I can somewhat imagine the correlation between the radius of curvature and magnitude of a vector like this:
If some curve C(s) is tangent at point C(p) to a circle with radius R centered at the origin, then if its radius of curvature Rc at p is greater than R, then its magnitude should increase after p because the osculating circle is bigger than the one centered at the origin. If you draw the circle tangent to C at C(p) with radius Rc then its magnitude will also be increasing in the direction of increasing p. Since the osculating circle's magnitude is increasing shortly after p, it should be reasonable to expect |C(p)| to also increase after p.
I realize this statement is very vague so:
- Is this true or are there counterexamples?
- If true, how can this be proved in terms of the properties of being tangent to the circle like the derivative wrt s of the magnitude being 0 at the point of tangency?