{"id":50,"date":"2021-07-30T17:06:07","date_gmt":"2021-07-30T17:06:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=50"},"modified":"2022-10-26T02:56:45","modified_gmt":"2022-10-26T02:56:45","slug":"summary-of-quadratic-surfaces","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-quadratic-surfaces\/","title":{"raw":"Summary of Quadric Surfaces","rendered":"Summary of Quadric Surfaces"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul id=\"fs-id1163723145121\" data-bullet-style=\"bullet\">\r\n \t<li>A set of lines parallel to a given line passing through a given curve is called a\u00a0<em data-effect=\"italics\">cylinder<\/em>, or a\u00a0<em data-effect=\"italics\">cylindrical surface<\/em>. The parallel lines are called\u00a0<em data-effect=\"italics\">rulings<\/em>.<\/li>\r\n \t<li>The intersection of a three-dimensional surface and a plane is called a\u00a0<em data-effect=\"italics\">trace<\/em>. To find the trace in the [latex]xy[\/latex]-, [latex]yz[\/latex]-, or [latex]xz[\/latex]-planes, set [latex]z=0[\/latex], [latex]x=0[\/latex], or [latex]y=0[\/latex], respectively.<\/li>\r\n \t<li>Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form [latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/li>\r\n \t<li>To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.<\/li>\r\n \t<li>Important quadric surfaces are summarized in\u00a0Figure 8\u00a0and\u00a0Figure 9.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>cylinder<\/dt>\r\n \t<dd>a set of lines parallel to a given line passing through a given curve<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>ellipsoid<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]\u00a0all traces of this surface are ellipses<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>elliptic cone<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0[\/latex]\u00a0traces of this surface include ellipses and intersecting lines<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>elliptic paraboloid<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]z=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>hyperboloid of one sheet<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>hyperboloid of two sheets<\/dt>\r\n \t<dd>a three-dimensional surface described by an equation of the form [latex]\\frac{z^2}{c^2}-\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>quadric surfaces<\/dt>\r\n \t<dd>surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>rulings<\/dt>\r\n \t<dd>parallel lines that make up a cylindrical surface<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>trace<\/dt>\r\n \t<dd>the intersection of a three-dimensional surface with a coordinate plane<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul id=\"fs-id1163723145121\" data-bullet-style=\"bullet\">\n<li>A set of lines parallel to a given line passing through a given curve is called a\u00a0<em data-effect=\"italics\">cylinder<\/em>, or a\u00a0<em data-effect=\"italics\">cylindrical surface<\/em>. The parallel lines are called\u00a0<em data-effect=\"italics\">rulings<\/em>.<\/li>\n<li>The intersection of a three-dimensional surface and a plane is called a\u00a0<em data-effect=\"italics\">trace<\/em>. To find the trace in the [latex]xy[\/latex]-, [latex]yz[\/latex]-, or [latex]xz[\/latex]-planes, set [latex]z=0[\/latex], [latex]x=0[\/latex], or [latex]y=0[\/latex], respectively.<\/li>\n<li>Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form [latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/li>\n<li>To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.<\/li>\n<li>Important quadric surfaces are summarized in\u00a0Figure 8\u00a0and\u00a0Figure 9.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>cylinder<\/dt>\n<dd>a set of lines parallel to a given line passing through a given curve<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>ellipsoid<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]\u00a0all traces of this surface are ellipses<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>elliptic cone<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=0[\/latex]\u00a0traces of this surface include ellipses and intersecting lines<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>elliptic paraboloid<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]z=\\frac{x^2}{a^2}+\\frac{y^2}{b^2}[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>hyperboloid of one sheet<\/dt>\n<dd>a three-dimensional surface described by an equation of the form\u00a0[latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>hyperboloid of two sheets<\/dt>\n<dd>a three-dimensional surface described by an equation of the form [latex]\\frac{z^2}{c^2}-\\frac{x^2}{a^2}-\\frac{y^2}{b^2}=1[\/latex]<strong>\u00a0<\/strong>traces of this surface include ellipses and parabolas<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>quadric surfaces<\/dt>\n<dd>surfaces in three dimensions having the property that the traces of the surface are conic sections (ellipses, hyperbolas, and parabolas)<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>rulings<\/dt>\n<dd>parallel lines that make up a cylindrical surface<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>trace<\/dt>\n<dd>the intersection of a three-dimensional surface with a coordinate plane<\/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-50\">\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":27,"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-50","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/50","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":10,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions"}],"predecessor-version":[{"id":6037,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/50\/revisions\/6037"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/50\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=50"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=50"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=50"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}