Elliptic Torus

A surface of revolution which is generalization of the ring torus. It is produced by rotating an ellipse having horizontal semi-axis , vertical semi-axis , embedded in the -plane, and located a distance away from the -axis about the -axis. It is given by the parametric equations

			(1)
			(2)
			(3)

for .

This gives first fundamental form coefficients of

			(4)
			(5)
			(6)

second fundamental form coefficients of

			(7)
			(8)
			(9)

The Gaussian curvature and mean curvature are

			(10)
			(11)

By Pappus's centroid theorems, the surface area and volume are

			(12)
			(13)
			(14)
			(15)

where is a complete elliptic integral of the first kind and

(16)

is the eccentricity of the ellipse cross section.