The curvature of a curve in space r(t) is given by
k(t) = | r'(t) à? r''(t) | / | r'(t) |^3 .
Consider now the curve
r(u) = r(sigma(u)),
given by the reparametrization t = sigma(u) of the initial curve. Show that the curvature k of the curve r is given by
k(u) = k(sigma(u)),
where k is the curvature of the initial curve r.
Instead of sigma I will just write t=t(u) for simplicity of notation.
Note the following:
dr/du = dr/dt * dt/du, and notice that ...