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

From Length of Arc of Evolute equals Difference in Radii of Curvature:

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 $E$.

$\blacksquare$


Sources

  • 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): evolute
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): evolute
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