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?

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 from the origin increases. Assume that the distance is a constant multiple of the angle that the line segment OP makes with the positive x-axis. Therefore , where 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, , and is the second coordinate. Therefore the equation for the spiral becomes . Note that when we also have , 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 for arbitrary constants and . 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 . A graph of the function is given in Figure 2. This spiral describes the shell shape of the chambered nautilus.

Figure 2. A logarithmic spiral is similar to the shape of the chambered nautilus shell. (credit: modification of work by Jitze Couperus, Flickr)
Candela Citations
- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction