# Curve is Involute of Evolute

## Theorem

Let $C$ be a curve defined by a real function which is twice differentiable.

Let the curvature of $C$ be non-constant.

Let $E$ be the evolute $C$.

Then the involute of $E$ is $C$.

## Proof

the length of arc of the evolute $E$ of $C$ between any two points $Q_1$ and $Q_2$ of $C$ is equal to the difference between the radii of curvature at the corresponding points $P_1$ and $P_2$ of $C$.

Thus $C$ exhibits precisely the property of the involute of $E$.

$\blacksquare$