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Ellipse Evolute


The Evolute of an Ellipse is given by the parametric equations

$\displaystyle x$ $\textstyle =$ $\displaystyle {a^2-b^2\over a}\cos^3 t$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {b^2-a^2\over b}\sin^3 t,$ (2)

which can be combined and written

$(ax)^{2/3}+(by)^{2/3}= [(a^2-b^2)\cos^3t]^{2/3}+[(b^2-a^2)\sin^3t]^{2/3}$
$ = (a^2-b^2)^{2/3}(\sin^2t+\cos^2t)=(a^2-b^2)^{2/3}=c^{4/3},\quad$ (3)
which is a stretched Astroid called the Lamé Curve. From a point inside the Evolute, four Normals can be drawn to the ellipse, but from a point outside, only two Normals can be drawn.

See also Astroid, Ellipse, Evolute, Lamé Curve


Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, p. 77, 1993.

© 1996-9 Eric W. Weisstein