Let C be a smooth C2 arc with parametrization (ɸ, [a, b]), and suppose that s = (t) is given by (2). The natural parametrization of C is the pair (v, [0, L]), where
v(s) = (ɸ o -1)(s) and L = L(C).
a) Prove that ||v'(s) = 1 for all s ˆˆ [0, L] and the arc length of a subcurve (v, [c, d]) of C is d - c. (This is why (v, [0, L]) is called the natural parametrization.)
b) Show that v'(s) and v"(s) are orthogonal for each s ˆˆ [0, L].
c) Prove that the absolute curvature (see Exercise 13.1.10) of (v, [0, L]) at x0 = v(s0) is K(x0) = ||v"(s0)ll-
d) Show that if x0 = ɸ(t0) = v(s0) and m = 3, then

e) Prove that the absolute curvature of an explicit Cp curve y = f(x) at (x0, y0) under the trivial parametrization is

tol (1 (y(xo))2)3/2