# Space Curve

## Homework Statement

Suppose $$\alpha$$ is a regular curve in $$\mathbb{R}^3$$ with arc-length parametrization such that the torsion $$\tau(s)\neq 0$$, and suppose that there is a vector $$Y\in \mathbb{R}^3$$ such that $$<\alpha',Y>=A$$ for some constant A. Show that $$\frac{k(s)}{\tau(s)}=B$$ for some constant B, where k(s) is the curvature of alpha.

## The Attempt at a Solution

I think the Frenet formula in question that I can use is $$n'=-kt-\tau b$$, but I can't make it work.

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What about the converse? Suppose k/tau is constaint and tau is nonzero everywhere, show that there exists a nonzero vector Y such that <a', Y> is constant.

Dick