Evolute of Ellipse/Cartesian 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 $E$ is given by the Cartesian equation:

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

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: $11.29$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 9$: Special Plane Curves: Evolute of an Ellipse: $9.29.$
• Weisstein, Eric W. "Ellipse Evolute." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EllipseEvolute.html