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?