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From the math blog...

Golden Spiral

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Length of an Archimedean Spiral

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Archimedes and the area of a parabolic segment

Archimedes had a good understanding of the way calculus works, almost 2000 years before Newton and Leibniz....

Equiangular Spiral

By M. Bourne

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 = ae^{θ cot b}

Example

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

Then

r = 0.5 e^{θ cot 1.4}

Using polar plot in Scientific Notebook, we have:

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.

[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 of Spiral of Archimedes

r = 2θ

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

8. Curves in Polar Coordinates

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