Ellipse Involute

The ellipse involute is the involute of an ellipse. For an ellipse with specified parametrically by

			(1)
			(2)

the involute is given by the parametric equations

			(3)
			(4)

where is an incomplete elliptic integral of the second kind, and

(5)

is the eccentricity. Here is the semimajor axis, and is the arc length measured from .

For , the curvature and a local choice of tangential angle are given by

			(6)
			(7)

where the inverse tangent specifies only modulo . A continuous branch is chosen along the curve.