Horn Torus

A horn torus is one of the three standard tori, given by the parametric equations

			(1)
			(2)
			(3)

corresponding to the torus with . Hauser's "Dullo" surface has equation

(4)

which is the horn torus with .

It has coefficients of the first fundamental form given by

			(5)
			(6)
			(7)

and of the second fundamental form given by

			(8)
			(9)
			(10)

The area element is

(11)

and the surface area and volume are

			(12)
			(13)

The geometric centroid is at , and the moment of inertia tensor for a solid torus is given by

(14)

for a uniform density torus of mass .

The inversion of a horn torus is a horn cyclide. The above figures show a horn torus (left), a cutaway (middle), and a cross section of the horn torus through the -plane (right).