Ellipse Evolute

The evolute of an ellipse specified parametrically by

			(1)
			(2)

is given by the parametric equations

			(3)
			(4)

Eliminating allows this to be written

			(5)
			(6)
			(7)
			(8)

which is a stretched astroid sometimes known as the Lamé curve.

From a point inside the evolute, four normal vectors can be drawn to the ellipse, from a point on the evolute precisely, three normals can be drawn, and from a point outside, only two normal vectors can be drawn.

The arc length and area enclosed are

			(9)
			(10)

and the curvature, and tangential angle are

			(11)
			(12)