# Evolute of Ellipse

## Theorem

Let $E$ be an ellipse embedded in a Cartesian plane with the equation:

$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

### Cartesian Form

The evolute of $E$ is given by the Cartesian equation:

$\paren {a x}^{2 / 3} + \paren {b y}^{2 / 3} = \paren {a^2 - b^2}^{2 / 3}$

### Parametric Form

The evolute of $E$ can be expressed using the parametric equation:

$\begin {cases} a x = \paren {a^2 - b^2} \cos^3 \theta \\ b y = \paren {a^2 - b^2} \sin^3 \theta \end {cases}$