Let x: I → R2 be a path of class C2 that is not a straight line and such that x’...

Let x: I → R² be a path of class C2 that is not a straight line and such that x’(t) 0. Let

This is the path traced by the center of the osculating circle of the path x . The quantity ρ = 1/κ is the radius of the osculating circle and is called the radius of curvature of the path x. The path e is called the evolute of the path x. Exercises 20–25 involve evolutes of paths.

Let x(t) = (t, t²) be a parabolic path. (See Figure 3.50.)

(a) Find the unit tangent vector T, the unit normal vector N, and the curvature κ as functions of t.

(b) Calculate the evolute of x.

(c) Use a computer to plot x(t) and e(t) on the same set of axes.