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given the parameterization: exp(t)*(cos(t), sin(t))

t $\in [0, 2\pi$]

how do I calculate the total curvature?

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Apparently there are two different definitions of total curvature. You probably should have specified the definition you're using... – J. M. 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.

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