Kiss Surface

The kiss surface is the quintic surface of revolution given by the equation

(1)

that is closely related to the ding-dong surface. It is so named because the shape of the lower portion resembles that of a Hershey's Chocolate Kiss.

It can be represented parametrically as

			(2)
			(3)
			(4)

The coefficients of the first fundamental form are

			(5)
			(6)
			(7)

and of the second fundamental form are

			(8)
			(9)
			(10)

The Gaussian and mean curvatures are given by

			(11)
			(12)

The Gaussian curvature can be given implicitly by

(13)

The surface area and volume enclosed of the top teardrop are given by

			(14)
			(15)

Its centroid is at and the moment of inertia tensor is

(16)

for a solid kiss with uniform density and mass .