{"id":1008,"date":"2025-06-20T17:28:30","date_gmt":"2025-06-20T17:28:30","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=1008"},"modified":"2025-07-30T14:39:35","modified_gmt":"2025-07-30T14:39:35","slug":"polar-coordinates-and-conic-sections-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/polar-coordinates-and-conic-sections-cheat-sheet\/","title":{"raw":"Polar Coordinates and Conic Sections: Cheat Sheet","rendered":"Polar Coordinates and Conic Sections: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Understanding Polar Coordinates<\/strong>\r\n<ul id=\"fs-id1167794324573\" data-bullet-style=\"bullet\">\r\n \t<li>The polar coordinate system provides an alternative way to locate points in the plane.<\/li>\r\n \t<li>Convert points between rectangular and polar coordinates using the formulas<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1167794324587\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]x=r\\cos\\theta \\text{ and }y=r\\sin\\theta [\/latex]\r\nand<span data-type=\"newline\">\r\n<\/span><\/div>\r\n<div id=\"fs-id1167794324627\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]r^2={x}^{2}+{y}^{2}\\text{ and }\\tan\\theta =\\frac{y}{x}[\/latex].<\/div><\/li>\r\n \t<li>To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.<\/li>\r\n \t<li>Use the conversion formulas to convert equations between rectangular and polar coordinates.<\/li>\r\n \t<li>Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.<\/li>\r\n<\/ul>\r\n<strong>Area and Arc Length in Polar Coordinates<\/strong>\r\n<ul id=\"fs-id1167793278125\" data-bullet-style=\"bullet\">\r\n \t<li>The area of a region in polar coordinates defined by the equation [latex]r=f\\left(\\theta \\right)[\/latex] with [latex]\\alpha \\le \\theta \\le \\beta [\/latex] is given by the integral [latex]A=\\frac{1}{2}{{\\displaystyle\\int }_{\\alpha }^{\\beta }\\left[f\\left(\\theta \\right)\\right]}^{2}d\\theta [\/latex].<\/li>\r\n \t<li>To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.<\/li>\r\n \t<li>The arc length of a polar curve defined by the equation [latex]r=f\\left(\\theta \\right)[\/latex] with [latex]\\alpha \\le \\theta \\le \\beta [\/latex] is given by the integral [latex]L={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{\\left[f\\left(\\theta \\right)\\right]}^{2}+{\\left[{f}^{\\prime }\\left(\\theta \\right)\\right]}^{2}}d\\theta ={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{r}^{2}+{\\left(\\frac{dr}{d\\theta }\\right)}^{2}}d\\theta [\/latex].<\/li>\r\n<\/ul>\r\n<strong>Conic Sections<\/strong>\r\n<ul id=\"fs-id1167793390029\" data-bullet-style=\"bullet\">\r\n \t<li>The equation of a vertical parabola in standard form with given focus and directrix is [latex]y=\\frac{1}{4p}{\\left(x-h\\right)}^{2}+k[\/latex] where <em data-effect=\"italics\">p<\/em> is the distance from the vertex to the focus and [latex]\\left(h,k\\right)[\/latex] are the coordinates of the vertex.<\/li>\r\n \t<li>The equation of a horizontal ellipse in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the major axis has length 2<em data-effect=\"italics\">a,<\/em> the minor axis has length 2<em data-effect=\"italics\">b<\/em>, and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n \t<li>The equation of a horizontal hyperbola in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the vertices are located at [latex]\\left(h\\pm a,k\\right)[\/latex], and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/li>\r\n \t<li>The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.<\/li>\r\n \t<li>The polar equation of a conic section with eccentricity <em data-effect=\"italics\">e<\/em> is [latex]r=\\frac{ep}{1\\pm e\\cos\\theta }[\/latex] or [latex]r=\\frac{ep}{1\\pm e\\sin\\theta }[\/latex], where <em data-effect=\"italics\">p<\/em> represents the focal parameter.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1167794122737\" data-bullet-style=\"bullet\">\r\n \t<li><strong data-effect=\"bold\">Area of a region bounded by a polar curve<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]A=\\frac{1}{2}{\\displaystyle\\int }_{\\alpha }^{\\beta }{\\left[f\\left(\\theta \\right)\\right]}^{2}d\\theta =\\frac{1}{2}{\\displaystyle\\int }_{\\alpha }^{\\beta }{r}^{2}d\\theta [\/latex]<\/li>\r\n \t<li><strong data-effect=\"bold\">Arc length of a polar curve<\/strong><span data-type=\"newline\">\r\n<\/span>\r\n[latex]L={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{\\left[f\\left(\\theta \\right)\\right]}^{2}+{\\left[{f}^{\\prime }\\left(\\theta \\right)\\right]}^{2}}d\\theta ={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{r}^{2}+{\\left(\\frac{dr}{d\\theta }\\right)}^{2}}d\\theta [\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1167794188366\">\r\n \t<dt>angular coordinate<\/dt>\r\n \t<dd id=\"fs-id1167794188372\">[latex]\\theta [\/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (<em data-effect=\"italics\">x<\/em>) axis, measured counterclockwise<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188386\">\r\n \t<dt>cardioid<\/dt>\r\n \t<dd id=\"fs-id1167794188392\">a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\\left(1+\\sin\\theta \\right)[\/latex] or [latex]r=a\\left(1+\\cos\\theta \\right)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188445\">\r\n \t<dt>\r\n<dl id=\"fs-id1167794049400\">\r\n \t<dt>conic section<\/dt>\r\n \t<dd id=\"fs-id1167794049405\">a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049410\">\r\n \t<dt>directrix<\/dt>\r\n \t<dd id=\"fs-id1167794049415\">a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049421\">\r\n \t<dt>discriminant<\/dt>\r\n \t<dd id=\"fs-id1167794049426\">the value [latex]4AC-{B}^{2}[\/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[\/latex], is called a discriminant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049471\">\r\n \t<dt>eccentricity<\/dt>\r\n \t<dd id=\"fs-id1167794049476\">the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049482\">\r\n \t<dt>focal parameter<\/dt>\r\n \t<dd id=\"fs-id1167794049487\">the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049460\">\r\n \t<dt>focus<\/dt>\r\n \t<dd id=\"fs-id1167794049465\">a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>lima\u00e7on<\/dt>\r\n \t<dd id=\"fs-id1167794188451\">the graph of the equation [latex]r=a+b\\sin\\theta [\/latex] or [latex]r=a+b\\cos\\theta [\/latex]. If [latex]a=b[\/latex] then the graph is a cardioid<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188504\">\r\n \t<dt>\r\n<dl id=\"fs-id1167794049502\">\r\n \t<dt>major axis<\/dt>\r\n \t<dd id=\"fs-id1167794049508\">the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049514\">\r\n \t<dt>minor axis<\/dt>\r\n \t<dd id=\"fs-id1167794049519\">the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049525\">\r\n \t<dt>nappe<\/dt>\r\n \t<dd id=\"fs-id1167794049530\">a nappe is one half of a double cone<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n \t<dt>polar axis<\/dt>\r\n \t<dd id=\"fs-id1167794188509\">the horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188522\">\r\n \t<dt>polar coordinate system<\/dt>\r\n \t<dd id=\"fs-id1167794188528\">a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta [\/latex], the angular coordinate<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188547\">\r\n \t<dt>polar equation<\/dt>\r\n \t<dd id=\"fs-id1167794188553\">an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188558\">\r\n \t<dt>pole<\/dt>\r\n \t<dd id=\"fs-id1167794188563\">the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188569\">\r\n \t<dt>radial coordinate<\/dt>\r\n \t<dd id=\"fs-id1167794188574\">[latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188583\">\r\n \t<dt>rose<\/dt>\r\n \t<dd id=\"fs-id1167794188588\">graph of the polar equation [latex]r=a\\cos{n}\\theta [\/latex] or [latex]r=a\\sin{n}\\theta [\/latex] for a positive constant [latex]a[\/latex] and an integer [latex]n \\ge 2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794188629\">\r\n \t<dt>space-filling curve<\/dt>\r\n \t<dd id=\"fs-id1167794188635\">a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049535\">\r\n \t<dt>standard form<\/dt>\r\n \t<dd id=\"fs-id1167794049540\">an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1167794049546\">\r\n \t<dt>vertex<\/dt>\r\n \t<dd id=\"fs-id1167794049551\">a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Understanding Polar Coordinates<\/strong><\/p>\n<ul id=\"fs-id1167794324573\" data-bullet-style=\"bullet\">\n<li>The polar coordinate system provides an alternative way to locate points in the plane.<\/li>\n<li>Convert points between rectangular and polar coordinates using the formulas<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1167794324587\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]x=r\\cos\\theta \\text{ and }y=r\\sin\\theta[\/latex]<br \/>\nand<span data-type=\"newline\"><br \/>\n<\/span><\/div>\n<div id=\"fs-id1167794324627\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">[latex]r^2={x}^{2}+{y}^{2}\\text{ and }\\tan\\theta =\\frac{y}{x}[\/latex].<\/div>\n<\/li>\n<li>To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.<\/li>\n<li>Use the conversion formulas to convert equations between rectangular and polar coordinates.<\/li>\n<li>Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.<\/li>\n<\/ul>\n<p><strong>Area and Arc Length in Polar Coordinates<\/strong><\/p>\n<ul id=\"fs-id1167793278125\" data-bullet-style=\"bullet\">\n<li>The area of a region in polar coordinates defined by the equation [latex]r=f\\left(\\theta \\right)[\/latex] with [latex]\\alpha \\le \\theta \\le \\beta[\/latex] is given by the integral [latex]A=\\frac{1}{2}{{\\displaystyle\\int }_{\\alpha }^{\\beta }\\left[f\\left(\\theta \\right)\\right]}^{2}d\\theta[\/latex].<\/li>\n<li>To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.<\/li>\n<li>The arc length of a polar curve defined by the equation [latex]r=f\\left(\\theta \\right)[\/latex] with [latex]\\alpha \\le \\theta \\le \\beta[\/latex] is given by the integral [latex]L={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{\\left[f\\left(\\theta \\right)\\right]}^{2}+{\\left[{f}^{\\prime }\\left(\\theta \\right)\\right]}^{2}}d\\theta ={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{r}^{2}+{\\left(\\frac{dr}{d\\theta }\\right)}^{2}}d\\theta[\/latex].<\/li>\n<\/ul>\n<p><strong>Conic Sections<\/strong><\/p>\n<ul id=\"fs-id1167793390029\" data-bullet-style=\"bullet\">\n<li>The equation of a vertical parabola in standard form with given focus and directrix is [latex]y=\\frac{1}{4p}{\\left(x-h\\right)}^{2}+k[\/latex] where <em data-effect=\"italics\">p<\/em> is the distance from the vertex to the focus and [latex]\\left(h,k\\right)[\/latex] are the coordinates of the vertex.<\/li>\n<li>The equation of a horizontal ellipse in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}+\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the major axis has length 2<em data-effect=\"italics\">a,<\/em> the minor axis has length 2<em data-effect=\"italics\">b<\/em>, and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\n<li>The equation of a horizontal hyperbola in standard form is [latex]\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1[\/latex] where the center has coordinates [latex]\\left(h,k\\right)[\/latex], the vertices are located at [latex]\\left(h\\pm a,k\\right)[\/latex], and the coordinates of the foci are [latex]\\left(h\\pm c,k\\right)[\/latex], where [latex]{c}^{2}={a}^{2}+{b}^{2}[\/latex].<\/li>\n<li>The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.<\/li>\n<li>The polar equation of a conic section with eccentricity <em data-effect=\"italics\">e<\/em> is [latex]r=\\frac{ep}{1\\pm e\\cos\\theta }[\/latex] or [latex]r=\\frac{ep}{1\\pm e\\sin\\theta }[\/latex], where <em data-effect=\"italics\">p<\/em> represents the focal parameter.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1167794122737\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Area of a region bounded by a polar curve<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]A=\\frac{1}{2}{\\displaystyle\\int }_{\\alpha }^{\\beta }{\\left[f\\left(\\theta \\right)\\right]}^{2}d\\theta =\\frac{1}{2}{\\displaystyle\\int }_{\\alpha }^{\\beta }{r}^{2}d\\theta[\/latex]<\/li>\n<li><strong data-effect=\"bold\">Arc length of a polar curve<\/strong><span data-type=\"newline\"><br \/>\n<\/span><br \/>\n[latex]L={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{\\left[f\\left(\\theta \\right)\\right]}^{2}+{\\left[{f}^{\\prime }\\left(\\theta \\right)\\right]}^{2}}d\\theta ={\\displaystyle\\int }_{\\alpha }^{\\beta }\\sqrt{{r}^{2}+{\\left(\\frac{dr}{d\\theta }\\right)}^{2}}d\\theta[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1167794188366\">\n<dt>angular coordinate<\/dt>\n<dd id=\"fs-id1167794188372\">[latex]\\theta[\/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (<em data-effect=\"italics\">x<\/em>) axis, measured counterclockwise<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188386\">\n<dt>cardioid<\/dt>\n<dd id=\"fs-id1167794188392\">a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\\left(1+\\sin\\theta \\right)[\/latex] or [latex]r=a\\left(1+\\cos\\theta \\right)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188445\">\n<dt>\n<\/dt>\n<dt>conic section<\/dt>\n<dd id=\"fs-id1167794049405\">a conic section is any curve formed by the intersection of a plane with a cone of two nappes<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049410\">\n<dt>directrix<\/dt>\n<dd id=\"fs-id1167794049415\">a directrix (plural: directrices) is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049421\">\n<dt>discriminant<\/dt>\n<dd id=\"fs-id1167794049426\">the value [latex]4AC-{B}^{2}[\/latex], which is used to identify a conic when the equation contains a term involving [latex]xy[\/latex], is called a discriminant<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049471\">\n<dt>eccentricity<\/dt>\n<dd id=\"fs-id1167794049476\">the eccentricity is defined as the distance from any point on the conic section to its focus divided by the perpendicular distance from that point to the nearest directrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049482\">\n<dt>focal parameter<\/dt>\n<dd id=\"fs-id1167794049487\">the focal parameter is the distance from a focus of a conic section to the nearest directrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049460\">\n<dt>focus<\/dt>\n<dd id=\"fs-id1167794049465\">a focus (plural: foci) is a point used to construct and define a conic section; a parabola has one focus; an ellipse and a hyperbola have two<\/dd>\n<\/dl>\n<p> \tlima\u00e7on<br \/>\n \tthe graph of the equation [latex]r=a+b\\sin\\theta[\/latex] or [latex]r=a+b\\cos\\theta[\/latex]. If [latex]a=b[\/latex] then the graph is a cardioid<\/p>\n<dl id=\"fs-id1167794188504\">\n<dt>\n<\/dt>\n<dt>major axis<\/dt>\n<dd id=\"fs-id1167794049508\">the major axis of a conic section passes through the vertex in the case of a parabola or through the two vertices in the case of an ellipse or hyperbola; it is also an axis of symmetry of the conic; also called the transverse axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049514\">\n<dt>minor axis<\/dt>\n<dd id=\"fs-id1167794049519\">the minor axis is perpendicular to the major axis and intersects the major axis at the center of the conic, or at the vertex in the case of the parabola; also called the conjugate axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049525\">\n<dt>nappe<\/dt>\n<dd id=\"fs-id1167794049530\">a nappe is one half of a double cone<\/dd>\n<\/dl>\n<p> \tpolar axis<br \/>\n \tthe horizontal axis in the polar coordinate system corresponding to [latex]r\\ge 0[\/latex]<\/p>\n<dl id=\"fs-id1167794188522\">\n<dt>polar coordinate system<\/dt>\n<dd id=\"fs-id1167794188528\">a system for locating points in the plane. The coordinates are [latex]r[\/latex], the radial coordinate, and [latex]\\theta[\/latex], the angular coordinate<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188547\">\n<dt>polar equation<\/dt>\n<dd id=\"fs-id1167794188553\">an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188558\">\n<dt>pole<\/dt>\n<dd id=\"fs-id1167794188563\">the central point of the polar coordinate system, equivalent to the origin of a Cartesian system<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188569\">\n<dt>radial coordinate<\/dt>\n<dd id=\"fs-id1167794188574\">[latex]r[\/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188583\">\n<dt>rose<\/dt>\n<dd id=\"fs-id1167794188588\">graph of the polar equation [latex]r=a\\cos{n}\\theta[\/latex] or [latex]r=a\\sin{n}\\theta[\/latex] for a positive constant [latex]a[\/latex] and an integer [latex]n \\ge 2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794188629\">\n<dt>space-filling curve<\/dt>\n<dd id=\"fs-id1167794188635\">a curve that completely occupies a two-dimensional subset of the real plane<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049535\">\n<dt>standard form<\/dt>\n<dd id=\"fs-id1167794049540\">an equation of a conic section showing its properties, such as location of the vertex or lengths of major and minor axes<\/dd>\n<\/dl>\n<dl id=\"fs-id1167794049546\">\n<dt>vertex<\/dt>\n<dd id=\"fs-id1167794049551\">a vertex is an extreme point on a conic section; a parabola has one vertex at its turning point. An ellipse has two vertices, one at each end of the major axis; a hyperbola has two vertices, one at the turning point of each branch<\/dd>\n<\/dl>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":677,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1008"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1008\/revisions"}],"predecessor-version":[{"id":1721,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1008\/revisions\/1721"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/677"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/1008\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=1008"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=1008"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=1008"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=1008"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}