Logarithmic Spiral is Equiangular

Theorem
The logarithmic spiral is equiangular, in the following sense:

Let $P = \polar {r, \theta}$ be a point on a logarithmic spiral $S$ expressed in polar coordinates as:
 * $r = a e^{b \theta}$

Then the angle $\psi$ that the tangent makes to the radius vector of $S$ is constant.

Proof
Consider the logarithmic spiral $S$ expressed as:
 * $r = a e^{b \theta}$


 * LogarithmicSpiralAngle.png

Let $\psi$ be the angle between the tangent to $S$ and the radius vector.

The derivative of $r$  $\theta$ is:


 * $\dfrac {\mathrm d r} {\mathrm d \theta} = a b e^{b \theta} = b r$

and thus:

Thus for a given logarithmic spiral, $\psi$ is constant and equal to $\operatorname{arccot} b$.