### 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?

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 [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:

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.