Involute
= Involute
{wiki}
An <involute> is the curve traced by the end of a taut string as it is unwound from a fixed curve. For a <circle> of radius $R$, one parametrization is $\mathbf x(\theta)=R(\cos\theta+\theta\sin\theta,\sin\theta-\theta\cos\theta)$. The straight free <length> is $R\theta$, and differentiation shows that the traced <velocity> is perpendicular to the taut string. Reversing the motion winds the string back around the circle.