11. Let G be a smooth C2 arc with parametrization (¢, [a,b]), and let s = let) be given by (2). The natuml pammetrization of G is the pair (v, [0, L]), where v(s) = (¢ol-l)(S) and L = L(G).

(a) Prove that Ilv'(s)11 = 1 for all s E [0, L] and the arc length of a sub curve (v, [c, d]) of Cis 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 E [0, L].

(c) Prove that the absolute curvature (see Exercise 10 above) of (v, [0, L]) at Xo = v(so) is tI;(xo) = Ilv//(so)ll·

(d) Show that ifxo = ¢(to) = v(so) and m = 3, then , // II IW(to) x ¢//(to)II tI;(xo) = Ilv (so) x v (so) = IW(to)113 .

(e) Prove that the absolute curvature of an explicit CP curve y = f (x) at (xo, Yo)
under the trivial parametrization is ly//(xo)1 tI;= .
(1 + (Y'(XO))2)3/2