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 unit tangent vector to the involute at t is the opposite of the unit normal vector N(t) to the original path x. (Hint: Use the Frenet–Serret formulas and the fact that a plane curve has torsion equal to zero everywhere.)