# What is the total curvature of the logarithmic spiral?

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

• Apparently there are two different definitions of total curvature. You probably should have specified the definition you're using... – J. M. is a poor mathematician Feb 9 '12 at 11:12

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.

I would like to expand on the previous answer for total curvature to point out that it applies to many spirals, not just the logarithmic spiral. Specifically, for any spiral that can be represented in the complex plane by

$$z(s)=\int e^{i\int \kappa (s) ds} ds$$

where $\kappa(s)$ is the curvature of the spiral and $s$ is the arc length. It is also known that the tangent angle is given by

$$\theta=\int \kappa (s) ds \ \ \ \text{or} \ \ \ \kappa (s)=\frac{d\theta}{ds}$$

I digress for a moment to point out that

$$z(s)=\int e^{i\theta (s)} ds \ \ \ \ \text{and} \ \ \ \ z(\theta)=\int \rho(\theta)e^{i\theta} d\theta$$

where $\rho=1/\kappa$ is the radius of curvature.

Returning to the total curvature, we'll use the standard definition given by

$$K=\int_{s_1}^{s_2}\kappa(s)ds=\int_{\theta_1}^{\theta_2}d\theta=\theta_2-\theta_1$$

independent of the spiral. Of course, $\theta$ is different for each particular spiral.

The reference for this work is: Zwikker, C. (1968). The Advanced Geometry of Plane Curves and Their Applications, Dover Press.