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).

Figure 1. How can we describe a spiral graph mathematically?

Solution:

As the point P travels around the spiral in a counterclockwise direction, its distance [latex]d[/latex] from the origin increases. Assume that the distance [latex]d[/latex] is a constant multiple [latex]k[/latex] of the angle [latex]\theta[/latex] that the line segment OP makes with the positive x-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:

[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]

 

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, [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, after the Greek mathematician Archimedes.

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.

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).

Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)