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I've been trying to solve the following problem:

If $C = \alpha(I)$ is a line of curvature, and $k$ is its curvature at $p$, then $k = \lvert k_n k_N \rvert$, where $k_n$ is the normal curvature at $p$ along the tangent line of $C$ and $k_N$ is the curvature of the spherical image $N(C) \subseteq S^2$ at $N(p)$.

I try working with a definition of curvature not involving parametrization by arclength but no luck. Please any help would be awesome!

Katherine
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1 Answers1

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HINT: Assume $\alpha$ is arclength parametrized. Let $\beta = N\circ\alpha$. Then $\beta'= k_n\alpha'$, so the unit tangent vector of $\beta$ is equal to the unit tangent vector of $\alpha$. Now compute the curvature of $\beta$ by the usual chain rule computation.

Ted Shifrin
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