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. Choose some t₀ ∈ I and let

y(t) = x(t) − s(t)T(t),

where is the arclength function and T is the unit tangent vector. The path y: I → R² is called the involute of x. Exercises 17–19 concern involutes of paths.

(a) Calculate the involute of the circular path of radius a, that is, x(t)=(a cos t, a sin t). (Take t₀ to be 0.)

(b) Let a = 1 and use a computer to graph the path x and the involute path y on the same set of axes.