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.

Show that the involute y of the path x is formed by unwinding a taut string that has been wrapped around x as follows:

(a) Show that the distance in R² between a point x(t) on the original path and the corresponding point y(t) on the involute is equal to the distance traveled from x(t₀) to x(t) along the underlying curve of x.

(b) Show that the distance between a point x(t) on the path and the corresponding point y(t) on the involute is equal to the distance from x(t) to y(t) measured along the tangent emanating from x(t). Then finish the argument.