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 d from the origin increases. Assume that the distance d is a constant multiple k of the angle θ that the line segment OP makes with the positive x-axis. Therefore d(P,O)=kθ, where O is the origin. Now use the distance formula and some trigonometry:

d(P,O)=kθ(x0)2+(y0)2=karctan(yx)x2+y2=karctan(yx)arctan(yx)=x2+y2ky=xtan(x2+y2k).

 

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(P,O)=r, and θ is the second coordinate. Therefore the equation for the spiral becomes r=kθ. Note that when θ=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+kθ 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=abθ. A graph of the function r=1.2(1.25θ) 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)