Equiangular Spiral

By M. Bourne


Equiangular_spiral__1.png
Nautilus Shell
(Image from Tree of Life)

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 = aeθ cot b

Example

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

Then

r = 0.5 eθ cot 1.4

Using polar plot in Scientific Notebook, we have:

Equiangular_spiral__8.png
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 MATH.

We can see that any radius vector makes the angle 80° with the curve.

Equiangular_spiral__12.png

[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

Example of Spiral of Archimedes

r = 2θ

Equiangular_spiral__15.png

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