 Involute

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.

The term "involute" can have different meanings depending on the context in which it is used. Here are a few key definitions: 1. **In Geometry**: An involute of a curve is a type of curve that is derived from the original curve.