Ding-Dong Surface

The ding-dong surface is the cubic surface of revolution given by the equation

(1)

(Hauser 2003) that is closely related to the kiss surface.

The surface can be represented in parametric form as

			(2)
			(3)
			(4)

for and . In this parametrization, 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 by the upper teardrop are

			(14)
			(15)

It has centroid at , and moment of inertia tensor

(16)

for a solid teardrop with uniform density and mass .