{"id":986,"date":"2025-06-20T17:26:55","date_gmt":"2025-06-20T17:26:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=986"},"modified":"2025-09-10T17:43:03","modified_gmt":"2025-09-10T17:43:03","slug":"fundamentals-of-parametric-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/fundamentals-of-parametric-equations-fresh-take\/","title":{"raw":"Fundamentals of Parametric Equations: Fresh Take","rendered":"Fundamentals of Parametric Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Create and sketch graphs of curves given their parametric equations<\/li>\r\n \t<li>Convert parametric equations into a regular y = f(x) form by eliminating the parameter<\/li>\r\n \t<li>Recognize and describe the curve called a cycloid<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Parametric Equations and Their Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Traditional functions [latex]y = f(x)[\/latex] work great for many curves, but what about loops, vertical lines, or paths that double back on themselves? Parametric equations solve this by letting both [latex]x[\/latex] and [latex]y[\/latex] depend on a third variable called a parameter.<\/p>\r\n<p class=\"whitespace-normal break-words\">In parametric equations, we write [latex]x = x(t)[\/latex] and [latex]y = y(t)[\/latex], where [latex]t[\/latex] is the parameter. Think of [latex]t[\/latex] as time on a stopwatch\u2014as [latex]t[\/latex] changes, both coordinates change simultaneously, tracing out a path through the plane.<\/p>\r\n<p class=\"whitespace-normal break-words\">The parameter [latex]t[\/latex] gives curves an orientation\u2014a direction of travel. As [latex]t[\/latex] increases, you can follow the path from start to finish with arrows showing which way you're moving.<\/p>\r\n<strong>The Process:<\/strong> Make a table with [latex]t[\/latex] values, calculate corresponding [latex]x[\/latex] and [latex]y[\/latex] coordinates, then plot and connect the points. The arrows show the direction as [latex]t[\/latex] increases.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1169295589408\" data-type=\"problem\">\r\n<p id=\"fs-id1169295759487\">Sketch the curve described by the parametric equations<\/p>\r\n\r\n<div id=\"fs-id1169295431091\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=3t+2,y\\left(t\\right)={t}^{2}-1,-3\\le t\\le 2[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558898\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1169295557869\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1169295652237\">Make a table of values for [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] using <em data-effect=\"italics\">t<\/em> values from \u22123 to 2.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1169295453647\" data-type=\"solution\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"642\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234620\/CNX_Calc_Figure_11_01_006.jpg\" alt=\"A curved line going from (\u22127, 8) through (\u22121, 0) and (2, \u22121) to (8, 3) with arrow going in that order. The point (\u22127, 8) is marked t = \u22123, the point (2, \u22121) is marked t = 0, and the point (8, 3) is marked t = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 \u2212 1, and \u22123 \u2264 t \u2264 2.\" width=\"642\" height=\"423\" data-media-type=\"image\/jpeg\" \/> Figure 6.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Eliminating the Parameter<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Sometimes you want to figure out what familiar curve your parametric equations represent. Eliminating the parameter means getting rid of [latex]t[\/latex] to find a direct relationship between [latex]x[\/latex] and [latex]y[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Basic Strategy:<\/strong> Solve one parametric equation for [latex]t[\/latex], then substitute that expression into the other equation. Choose whichever equation makes the algebra easier.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Standard Approach:<\/strong><\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">From [latex]y = 2t + 1[\/latex], solve for [latex]t[\/latex]: [latex]t = \\frac{y-1}{2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Substitute into [latex]x = t^2 - 3[\/latex]: [latex]x = \\left(\\frac{y-1}{2}\\right)^2 - 3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify to get the final [latex]x[\/latex]-[latex]y[\/latex] relationship<\/li>\r\n<\/ul>\r\nThe parameter restrictions matter! If [latex]-2 \\leq t \\leq 3[\/latex], your curve is only a piece of the full equation, not the entire parabola or circle.\r\n\r\nYou can also start with a regular equation like [latex]y = 2x^2 - 3[\/latex] and create parametric equations. The simplest way is [latex]x = t, y = 2t^2 - 3[\/latex], but you could also choose [latex]x = 3t - 2[\/latex] and adjust [latex]y[\/latex] accordingly.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1169295362514\" data-type=\"problem\">\r\n<p id=\"fs-id1169295555110\">Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.<\/p>\r\n\r\n<div id=\"fs-id1169295555115\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=2+\\frac{3}{t},y\\left(t\\right)=t - 1,2\\le t\\le 6[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558893\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558893\"]\r\n<div id=\"fs-id1169293185389\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1169293313963\">Solve one of the equations for <em data-effect=\"italics\">t<\/em> and substitute into the other equation.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1169293397158\" data-type=\"solution\">\r\n<p id=\"fs-id1169293397160\">[latex]x=2+\\frac{3}{y+1}[\/latex], or [latex]y=-1+\\frac{3}{x - 2}[\/latex]. This equation describes a portion of a rectangular hyperbola centered at [latex]\\left(2,-1\\right)[\/latex]. <span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"330\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234629\/CNX_Calc_Figure_11_01_009.jpg\" alt=\"A curved line going from (3.5, 1) to (2.5, 5) with arrow going in that order. The point (3.5, 1) is marked t = 2 and the point (2.5, 5) is marked t = 6. On the graph there are also written three equations: x(t) = 2 + 3\/t, y(t) = t \u2212 1, and 2 \u2264 t \u2264 6.\" width=\"330\" height=\"310\" data-media-type=\"image\/jpeg\" \/> Figure 9.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1169295461544\" data-type=\"problem\">\r\n<p id=\"fs-id1169293398615\">Find two different sets of parametric equations to represent the graph of [latex]y={x}^{2}+2x[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558890\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1169293392950\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1169293253923\">Follow the steps from the example. Remember we have freedom in choosing the parameterization for [latex]x\\left(t\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558891\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558891\"]\r\n<div id=\"fs-id1169293390102\" data-type=\"solution\">\r\n<p id=\"fs-id1169293390104\">One possibility is [latex]x\\left(t\\right)=t,y\\left(t\\right)={t}^{2}+2t[\/latex]. Another possibility is [latex]x\\left(t\\right)=2t - 3,y\\left(t\\right)={\\left(2t - 3\\right)}^{2}+2\\left(2t - 3\\right)=4{t}^{2}-8t+3[\/latex].<\/p>\r\n<p id=\"fs-id1169295712634\">There are, in fact, an infinite number of possibilities.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Cycloids and Other Parametric Curves<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Imagine a piece of gum stuck to your bicycle tire. As you ride forward, that gum traces out a specific curve through space\u2014and that curve has a name: a <strong>cycloid<\/strong>. This isn't just abstract math; it's the actual path traced by any point on a rolling wheel.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Cycloid:<\/strong> For a wheel of radius [latex]a[\/latex] rolling along a straight line, the parametric equations are:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x(t) = a(t - \\sin t)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]y(t) = a(1 - \\cos t)[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The motion combines two parts\u2014the wheel's center moving forward at constant height [latex]a[\/latex], plus the point rotating around that moving center. The [latex]t - \\sin t[\/latex] and [latex]1 - \\cos t[\/latex] terms capture this combined motion perfectly.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Hypocycloids:<\/strong> Now imagine a smaller circle rolling inside a larger circle. The curves become even more fascinating, creating star-like shapes with sharp points called <strong>cusps<\/strong>.<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Magic Ratio:<\/strong> The ratio [latex]\\frac{a}{b}[\/latex] (big circle radius \u00f7 small circle radius) determines everything:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{a}{b} = 3[\/latex] \u2192 3-pointed star (deltoid)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]\\frac{a}{b} = 4[\/latex] \u2192 4-pointed star (astroid)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Rational ratios \u2192 finite cusps that eventually close<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Irrational ratios \u2192 infinite cusps that never repeat<\/li>\r\n<\/ul>\r\n<\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Create and sketch graphs of curves given their parametric equations<\/li>\n<li>Convert parametric equations into a regular y = f(x) form by eliminating the parameter<\/li>\n<li>Recognize and describe the curve called a cycloid<\/li>\n<\/ul>\n<\/section>\n<h2>Parametric Equations and Their Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Traditional functions [latex]y = f(x)[\/latex] work great for many curves, but what about loops, vertical lines, or paths that double back on themselves? Parametric equations solve this by letting both [latex]x[\/latex] and [latex]y[\/latex] depend on a third variable called a parameter.<\/p>\n<p class=\"whitespace-normal break-words\">In parametric equations, we write [latex]x = x(t)[\/latex] and [latex]y = y(t)[\/latex], where [latex]t[\/latex] is the parameter. Think of [latex]t[\/latex] as time on a stopwatch\u2014as [latex]t[\/latex] changes, both coordinates change simultaneously, tracing out a path through the plane.<\/p>\n<p class=\"whitespace-normal break-words\">The parameter [latex]t[\/latex] gives curves an orientation\u2014a direction of travel. As [latex]t[\/latex] increases, you can follow the path from start to finish with arrows showing which way you&#8217;re moving.<\/p>\n<p><strong>The Process:<\/strong> Make a table with [latex]t[\/latex] values, calculate corresponding [latex]x[\/latex] and [latex]y[\/latex] coordinates, then plot and connect the points. The arrows show the direction as [latex]t[\/latex] increases.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1169295589408\" data-type=\"problem\">\n<p id=\"fs-id1169295759487\">Sketch the curve described by the parametric equations<\/p>\n<div id=\"fs-id1169295431091\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=3t+2,y\\left(t\\right)={t}^{2}-1,-3\\le t\\le 2[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Hint<\/button><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295557869\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1169295652237\">Make a table of values for [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] using <em data-effect=\"italics\">t<\/em> values from \u22123 to 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Show Solution<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295453647\" data-type=\"solution\">\n<figure style=\"width: 642px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234620\/CNX_Calc_Figure_11_01_006.jpg\" alt=\"A curved line going from (\u22127, 8) through (\u22121, 0) and (2, \u22121) to (8, 3) with arrow going in that order. The point (\u22127, 8) is marked t = \u22123, the point (2, \u22121) is marked t = 0, and the point (8, 3) is marked t = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 \u2212 1, and \u22123 \u2264 t \u2264 2.\" width=\"642\" height=\"423\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 6.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2 data-type=\"title\">Eliminating the Parameter<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Sometimes you want to figure out what familiar curve your parametric equations represent. Eliminating the parameter means getting rid of [latex]t[\/latex] to find a direct relationship between [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Basic Strategy:<\/strong> Solve one parametric equation for [latex]t[\/latex], then substitute that expression into the other equation. Choose whichever equation makes the algebra easier.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Standard Approach:<\/strong><\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">From [latex]y = 2t + 1[\/latex], solve for [latex]t[\/latex]: [latex]t = \\frac{y-1}{2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Substitute into [latex]x = t^2 - 3[\/latex]: [latex]x = \\left(\\frac{y-1}{2}\\right)^2 - 3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Simplify to get the final [latex]x[\/latex]&#8211;[latex]y[\/latex] relationship<\/li>\n<\/ul>\n<p>The parameter restrictions matter! If [latex]-2 \\leq t \\leq 3[\/latex], your curve is only a piece of the full equation, not the entire parabola or circle.<\/p>\n<p>You can also start with a regular equation like [latex]y = 2x^2 - 3[\/latex] and create parametric equations. The simplest way is [latex]x = t, y = 2t^2 - 3[\/latex], but you could also choose [latex]x = 3t - 2[\/latex] and adjust [latex]y[\/latex] accordingly.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1169295362514\" data-type=\"problem\">\n<p id=\"fs-id1169295555110\">Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.<\/p>\n<div id=\"fs-id1169295555115\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=2+\\frac{3}{t},y\\left(t\\right)=t - 1,2\\le t\\le 6[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558893\">Hint<\/button><\/p>\n<div id=\"q44558893\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293185389\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1169293313963\">Solve one of the equations for <em data-effect=\"italics\">t<\/em> and substitute into the other equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558894\">Show Solution<\/button><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293397158\" data-type=\"solution\">\n<p id=\"fs-id1169293397160\">[latex]x=2+\\frac{3}{y+1}[\/latex], or [latex]y=-1+\\frac{3}{x - 2}[\/latex]. This equation describes a portion of a rectangular hyperbola centered at [latex]\\left(2,-1\\right)[\/latex]. <span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure style=\"width: 330px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234629\/CNX_Calc_Figure_11_01_009.jpg\" alt=\"A curved line going from (3.5, 1) to (2.5, 5) with arrow going in that order. The point (3.5, 1) is marked t = 2 and the point (2.5, 5) is marked t = 6. On the graph there are also written three equations: x(t) = 2 + 3\/t, y(t) = t \u2212 1, and 2 \u2264 t \u2264 6.\" width=\"330\" height=\"310\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 9.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1169295461544\" data-type=\"problem\">\n<p id=\"fs-id1169293398615\">Find two different sets of parametric equations to represent the graph of [latex]y={x}^{2}+2x[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558890\">Hint<\/button><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293392950\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1169293253923\">Follow the steps from the example. Remember we have freedom in choosing the parameterization for [latex]x\\left(t\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Show Solution<\/button><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293390102\" data-type=\"solution\">\n<p id=\"fs-id1169293390104\">One possibility is [latex]x\\left(t\\right)=t,y\\left(t\\right)={t}^{2}+2t[\/latex]. Another possibility is [latex]x\\left(t\\right)=2t - 3,y\\left(t\\right)={\\left(2t - 3\\right)}^{2}+2\\left(2t - 3\\right)=4{t}^{2}-8t+3[\/latex].<\/p>\n<p id=\"fs-id1169295712634\">There are, in fact, an infinite number of possibilities.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Cycloids and Other Parametric Curves<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Imagine a piece of gum stuck to your bicycle tire. As you ride forward, that gum traces out a specific curve through space\u2014and that curve has a name: a <strong>cycloid<\/strong>. This isn&#8217;t just abstract math; it&#8217;s the actual path traced by any point on a rolling wheel.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Cycloid:<\/strong> For a wheel of radius [latex]a[\/latex] rolling along a straight line, the parametric equations are:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]x(t) = a(t - \\sin t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]y(t) = a(1 - \\cos t)[\/latex]<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The motion combines two parts\u2014the wheel&#8217;s center moving forward at constant height [latex]a[\/latex], plus the point rotating around that moving center. The [latex]t - \\sin t[\/latex] and [latex]1 - \\cos t[\/latex] terms capture this combined motion perfectly.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Hypocycloids:<\/strong> Now imagine a smaller circle rolling inside a larger circle. The curves become even more fascinating, creating star-like shapes with sharp points called <strong>cusps<\/strong>.<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Magic Ratio:<\/strong> The ratio [latex]\\frac{a}{b}[\/latex] (big circle radius \u00f7 small circle radius) determines everything:<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">[latex]\\frac{a}{b} = 3[\/latex] \u2192 3-pointed star (deltoid)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\frac{a}{b} = 4[\/latex] \u2192 4-pointed star (astroid)<\/li>\n<li class=\"whitespace-normal break-words\">Rational ratios \u2192 finite cusps that eventually close<\/li>\n<li class=\"whitespace-normal break-words\">Irrational ratios \u2192 infinite cusps that never repeat<\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":675,"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\/986"}],"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":7,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/986\/revisions"}],"predecessor-version":[{"id":2300,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/986\/revisions\/2300"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/675"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/986\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=986"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=986"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=986"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=986"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}