Evolute of Ellipse/Parametric Form

Theorem

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

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

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

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

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Evolute of an Ellipse: $11.30$

• Weisstein, Eric W. "Ellipse Evolute." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipseEvolute.html