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.