{"id":72,"date":"2021-07-30T17:08:54","date_gmt":"2021-07-30T17:08:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=72"},"modified":"2022-10-29T00:53:30","modified_gmt":"2022-10-29T00:53:30","slug":"summary-of-arc-length-and-curvature","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-arc-length-and-curvature\/","title":{"raw":"Summary of Arc Length and Curvature","rendered":"Summary of Arc Length and Curvature"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>The arc-length function for a vector-valued function is calculated using the integral formula [latex]s(t)=\\displaystyle\\int_{a}^{t} \\parallel{\\bf{r}}^{\\prime}(u)\\parallel{du}[\/latex].\u00a0This formula is valid in both two and three dimensions.<\/li>\r\n \t<li>The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.<\/li>\r\n \t<li>There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.<\/li>\r\n \t<li>The principal unit normal vector at [latex]t[\/latex]\u00a0is defined to be\u00a0[latex]{\\bf{N}}(t)=\\dfrac{{\\bf{T}}^{\\prime}(t)}{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}[\/latex].<\/li>\r\n \t<li>The binormal vector at\u00a0[latex]t[\/latex]\u00a0is defined as [latex]{\\bf{B}}(t)={\\bf{T}}(t)\\times{\\bf{N}}(t)[\/latex], where [latex]{\\bf{T}}(t)[\/latex] is the unit tangent vector.<\/li>\r\n \t<li>The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.<\/li>\r\n \t<li>The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Arc length of space curve\r\n<\/strong>[latex]s={\\displaystyle\\int_{a}^{b}} \\sqrt{\\left[f^{\\prime}(t)\\right]^{2}+\\left[g^{\\prime}(t)\\right]^{2}+\\left[h^{\\prime}(t)\\right]^{2}} dt =\\displaystyle\\int_{a}^{b} \\parallel{\\bf{r}}^{\\prime}(t)\\parallel{dt}[\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Arc-length function<\/strong>\r\n[latex]s(t)={\\displaystyle\\int_{a}^{t}} \\sqrt{\\left(f^{\\prime}(u)\\right)^{2}+\\left(g^{\\prime}(u)\\right)^{2}+\\left(h^{\\prime}(u)\\right)^{2}} du[\/latex] or [latex]s(t)=\\displaystyle\\int_{a}^{t} \\parallel{\\bf{r}}^{\\prime}(u)\\parallel{du}[\/latex]<\/li>\r\n \t<li><strong>Curvature<\/strong>\r\n[latex]\\kappa=\\dfrac{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}{\\parallel{\\bf{r}}^{\\prime}(t)\\parallel}[\/latex] or\u00a0[latex]\\kappa=\\dfrac{\\parallel{\\bf{r}}^{\\prime}(t)\\times{\\bf{r}}^{\\prime\\prime}(t)\\parallel}{\\parallel{\\bf{r}}^{\\prime}(t)\\parallel^{3}}[\/latex] or\u00a0[latex]\\kappa=\\dfrac{|y^{\\sigma}|}{[1+(y^{\\prime})^{2}]^{3{\/}2}}[\/latex]<\/li>\r\n \t<li><strong>Principal unit normal vector<\/strong>\r\n[latex]{\\bf{N}}(t)=\\dfrac{{\\bf{T}}^{\\prime}(t)}{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Binormal vector<\/strong>\r\n[latex]{\\bf{B}}(t)={\\bf{T}}(t)\\times{\\bf{N}}(t)[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>arc-length function<\/dt>\r\n \t<dd>a function\u00a0[latex]s(t)[\/latex]\u00a0that describes the arc length of curve\u00a0[latex]C[\/latex]\u00a0as a function of\u00a0[latex]t[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>arc-length parameterization<\/dt>\r\n \t<dd>a reparameterization of a vector-valued function in which the parameter is equal to the arc length<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>binormal vector<\/dt>\r\n \t<dd>a unit vector orthogonal to the unit tangent vector and the unit normal vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>curvature<\/dt>\r\n \t<dd>the derivative of the unit tangent vector with respect to the arc-length parameter<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Frenet frame of reference<\/dt>\r\n \t<dd>(TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>normal plane<\/dt>\r\n \t<dd>a plane that is perpendicular to a curve at any point on the curve<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>osculating circle<\/dt>\r\n \t<dd>a circle that is tangent to a curve\u00a0[latex]C[\/latex]\u00a0<span style=\"font-size: 1em;\">at a point [latex]P[\/latex]\u00a0<\/span><span style=\"font-size: 1em;\">and that shares the same curvature<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>osculating plane<\/dt>\r\n \t<dd>the plane determined by the unit tangent and the unit normal vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>principal unit normal vector<\/dt>\r\n \t<dd>a vector orthogonal to the unit tangent vector, given by the formula [latex]\\frac{{\\bf{T}}^{\\prime}(t)}{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>radius of curvature<\/dt>\r\n \t<dd>the reciprocal of the curvature<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>smooth<\/dt>\r\n \t<dd>curves where the vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex] is differentiable with a non-zero derivative<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>The arc-length function for a vector-valued function is calculated using the integral formula [latex]s(t)=\\displaystyle\\int_{a}^{t} \\parallel{\\bf{r}}^{\\prime}(u)\\parallel{du}[\/latex].\u00a0This formula is valid in both two and three dimensions.<\/li>\n<li>The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.<\/li>\n<li>There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.<\/li>\n<li>The principal unit normal vector at [latex]t[\/latex]\u00a0is defined to be\u00a0[latex]{\\bf{N}}(t)=\\dfrac{{\\bf{T}}^{\\prime}(t)}{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}[\/latex].<\/li>\n<li>The binormal vector at\u00a0[latex]t[\/latex]\u00a0is defined as [latex]{\\bf{B}}(t)={\\bf{T}}(t)\\times{\\bf{N}}(t)[\/latex], where [latex]{\\bf{T}}(t)[\/latex] is the unit tangent vector.<\/li>\n<li>The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.<\/li>\n<li>The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.<\/li>\n<\/ul>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Arc length of space curve<br \/>\n<\/strong>[latex]s={\\displaystyle\\int_{a}^{b}} \\sqrt{\\left[f^{\\prime}(t)\\right]^{2}+\\left[g^{\\prime}(t)\\right]^{2}+\\left[h^{\\prime}(t)\\right]^{2}} dt =\\displaystyle\\int_{a}^{b} \\parallel{\\bf{r}}^{\\prime}(t)\\parallel{dt}[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Arc-length function<\/strong><br \/>\n[latex]s(t)={\\displaystyle\\int_{a}^{t}} \\sqrt{\\left(f^{\\prime}(u)\\right)^{2}+\\left(g^{\\prime}(u)\\right)^{2}+\\left(h^{\\prime}(u)\\right)^{2}} du[\/latex] or [latex]s(t)=\\displaystyle\\int_{a}^{t} \\parallel{\\bf{r}}^{\\prime}(u)\\parallel{du}[\/latex]<\/li>\n<li><strong>Curvature<\/strong><br \/>\n[latex]\\kappa=\\dfrac{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}{\\parallel{\\bf{r}}^{\\prime}(t)\\parallel}[\/latex] or\u00a0[latex]\\kappa=\\dfrac{\\parallel{\\bf{r}}^{\\prime}(t)\\times{\\bf{r}}^{\\prime\\prime}(t)\\parallel}{\\parallel{\\bf{r}}^{\\prime}(t)\\parallel^{3}}[\/latex] or\u00a0[latex]\\kappa=\\dfrac{|y^{\\sigma}|}{[1+(y^{\\prime})^{2}]^{3{\/}2}}[\/latex]<\/li>\n<li><strong>Principal unit normal vector<\/strong><br \/>\n[latex]{\\bf{N}}(t)=\\dfrac{{\\bf{T}}^{\\prime}(t)}{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Binormal vector<\/strong><br \/>\n[latex]{\\bf{B}}(t)={\\bf{T}}(t)\\times{\\bf{N}}(t)[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>arc-length function<\/dt>\n<dd>a function\u00a0[latex]s(t)[\/latex]\u00a0that describes the arc length of curve\u00a0[latex]C[\/latex]\u00a0as a function of\u00a0[latex]t[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>arc-length parameterization<\/dt>\n<dd>a reparameterization of a vector-valued function in which the parameter is equal to the arc length<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>binormal vector<\/dt>\n<dd>a unit vector orthogonal to the unit tangent vector and the unit normal vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>curvature<\/dt>\n<dd>the derivative of the unit tangent vector with respect to the arc-length parameter<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Frenet frame of reference<\/dt>\n<dd>(TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>normal plane<\/dt>\n<dd>a plane that is perpendicular to a curve at any point on the curve<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>osculating circle<\/dt>\n<dd>a circle that is tangent to a curve\u00a0[latex]C[\/latex]\u00a0<span style=\"font-size: 1em;\">at a point [latex]P[\/latex]\u00a0<\/span><span style=\"font-size: 1em;\">and that shares the same curvature<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>osculating plane<\/dt>\n<dd>the plane determined by the unit tangent and the unit normal vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>principal unit normal vector<\/dt>\n<dd>a vector orthogonal to the unit tangent vector, given by the formula [latex]\\frac{{\\bf{T}}^{\\prime}(t)}{\\parallel{\\bf{T}}^{\\prime}(t)\\parallel}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>radius of curvature<\/dt>\n<dd>the reciprocal of the curvature<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>smooth<\/dt>\n<dd>curves where the vector-valued function\u00a0[latex]{\\bf{r}}(t)[\/latex] is differentiable with a non-zero derivative<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-72\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-72","chapter","type-chapter","status-publish","hentry"],"part":21,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/72","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/72\/revisions"}],"predecessor-version":[{"id":3697,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/72\/revisions\/3697"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/21"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/72\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=72"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=72"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=72"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=72"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}