Prove that a SRAN curve is part of a circle

Let *alpha*(s): I -> R^3 be smooth, regular, non-singular curve parameterized by arc length with the property that all normal lines to *alpha(*s) pass through the origin. We can assume that *alpha*(s) is non-zero for any s in I.

Prove that *alpha* is part of a circle. (i.e. the trace is contained in a plane and the distance from *alpha* to a fixed point is constant). Hint: explain why *alpha*(s)+f(s)n(s) = 0 for some f:I -> R, and use this together with the Frenet equations to prove the result.

From the hint, I see that since n(s) is the normal vector to *alpha* at s, f can be -*alpha*(s) to give us the zero. But what does this have to do with the Frenet equations? Do we want to prove that curvature is 1/r, thus the trace of *alpha *is a circle at that point?

Re: Prove that a SRAN curve is part of a circle

The normal line to is given by

.

Since every normal line passes through the origin, for every there exists an such that

.

The function is differentiable (why?). Differentiating the last relation gives

where is the Serret-Frenet trihedron. Linear independence now implies

, so the curve is planar and

, so the curvature is constant;

Thus is part of a circle.