{"id":289,"date":"2021-03-25T03:03:35","date_gmt":"2021-03-25T03:03:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/?post_type=chapter&p=289"},"modified":"2021-11-17T23:51:30","modified_gmt":"2021-11-17T23:51:30","slug":"putting-it-together-parametric-equations-and-polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/putting-it-together-parametric-equations-and-polar-coordinates\/","title":{"raw":"Putting It Together: Parametric Equations and Polar Coordinates","rendered":"Putting It Together: Parametric Equations and Polar Coordinates"},"content":{"raw":"

Describing a Spiral<\/h3>\r\n
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Recall the chambered nautilus<\/span> introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1 below shows a spiral in rectangular coordinates. How can we describe this curve mathematically?<\/p>\r\n\r\n

<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]\"A Figure 1. How can we describe a spiral graph mathematically?[\/caption]<\/figure>\r\n<\/div>\r\nSolution:\r\n
\r\n

As the point P<\/em> travels around the spiral in a counterclockwise direction, its distance d<\/em> from the origin increases. Assume that the distance d<\/em> is a constant multiple k<\/em> of the angle [latex]\\theta [\/latex] that the line segment OP<\/em> makes with the positive x<\/em>-axis. Therefore [latex]d\\left(P,O\\right)=k\\theta [\/latex], where [latex]O[\/latex] is the origin. Now use the distance formula and some trigonometry:<\/p>\r\n\r\n

[latex]\\begin{array}{ccc}\\hfill d\\left(P,O\\right)& =\\hfill & k\\theta \\hfill \\\\ \\hfill \\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y - 0\\right)}^{2}}& =\\hfill & k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\sqrt{{x}^{2}+{y}^{2}}& =\\hfill & k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\text{arctan}\\left(\\frac{y}{x}\\right)& =\\hfill & \\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\hfill \\\\ \\hfill y& =\\hfill & x\\tan\\left(\\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\right).\\hfill \\end{array}[\/latex]<\/div>\r\n \r\n

Although this equation describes the spiral, it is not possible to solve it directly for either x<\/em> or y<\/em>. However, if we use polar coordinates, the equation becomes much simpler. In particular, [latex]d\\left(P,O\\right)=r[\/latex], and [latex]\\theta [\/latex] is the second coordinate. Therefore the equation for the spiral becomes [latex]r=k\\theta [\/latex]. Note that when [latex]\\theta =0[\/latex] we also have [latex]r=0[\/latex], so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes [latex]r=a+k\\theta [\/latex] for arbitrary constants [latex]a[\/latex] and [latex]k[\/latex]. This is referred to as an Archimedean spiral<\/span>, after the Greek mathematician Archimedes.<\/p>\r\n

Another type of spiral is the logarithmic spiral, described by the function [latex]r=a\\cdot {b}^{\\theta }[\/latex]. A graph of the function [latex]r=1.2\\left({1.25}^{\\theta }\\right)[\/latex] is given in Figure 2. This spiral describes the shell shape of the chambered nautilus.<\/p>\r\n\r\n

<\/figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"858\"]\"This Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)[\/caption]<\/figure>\r\n<\/div>","rendered":"

Describing a Spiral<\/h3>\n
\n

Recall the chambered nautilus<\/span> introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. Figure 1 below shows a spiral in rectangular coordinates. How can we describe this curve mathematically?<\/p>\n

<\/figcaption>
\"A<\/p>\n

Figure 1. How can we describe a spiral graph mathematically?<\/p>\n<\/div>\n<\/figure>\n<\/div>\n

Solution:<\/p>\n

\n

As the point P<\/em> travels around the spiral in a counterclockwise direction, its distance d<\/em> from the origin increases. Assume that the distance d<\/em> is a constant multiple k<\/em> of the angle [latex]\\theta[\/latex] that the line segment OP<\/em> makes with the positive x<\/em>-axis. Therefore [latex]d\\left(P,O\\right)=k\\theta[\/latex], where [latex]O[\/latex] is the origin. Now use the distance formula and some trigonometry:<\/p>\n

[latex]\\begin{array}{ccc}\\hfill d\\left(P,O\\right)& =\\hfill & k\\theta \\hfill \\\\ \\hfill \\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y - 0\\right)}^{2}}& =\\hfill & k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\sqrt{{x}^{2}+{y}^{2}}& =\\hfill & k\\text{arctan}\\left(\\frac{y}{x}\\right)\\hfill \\\\ \\hfill \\text{arctan}\\left(\\frac{y}{x}\\right)& =\\hfill & \\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\hfill \\\\ \\hfill y& =\\hfill & x\\tan\\left(\\frac{\\sqrt{{x}^{2}+{y}^{2}}}{k}\\right).\\hfill \\end{array}[\/latex]<\/div>\n

 <\/p>\n

Although this equation describes the spiral, it is not possible to solve it directly for either x<\/em> or y<\/em>. However, if we use polar coordinates, the equation becomes much simpler. In particular, [latex]d\\left(P,O\\right)=r[\/latex], and [latex]\\theta[\/latex] is the second coordinate. Therefore the equation for the spiral becomes [latex]r=k\\theta[\/latex]. Note that when [latex]\\theta =0[\/latex] we also have [latex]r=0[\/latex], so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes [latex]r=a+k\\theta[\/latex] for arbitrary constants [latex]a[\/latex] and [latex]k[\/latex]. This is referred to as an Archimedean spiral<\/span>, after the Greek mathematician Archimedes.<\/p>\n

Another type of spiral is the logarithmic spiral, described by the function [latex]r=a\\cdot {b}^{\\theta }[\/latex]. A graph of the function [latex]r=1.2\\left({1.25}^{\\theta }\\right)[\/latex] is given in Figure 2. This spiral describes the shell shape of the chambered nautilus.<\/p>\n

<\/figcaption>
\"This<\/p>\n

Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)<\/p>\n<\/div>\n<\/figure>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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