{"id":715,"date":"2021-05-10T19:10:33","date_gmt":"2021-05-10T19:10:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=715"},"modified":"2022-03-14T17:17:46","modified_gmt":"2022-03-14T17:17:46","slug":"summary-of-parametric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/summary-of-parametric-equations\/","title":{"raw":"Summary of Parametric Equations","rendered":"Summary of Parametric Equations"},"content":{"raw":"<section id=\"fs-id1169293395741\" class=\"key-concepts\" data-depth=\"1\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1169293333646\" data-bullet-style=\"bullet\">\r\n \t<li>Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.<\/li>\r\n \t<li>It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.<\/li>\r\n \t<li>There is always more than one way to parameterize a curve.<\/li>\r\n \t<li>Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169293389862\" class=\"section-exercises\" data-depth=\"1\"><\/section>\r\n<div data-type=\"glossary\">\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169293394905\">\r\n \t<dt>cusp<\/dt>\r\n \t<dd id=\"fs-id1169293394910\">a pointed end or part where two curves meet<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394894\">\r\n \t<dt>cycloid<\/dt>\r\n \t<dd id=\"fs-id1169293394900\">the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394914\">\r\n \t<dt>orientation<\/dt>\r\n \t<dd id=\"fs-id1169293394920\">the direction that a point moves on a graph as the parameter increases<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394924\">\r\n \t<dt>parameter<\/dt>\r\n \t<dd id=\"fs-id1169293394929\">an independent variable that both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> depend on in a parametric curve; usually represented by the variable <em data-effect=\"italics\">t<\/em><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293394948\">\r\n \t<dt>parametric curve<\/dt>\r\n \t<dd id=\"fs-id1169293394954\">the graph of the parametric equations [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] over an interval [latex]a\\le t\\le b[\/latex] combined with the equations<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293296425\">\r\n \t<dt>parametric equations<\/dt>\r\n \t<dd id=\"fs-id1169293296430\">the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169293296469\">\r\n \t<dt>parameterization of a curve<\/dt>\r\n \t<dd id=\"fs-id1169293296474\">rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<section id=\"fs-id1169293395741\" class=\"key-concepts\" data-depth=\"1\">\n<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1169293333646\" data-bullet-style=\"bullet\">\n<li>Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.<\/li>\n<li>It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.<\/li>\n<li>There is always more than one way to parameterize a curve.<\/li>\n<li>Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section id=\"fs-id1169293389862\" class=\"section-exercises\" data-depth=\"1\"><\/section>\n<div data-type=\"glossary\">\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169293394905\">\n<dt>cusp<\/dt>\n<dd id=\"fs-id1169293394910\">a pointed end or part where two curves meet<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394894\">\n<dt>cycloid<\/dt>\n<dd id=\"fs-id1169293394900\">the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394914\">\n<dt>orientation<\/dt>\n<dd id=\"fs-id1169293394920\">the direction that a point moves on a graph as the parameter increases<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394924\">\n<dt>parameter<\/dt>\n<dd id=\"fs-id1169293394929\">an independent variable that both <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> depend on in a parametric curve; usually represented by the variable <em data-effect=\"italics\">t<\/em><\/dd>\n<\/dl>\n<dl id=\"fs-id1169293394948\">\n<dt>parametric curve<\/dt>\n<dd id=\"fs-id1169293394954\">the graph of the parametric equations [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] over an interval [latex]a\\le t\\le b[\/latex] combined with the equations<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293296425\">\n<dt>parametric equations<\/dt>\n<dd id=\"fs-id1169293296430\">the equations [latex]x=x\\left(t\\right)[\/latex] and [latex]y=y\\left(t\\right)[\/latex] that define a parametric curve<\/dd>\n<\/dl>\n<dl id=\"fs-id1169293296469\">\n<dt>parameterization of a curve<\/dt>\n<dd id=\"fs-id1169293296474\">rewriting the equation of a curve defined by a function [latex]y=f\\left(x\\right)[\/latex] as parametric equations<\/dd>\n<\/dl>\n<\/div>\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-715\">\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 2. <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-2\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-2\/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-2\/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":416434,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/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-715","chapter","type-chapter","status-publish","hentry"],"part":162,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/715","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/416434"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/715\/revisions"}],"predecessor-version":[{"id":2715,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/715\/revisions\/2715"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/162"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/715\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=715"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=715"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=715"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=715"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}