. The rotation then adjusts the orientation of the curve <span class="texhtml mvar" style="font-style:italic;">C</span> to line up with that of <span class="texhtml mvar" style="font-style:italic;">C'</span>. Such a combination of translation and rotation is called a <a href="/wiki/Euclidean_transformation" class="mw-redirect" title="Euclidean transformation">Euclidean motion</a>. In terms of the parametrization <span class="texhtml"><b>r</b>(<i>t</i>)</span> defining the first curve <span class="texhtml mvar" style="font-style:italic;">C</span>, a general Euclidean motion of <span class="texhtml mvar" style="font-style:italic;">C</span> is a composite of the following operations: </p> <ul><li>(<i>Translation</i>) <span class="texhtml"><b>r</b>(<i>t</i>) → <b>r</b>(<i>t</i>) + <b>v</b></span>, where <span class="texhtml"><b>v</b></span> is a constant vector.</li> <li>(<i>Rotation</i>) <span class="texhtml"><b>r</b>(<i>t</i>) + <b>v</b> → <i>M</i>(<b>r</b>(<i>t</i>) + <b>v</b>)</span>, where <span class="texhtml mvar" style="font-style:italic;">M</span> is the matrix of a rotation.</li></ul> <p>The Frenet–Serret frame is particularly well-behaved with regard to Euclidean motions. First, since <span class="texhtml"><b>T</b></span>, <span class="texhtml"><b>N</b></span>, and <span class="texhtml"><b>B</b></span> can all be given as successive derivatives of the parametrization of the curve, each of them is insensitive to the addition of a constant vector to <span class="texhtml"><b>r</b>(<i>t</i>)</span>. Intuitively, the <span class="texhtml"><b>TNB</b></span> frame attached to <span class="texhtml"><b>r</b>(<i>t</i>)</span> is the same as the <span class="texhtml"><b>TNB</b></span> frame attached to the new curve <span class="texhtml"><b>r</b>(<i>t</i>) + <b>v</b></span>. </p><p>This leaves only the rotations to consider. Intuitively, if we apply a rotation <span class="texhtml mvar" style="font-style:italic;">M</span> to the curve, then the <span class="texhtml"><b>TNB</b></span> frame also rotates. More precisely, the matrix <span class="texhtml mvar" style="font-style:italic;">Q</span> whose rows are the <span class="texhtml"><b>TNB</b></span> vectors of the Frenet–Serret frame changes by the matrix of a rotation </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q\rightarrow QM.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">→</mo> <mi>Q</mi> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\rightarrow QM.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21eccab2af4225beab6b90ec66ddb7370cd953f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.38ex; height:2.509ex;" alt="{\displaystyle Q\rightarrow QM.}"></span> </p><p><i>A fortiori</i>, the matrix <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 {\tfrac {dQ}{ds}}Q^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mstyle> </mrow> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {dQ}{ds}}Q^{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094971e7e191bd9aa1fe5b065900fa41ef34c729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.253ex; height:4.176ex;" alt="{\displaystyle {\tfrac {dQ}{ds}}Q^{\mathrm {T} }}"></span> is unaffected by a rotation: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} (QM)}{\mathrm {d} s}}(QM)^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}MM^{\top }Q^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}Q^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mi>M</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>Q</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <mi>M</mi> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>Q</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} (QM)}{\mathrm {d} s}}(QM)^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}MM^{\top }Q^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}Q^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdfd89caa903eb1d1553248b5285b504f81281f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.1ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} (QM)}{\mathrm {d} s}}(QM)^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}MM^{\top }Q^{\top }={\frac {\mathrm {d} Q}{\mathrm {d} s}}Q^{\top }}"></span> </p><p>since <span class="texhtml"><i>MM</i><sup>T</sup> = <i>I</i></span> for the matrix of a rotation. </p><p>Hence the entries <span class="texhtml mvar" style="font-style:italic;">κ</span> and <span class="texhtml mvar" style="font-style:italic;">τ</span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {dQ}{ds}}Q^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>d</mi> <mi>Q</mi> </mrow> <mrow> <mi>d</mi> <mi>s</mi> </mrow> </mfrac> </mstyle> </mrow> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {dQ}{ds}}Q^{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/094971e7e191bd9aa1fe5b065900fa41ef34c729" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.253ex; height:4.176ex;" alt="{\displaystyle {\tfrac {dQ}{ds}}Q^{\mathrm {T} }}"></span> are <i>invariants</i> of the curve under Euclidean motions: if a Euclidean motion is applied to a curve, then the resulting curve has <i>the same</i> curvature and torsion. </p><p>Moreover, using the Frenet–Serret frame, one can also prove the converse: any two curves having the same curvature and torsion functions must be congruent by a Euclidean motion. Roughly speaking, the Frenet–Serret formulas express the <a href="/wiki/Darboux_derivative" title="Darboux derivative">Darboux derivative</a> of the <span class="texhtml"><b>TNB</b></span> frame. If the Darboux derivatives of two frames are equal, then a version of the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">fundamental theorem of calculus</a> asserts that the curves are congruent. In particular, the curvature and torsion are a <i>complete</i> set of invariants for a curve in three-dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Other_expressions_of_the_frame">Other expressions of the frame</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=12" title="Edit section: Other expressions of the frame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formulas given above for <span class="texhtml"><b>T</b></span>, <span class="texhtml"><b>N</b></span>, and <span class="texhtml"><b>B</b></span> depend on the curve being given in terms of the arclength parameter. This is a natural assumption in <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of <a href="/wiki/Gauge_theory" title="Gauge theory">gauge</a>. However, it may be awkward to work with in practice. A number of other equivalent expressions are available. </p><p>Suppose that the curve is given by <span class="texhtml"><b>r</b>(<i>t</i>)</span>, where the parameter <span class="texhtml mvar" style="font-style:italic;">t</span> need no longer be arclength. Then the unit tangent vector <span class="texhtml"><b>T</b></span> may be written as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\|\mathbf {r} '(t)\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\|\mathbf {r} '(t)\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e166624cc8a7262ac2d85a14ef8c285a85c03c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.203ex; height:6.509ex;" alt="{\displaystyle \mathbf {T} (t)={\frac {\mathbf {r} '(t)}{\|\mathbf {r} '(t)\|}}}"></span> </p><p>The normal vector <span class="texhtml"><b>N</b></span> takes the form </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} (t)={\frac {\mathbf {T} '(t)}{\|\mathbf {T} '(t)\|}}={\frac {\mathbf {r} '(t)\times \left(\mathbf {r} ''(t)\times \mathbf {r} '(t)\right)}{\left\|\mathbf {r} '(t)\right\|\,\left\|\mathbf {r} ''(t)\times \mathbf {r} '(t)\right\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> <mspace width="thinmathspace" /> <mrow> <mo symmetric="true">‖</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo symmetric="true">‖</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} (t)={\frac {\mathbf {T} '(t)}{\|\mathbf {T} '(t)\|}}={\frac {\mathbf {r} '(t)\times \left(\mathbf {r} ''(t)\times \mathbf {r} '(t)\right)}{\left\|\mathbf {r} '(t)\right\|\,\left\|\mathbf {r} ''(t)\times \mathbf {r} '(t)\right\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2d8fe5022eca69285fa084b0f41a2d192bbe76" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:42.151ex; height:6.509ex;" alt="{\displaystyle \mathbf {N} (t)={\frac {\mathbf {T} '(t)}{\|\mathbf {T} '(t)\|}}={\frac {\mathbf {r} '(t)\times \left(\mathbf {r} ''(t)\times \mathbf {r} '(t)\right)}{\left\|\mathbf {r} '(t)\right\|\,\left\|\mathbf {r} ''(t)\times \mathbf {r} '(t)\right\|}}}"></span> </p><p>The binormal <span class="texhtml"><b>B</b></span> is then </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/400caf2b46ba0bb3816061dd6d4550a8818828bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.161ex; height:6.509ex;" alt="{\displaystyle \mathbf {B} (t)=\mathbf {T} (t)\times \mathbf {N} (t)={\frac {\mathbf {r} '(t)\times \mathbf {r} ''(t)}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}}}"></span> </p><p>An alternative way to arrive at the same expressions is to take the first three derivatives of the curve <span class="texhtml"><b>r</b>′(<i>t</i>), <b>r</b>′′(<i>t</i>), <b>r</b>′′′(<i>t</i>)</span>, and to apply the <a href="/wiki/Gram-Schmidt_process" class="mw-redirect" title="Gram-Schmidt process">Gram-Schmidt process</a>. The resulting ordered <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> is precisely the <span class="texhtml"><b>TNB</b></span> frame. This procedure also generalizes to produce Frenet frames in higher dimensions. </p><p>In terms of the parameter <span class="texhtml mvar" style="font-style:italic;">t</span>, the Frenet–Serret formulas pick up an additional factor of <span class="texhtml">||<b>r</b>′(<i>t</i>)||</span> because of the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}=\|\mathbf {r} '(t)\|{\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>κ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>κ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mi>τ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}=\|\mathbf {r} '(t)\|{\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c17126caae1dd93af5e5a8aa91a88895f0e5094" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:40.571ex; height:9.176ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}=\|\mathbf {r} '(t)\|{\begin{bmatrix}0&\kappa &0\\-\kappa &0&\tau \\0&-\tau &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}}"></span> </p><p>Explicit expressions for the curvature and torsion may be computed. For example, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\frac {\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}{\|\mathbf {r} '(t)\|^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}{\|\mathbf {r} '(t)\|^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1733ed2ae606ff64f8139d41ac44729e7d6e11d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.763ex; height:6.509ex;" alt="{\displaystyle \kappa ={\frac {\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|}{\|\mathbf {r} '(t)\|^{3}}}}"></span> </p><p>The torsion may be expressed using a <a href="/wiki/Scalar_triple_product" class="mw-redirect" title="Scalar triple product">scalar triple product</a> as follows, </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\frac {[\mathbf {r} '(t),\mathbf {r} ''(t),\mathbf {r} '''(t)]}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>‴</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {[\mathbf {r} '(t),\mathbf {r} ''(t),\mathbf {r} '''(t)]}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6c72460a68ff77535c83f91d4197a09488d0d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.163ex; height:6.509ex;" alt="{\displaystyle \tau ={\frac {[\mathbf {r} '(t),\mathbf {r} ''(t),\mathbf {r} '''(t)]}{\|\mathbf {r} '(t)\times \mathbf {r} ''(t)\|^{2}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=13" title="Edit section: Special cases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the curvature is always zero then the curve will be a straight line. Here the vectors <span class="texhtml"><b>N</b>, <b>B</b></span> and the torsion are not well defined. </p><p>If the torsion is always zero then the curve will lie in a plane. </p><p>A curve may have nonzero curvature and zero torsion. For example, the <a href="/wiki/Circle" title="Circle">circle</a> of radius <span class="texhtml mvar" style="font-style:italic;">R</span> given by <span class="texhtml"><b>r</b>(<i>t</i>) = (<i>R</i> cos <i>t</i>, <i>R</i> sin <i>t</i>, 0)</span> in the <span class="texhtml"><i>z</i> = 0</span> plane has zero torsion and curvature equal to <span class="texhtml">1/<i>R</i></span>. The converse, however, is false. That is, a regular curve with nonzero torsion must have nonzero curvature. This is just the contrapositive of the fact that zero curvature implies zero torsion. </p><p>A <a href="/wiki/Helix" title="Helix">helix</a> has constant curvature and constant torsion. </p> <div class="mw-heading mw-heading3"><h3 id="Plane_curves">Plane curves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=14" title="Edit section: Plane curves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Plane_curve" title="Plane curve">Plane curve</a></div> <p>If a curve <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 {\bf {r}}(t)=\langle x(t),y(t),0\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bf {r}}(t)=\langle x(t),y(t),0\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec69c43938ed21c04b9827a98ea0534c21d3c4ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.672ex; height:2.843ex;" alt="{\displaystyle {\bf {r}}(t)=\langle x(t),y(t),0\rangle }"></span> is contained in the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane, then its tangent vector <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 \mathbf {T} ={\tfrac {\mathbf {r} '(t)}{||\mathbf {r} '(t)||}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ={\tfrac {\mathbf {r} '(t)}{||\mathbf {r} '(t)||}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46af1d5cd14a793c05f2cb7f0ddb6b9901cf4627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.807ex; height:4.843ex;" alt="{\displaystyle \mathbf {T} ={\tfrac {\mathbf {r} '(t)}{||\mathbf {r} '(t)||}}}"></span> and principal unit normal vector <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 \mathbf {N} ={\tfrac {\mathbf {T} '(t)}{||\mathbf {T} '(t)||}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} ={\tfrac {\mathbf {T} '(t)}{||\mathbf {T} '(t)||}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbff2c5afb9fc40d6754185827eec2a67ad5100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.575ex; height:4.843ex;" alt="{\displaystyle \mathbf {N} ={\tfrac {\mathbf {T} '(t)}{||\mathbf {T} '(t)||}}}"></span> will also lie in the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane. As a result, the unit binormal vector <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 \mathbf {B} =\mathbf {T} \times \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {T} \times \mathbf {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f770d60a50f1432747399655fe3d6c9b31add603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.791ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} =\mathbf {T} \times \mathbf {N} }"></span> is perpendicular to the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane and thus must be either <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 \langle 0,0,1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0,0,1\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85363e2460790d5f13e4cd425b9ac3805b2a38f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle \langle 0,0,1\rangle }"></span> or <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 \langle 0,0,-1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0,0,-1\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9855f25cd0dbb51307501cb3d1acfedaefee2421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle \langle 0,0,-1\rangle }"></span>. By the right-hand rule <span class="texhtml"><b>B</b></span> will be <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 \langle 0,0,1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0,0,1\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85363e2460790d5f13e4cd425b9ac3805b2a38f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.365ex; height:2.843ex;" alt="{\displaystyle \langle 0,0,1\rangle }"></span> if, when viewed from above, the curve's trajectory is turning leftward, and will be <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 \langle 0,0,-1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle 0,0,-1\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9855f25cd0dbb51307501cb3d1acfedaefee2421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.173ex; height:2.843ex;" alt="{\displaystyle \langle 0,0,-1\rangle }"></span> if it is turning rightward. As a result, the torsion <span class="texhtml mvar" style="font-style:italic;">τ</span> will always be zero and the formula <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 {\tfrac {||\mathbf {r} '(t)\times \mathbf {r} ''(t)||}{||\mathbf {r} '(t)||^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {||\mathbf {r} '(t)\times \mathbf {r} ''(t)||}{||\mathbf {r} '(t)||^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8828a785dacdd6ae9a8d9ae42965b0feb717e3a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.679ex; height:5.176ex;" alt="{\displaystyle {\tfrac {||\mathbf {r} '(t)\times \mathbf {r} ''(t)||}{||\mathbf {r} '(t)||^{3}}}}"></span> for the curvature <span class="texhtml mvar" style="font-style:italic;">κ</span> becomes <br /> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa ={\frac {|x'(t)y''(t)-y'(t)x''(t)|}{{\bigl [}(x'(t))^{2}+(y'(t))^{2}{\bigr ]}^{3/2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>y</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <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> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa ={\frac {|x'(t)y''(t)-y'(t)x''(t)|}{{\bigl [}(x'(t))^{2}+(y'(t))^{2}{\bigr ]}^{3/2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbd78d0dd11b7517e21181c535483a92d478804" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:28.627ex; height:7.509ex;" alt="{\displaystyle \kappa ={\frac {|x'(t)y''(t)-y'(t)x''(t)|}{{\bigl [}(x'(t))^{2}+(y'(t))^{2}{\bigr ]}^{3/2}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, 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transverse, normal">Radial, transverse, normal</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFKühnel2002">Kühnel 2002</a>, §1.9</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Only the first <span class="texhtml"><i>n</i> − 1</span> actually need to be linearly independent, as the final remaining frame vector <span class="texhtml"><b>e</b><sub><i>n</i></sub></span> can be chosen as the unit vector orthogonal to the span of the others, such that the resulting frame is positively oriented.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Crenshaw (1993).</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Iyer and Vishveshwara (1993).</span> </li> <li id="cite_note-rucker-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-rucker_5-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRucker1999" class="citation web cs1">Rucker, Rudy (1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20041015020304/http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm">"Watching Flies Fly: Kappatau Space Curves"</a>. San Jose State University. Archived from <a rel="nofollow" class="external text" href="http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm">the original</a> on 15 October 2004.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Watching+Flies+Fly%3A+Kappatau+Space+Curves&rft.pub=San+Jose+State+University&rft.date=1999&rft.aulast=Rucker&rft.aufirst=Rudy&rft_id=http%3A%2F%2Fwww.mathcs.sjsu.edu%2Ffaculty%2Frucker%2Fkaptaudoc%2Fktpaper.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFKühnel2002">Kühnel 2002</a>, p. 19</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Goriely <i>et al.</i> (2006).</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Hanson.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">For terminology, see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSternberg1964" class="citation book cs1">Sternberg (1964). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/lecturesondiffer00ster_853"><i>Lectures on Differential Geometry</i></a></span>. Englewood Cliffs, N.J., Prentice-Hall. p. <a rel="nofollow" class="external text" href="https://archive.org/details/lecturesondiffer00ster_853/page/n263">252</a>-254. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780135271506" title="Special:BookSources/9780135271506"><bdi>9780135271506</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Differential+Geometry&rft.pages=252-254&rft.pub=Englewood+Cliffs%2C+N.J.%2C+Prentice-Hall&rft.date=1964&rft.isbn=9780135271506&rft.au=Sternberg&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flecturesondiffer00ster_853&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrenshawEdelstein-Keshet1993" class="citation cs2">Crenshaw, H.C.; Edelstein-Keshet, L. (1993), "Orientation by Helical Motion II. Changing the direction of the axis of motion", <i>Bulletin of Mathematical Biology</i>, <b>55</b> (1): 213–230, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0092-8240%2805%2980070-9">10.1016/s0092-8240(05)80070-9</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:50734771">50734771</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+Mathematical+Biology&rft.atitle=Orientation+by+Helical+Motion+II.+Changing+the+direction+of+the+axis+of+motion&rft.volume=55&rft.issue=1&rft.pages=213-230&rft.date=1993&rft_id=info%3Adoi%2F10.1016%2Fs0092-8240%2805%2980070-9&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A50734771%23id-name%3DS2CID&rft.aulast=Crenshaw&rft.aufirst=H.C.&rft.au=Edelstein-Keshet%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEtgenHilleSalas1995" class="citation cs2">Etgen, Garret; Hille, Einar; Salas, Saturnino (1995), <i>Salas and Hille's Calculus — One and Several Variables</i> (7th ed.), John Wiley & Sons, p. 896</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Salas+and+Hille%27s+Calculus+%26mdash%3B+One+and+Several+Variables&rft.pages=896&rft.edition=7th&rft.pub=John+Wiley+%26+Sons&rft.date=1995&rft.aulast=Etgen&rft.aufirst=Garret&rft.au=Hille%2C+Einar&rft.au=Salas%2C+Saturnino&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrenet1847" class="citation cs2">Frenet, F. (1847), <a rel="nofollow" class="external text" href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A22_0.pdf"><i>Sur les courbes à double courbure</i></a> <span class="cs1-format">(PDF)</span>, Thèse, Toulouse</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sur+les+courbes+%C3%A0+double+courbure&rft.place=Th%C3%A8se%2C+Toulouse&rft.date=1847&rft.aulast=Frenet&rft.aufirst=F.&rft_id=http%3A%2F%2Fsites.mathdoc.fr%2FJMPA%2FPDF%2FJMPA_1852_1_17_A22_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>. Abstract in <i>Journal de Mathématiques Pures et Appliquées</i> <b>17</b>, 1852.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGorielyRobertson-TessiTaborVandiver2006" class="citation cs2">Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006), "Elastic growth models", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061229011619/http://math.arizona.edu/~goriely/Papers/2006-biomat.pdf"><i>BIOMAT-2006</i></a> <span class="cs1-format">(PDF)</span>, Springer-Verlag, archived from <a rel="nofollow" class="external text" href="http://math.arizona.edu/~goriely/Papers/2006-biomat.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2006-12-29</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Elastic+growth+models&rft.btitle=BIOMAT-2006&rft.pub=Springer-Verlag&rft.date=2006&rft.aulast=Goriely&rft.aufirst=A.&rft.au=Robertson-Tessi%2C+M.&rft.au=Tabor%2C+M.&rft.au=Vandiver%2C+R.&rft_id=http%3A%2F%2Fmath.arizona.edu%2F~goriely%2FPapers%2F2006-biomat.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1974" class="citation cs2"><a href="/wiki/Phillip_Griffiths" title="Phillip Griffiths">Griffiths, Phillip</a> (1974), "On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry", <i><a href="/wiki/Duke_Mathematical_Journal" title="Duke Mathematical Journal">Duke Mathematical Journal</a></i>, <b>41</b> (4): 775–814, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1215%2FS0012-7094-74-04180-5">10.1215/S0012-7094-74-04180-5</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12966544">12966544</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Duke+Mathematical+Journal&rft.atitle=On+Cartan%27s+method+of+Lie+groups+and+moving+frames+as+applied+to+uniqueness+and+existence+questions+in+differential+geometry&rft.volume=41&rft.issue=4&rft.pages=775-814&rft.date=1974&rft_id=info%3Adoi%2F10.1215%2FS0012-7094-74-04180-5&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12966544%23id-name%3DS2CID&rft.aulast=Griffiths&rft.aufirst=Phillip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuggenheimer1977" class="citation cs2">Guggenheimer, Heinrich (1977), <i>Differential Geometry</i>, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-63433-7" title="Special:BookSources/0-486-63433-7"><bdi>0-486-63433-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Geometry&rft.pub=Dover&rft.date=1977&rft.isbn=0-486-63433-7&rft.aulast=Guggenheimer&rft.aufirst=Heinrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanson2007" class="citation cs2">Hanson, A.J. (2007), <a rel="nofollow" class="external text" href="http://www.cs.indiana.edu/pub/techreports/TR407.pdf">"Quaternion Frenet Frames: Making Optimal Tubes and Ribbons from Curves"</a> <span class="cs1-format">(PDF)</span>, <i>Indiana University Technical Report</i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Indiana+University+Technical+Report&rft.atitle=Quaternion+Frenet+Frames%3A+Making+Optimal+Tubes+and+Ribbons+from+Curves&rft.date=2007&rft.aulast=Hanson&rft.aufirst=A.J.&rft_id=http%3A%2F%2Fwww.cs.indiana.edu%2Fpub%2Ftechreports%2FTR407.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIyerVishveshwara1993" class="citation cs2">Iyer, B.R.; Vishveshwara, C.V. (1993), "Frenet-Serret description of gyroscopic precession", <i>Phys. Rev.</i>, D, <b>48</b> (12): 5706–5720, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/9310019">gr-qc/9310019</a></span>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhRvD..48.5706I">1993PhRvD..48.5706I</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysrevd.48.5706">10.1103/physrevd.48.5706</a>, <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/10016237">10016237</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119458843">119458843</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Phys.+Rev.&rft.atitle=Frenet-Serret+description+of+gyroscopic+precession&rft.volume=48&rft.issue=12&rft.pages=5706-5720&rft.date=1993&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119458843%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1993PhRvD..48.5706I&rft_id=info%3Aarxiv%2Fgr-qc%2F9310019&rft_id=info%3Apmid%2F10016237&rft_id=info%3Adoi%2F10.1103%2Fphysrevd.48.5706&rft.aulast=Iyer&rft.aufirst=B.R.&rft.au=Vishveshwara%2C+C.V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJordan1874" class="citation cs2">Jordan, Camille (1874), "Sur la théorie des courbes dans l'espace à n dimensions", <i>C. R. Acad. Sci. Paris</i>, <b>79</b>: 795–797</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=C.+R.+Acad.+Sci.+Paris&rft.atitle=Sur+la+th%C3%A9orie+des+courbes+dans+l%27espace+%C3%A0+n+dimensions&rft.volume=79&rft.pages=795-797&rft.date=1874&rft.aulast=Jordan&rft.aufirst=Camille&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKühnel2002" class="citation cs2">Kühnel, Wolfgang (2002), <i>Differential geometry</i>, Student Mathematical Library, vol. 16, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2656-0" title="Special:BookSources/978-0-8218-2656-0"><bdi>978-0-8218-2656-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1882174">1882174</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+geometry&rft.place=Providence%2C+R.I.&rft.series=Student+Mathematical+Library&rft.pub=American+Mathematical+Society&rft.date=2002&rft.isbn=978-0-8218-2656-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1882174%23id-name%3DMR&rft.aulast=K%C3%BChnel&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerret1851" class="citation cs2">Serret, J. A. (1851), <a rel="nofollow" class="external text" href="http://sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A12_0.pdf">"Sur quelques formules relatives à la théorie des courbes à double courbure"</a> <span class="cs1-format">(PDF)</span>, <i>Journal de Mathématiques Pures et Appliquées</i>, <b>16</b></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+Math%C3%A9matiques+Pures+et+Appliqu%C3%A9es&rft.atitle=Sur+quelques+formules+relatives+%C3%A0+la+th%C3%A9orie+des+courbes+%C3%A0+double+courbure&rft.volume=16&rft.date=1851&rft.aulast=Serret&rft.aufirst=J.+A.&rft_id=http%3A%2F%2Fsites.mathdoc.fr%2FJMPA%2FPDF%2FJMPA_1851_1_16_A12_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpivak1999" class="citation cs2"><a href="/wiki/Michael_Spivak" title="Michael Spivak">Spivak, Michael</a> (1999), <i>A Comprehensive Introduction to Differential Geometry (Volume Two)</i>, Publish or Perish, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Comprehensive+Introduction+to+Differential+Geometry+%28Volume+Two%29&rft.pub=Publish+or+Perish%2C+Inc.&rft.date=1999&rft.aulast=Spivak&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSternberg1964" class="citation cs2">Sternberg, Shlomo (1964), <i>Lectures on Differential Geometry</i>, Prentice-Hall</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Differential+Geometry&rft.pub=Prentice-Hall&rft.date=1964&rft.aulast=Sternberg&rft.aufirst=Shlomo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStruik1961" class="citation cs2">Struik, Dirk J. (1961), <i>Lectures on Classical Differential Geometry</i>, Reading, Mass: Addison-Wesley</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Classical+Differential+Geometry&rft.place=Reading%2C+Mass&rft.pub=Addison-Wesley&rft.date=1961&rft.aulast=Struik&rft.aufirst=Dirk+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFrenet%E2%80%93Serret+formulas" class="Z3988"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Frenet%E2%80%93Serret_formulas&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.math.uni-muenster.de/u/urs.hartl/gifs/CurvatureAndTorsionOfCurves.mw">Create your own animated illustrations of moving Frenet-Serret frames, curvature and torsion functions</a> (<a href="/wiki/Maple_(software)" title="Maple (software)">Maple</a> Worksheet)</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20041015020304/http://www.mathcs.sjsu.edu/faculty/rucker/kaptaudoc/ktpaper.htm">Rudy Rucker's KappaTau Paper</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.math.byu.edu/~math302/content/learningmod/trihedron/trihedron.html">Very nice visual representation for the trihedron</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist 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title="Template talk:Curvature"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Curvature" title="Special:EditPage/Template:Curvature"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Various_notions_of_curvature_defined_in_differential_geometry" style="font-size:114%;margin:0 4em">Various notions of <a href="/wiki/Curvature" title="Curvature">curvature</a> defined in <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_geometry_of_curves" class="mw-redirect" title="Differential geometry of curves">Differential geometry <br />of curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Curvature" title="Curvature">Curvature</a></li> <li><a href="/wiki/Torsion_of_a_curve" title="Torsion of a curve">Torsion of a curve</a></li> <li><a class="mw-selflink selflink">Frenet–Serret formulas</a></li> <li><a href="/wiki/Radius_of_curvature_(applications)" class="mw-redirect" title="Radius of curvature (applications)">Radius of curvature (applications)</a></li> <li><a href="/wiki/Affine_curvature" title="Affine curvature">Affine curvature</a></li> <li><a href="/wiki/Total_curvature" title="Total curvature">Total curvature</a></li> <li><a href="/wiki/Total_absolute_curvature" title="Total absolute curvature">Total absolute curvature</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Differential_geometry_of_surfaces" title="Differential geometry of surfaces">Differential geometry <br />of surfaces</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Principal_curvature" title="Principal curvature">Principal curvatures</a></li> <li><a href="/wiki/Gaussian_curvature" title="Gaussian curvature">Gaussian curvature</a></li> <li><a href="/wiki/Mean_curvature" title="Mean curvature">Mean curvature</a></li> <li><a href="/wiki/Darboux_frame" title="Darboux frame">Darboux frame</a></li> <li><a href="/wiki/Gauss%E2%80%93Codazzi_equations" title="Gauss–Codazzi equations">Gauss–Codazzi equations</a></li> <li><a href="/wiki/First_fundamental_form" title="First fundamental form">First fundamental form</a></li> <li><a href="/wiki/Second_fundamental_form" title="Second fundamental form">Second fundamental form</a></li> <li><a href="/wiki/Third_fundamental_form" title="Third fundamental form">Third fundamental form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Curvature_of_Riemannian_manifolds" title="Curvature of Riemannian manifolds">Curvature of Riemannian manifolds</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Scalar_curvature" title="Scalar curvature">Scalar curvature</a></li> <li><a href="/wiki/Sectional_curvature" title="Sectional curvature">Sectional curvature</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Curvature of connections</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Curvature_form" title="Curvature form">Curvature form</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Cocurvature" title="Cocurvature">Cocurvature</a></li> <li><a href="/wiki/Holonomy" title="Holonomy">Holonomy</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Frenet–Serret_formulas&oldid=1253791784">https://en.wikipedia.org/w/index.php?title=Frenet–Serret_formulas&oldid=1253791784</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Differential_geometry" title="Category:Differential 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