Putting It Together: Parametric Equations and Polar Coordinates

Describing a Spiral

Recall the chambered nautilus 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?

A spiral starting at the origin and continually increasing its radius to a point P(x, y).

Solution:

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

\begin{matrix} d (P, O) & = & k θ \sqrt{{(x - 0)}^{2} + {(y - 0)}^{2}} & = & k arctan (\frac{y}{x}) \sqrt{x^{2} + y^{2}} & = & k arctan (\frac{y}{x}) arctan (\frac{y}{x}) & = & \frac{\sqrt{x^{2} + y^{2}}}{k} y & = & x tan (\frac{\sqrt{x^{2} + y^{2}}}{k}) . \end{matrix}

$\begin{array}{ccc}\hfill d\left(P,O\right)& =\hfill & k\theta \hfill \\ \hfill \sqrt{{\left(x - 0\right)}^{2}+{\left(y - 0\right)}^{2}}& =\hfill & k\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \sqrt{{x}^{2}+{y}^{2}}& =\hfill & k\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \text{arctan}\left(\frac{y}{x}\right)& =\hfill & \frac{\sqrt{{x}^{2}+{y}^{2}}}{k}\hfill \\ \hfill y& =\hfill & x\tan\left(\frac{\sqrt{{x}^{2}+{y}^{2}}}{k}\right).\hfill \end{array}$

Although this equation describes the spiral, it is not possible to solve it directly for either x or y. However, if we use polar coordinates, the equation becomes much simpler. In particular, $d\left(P,O\right)=r$ , and $\theta$ is the second coordinate. Therefore the equation for the spiral becomes $r=k\theta$ . Note that when $\theta =0$ we also have $r=0$ , 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 $r=a+k\theta$ for arbitrary constants $a$ and $k$ . This is referred to as an Archimedean spiral, after the Greek mathematician Archimedes.

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

This figure has two figures. The first is a shell with many chambers that increase in size from the center out. The second is a spiral with equation r = 1.2(1.25θ).

Module 7: Parametric Equations and Polar Coordinates

Putting It Together: Parametric Equations and Polar Coordinates

Describing a Spiral

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