# Equiangular Spiral

By M. Bourne

The equation for the equiangular spiral was developed by Rene Descartes (1596-1650) in 1638.

This spiral occurs naturally in many places like sea-shells where the growth of an organism is proportional to the size of the organism.

It's also known as the **Logarithmic Spiral** due to the way the spiral arms increase in distance from the center at the same ratio.

The general polar equation for the equiangular spiral curve is

r=ae^{θ cot b}

### Example 1: Equiangular Spiral

Let *a* = 0.5 and *b* = 1.4 (radians).

Then

r= 0.5e^{θ cot 1.4}

Here's the graph:

Graph of *r* = 0.5 *e*^{θ cot 1.4}, an equiangular spiral.

It is called an "equiangular" spiral because any radius vector makes the same angle with the curve. In this example, *b* = 1.4 radians, or in degrees, `1.4xx180/pi=80.214^text(o)`.

We can see that any radius vector makes the angle `80^@` with the curve.

Graph of equiangular spiral showing equal angles cut by radius vectors.

[For background, see the vectors chapter.]

## Spiral of Archimedes

The equiangular spiral is not the same as the "Spiral of Archimedes" which has the form:

r=aθ

This spiral has arms which are equidistant.

### Example 2: Spiral of Archimedes

r= 2θ

Graph of *r* = 2*θ*, an Archimedean Spiral.

See this article on finding the length of an Archimedean Sprial, using calculus.