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.

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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