What is the total curvature of the logarithmic spiral?

Question

Given the parameterization $\exp(t) * (\cos(t), \sin(t)), t \in [0, 2\pi]$, how do I calculate the total curvature?

Answer 1

The total curvature is the total turning angle of the tangent vector $\dot{\bf z}(t)$ during the given time interval. Since we are talking of a logarithmic spiral here $\dot{\bf z}(t)$ encloses a constant angle with the position vector ${\bf z}(t)$. The latter turns by $2\pi$ counterclockwise; therefore the total curvature of the considered arc is $2\pi$ either.