Ellipse Evolute

$\begin{figure}\begin{center}\BoxedEPSF{ellipse_evolute.epsf}\end{center}\end{figure}$

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

References

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

$(ax)^{2/3}+(by)^{2/3}= [(a^2-b^2)\cos^3t]^{2/3}+[(b^2-a^2)\sin^3t]^{2/3}$
$= (a^2-b^2)^{2/3}(\sin^2t+\cos^2t)=(a^2-b^2)^{2/3}=c^{4/3},\quad$	(3)