Curve is Involute of Evolute
Jump to navigation
Jump to search
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