The equiangular spiral

The polar equation of the spiral to equiangular is:

It receives the name of equiangular because the angle formed by the radius vector and the tangent is constant.

If is that angle, the equiangular spiral can be expressed with the equation

It is the locus of the transform of (a,0) by a dilative rotation (Coxeter).

In the applet different equiangular spirals can be seen. For example, dragging the indicated points.

The property that gives name to the equiangular spiral can be verified pressing the bottons "Step +" or "Step -".

This spiral was called "spiral mirabilis" by Jacob Bernouilli.

REFERENCES

Coxeter - Introduction to Geometry (John Whiley and sons)

Steinhaus - Mathematical Snapshots.

D'Arcy Thompson - On Growth and Form. (Cambridge University Press)