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:

$r = a e^{b \theta}$

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

The derivative of $r$ with respect to $\theta$ is:

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

and thus:

\(\ds \tan \psi\)

\(=\)

\(\ds \frac r {\frac {\d r} {\d \theta} }\)

\(\ds \)

\(=\)

\(\ds \frac r {b r}\)

\(\ds \)

\(=\)

\(\ds \frac 1 b\)

\(\ds \leadsto \ \ \)

\(\ds \cot \psi\)

\(=\)

\(\ds b\)

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

$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.15$: Torricelli ($\text {1608}$ – $\text {1647}$)