Torispherical Dome

A torispherical dome is the surface obtained from the intersection of a spherical cap with a tangent torus, as illustrated above. The radius of the sphere is called the "crown radius," and the radius of the torus is called the "knuckle radius." Torispherical domes are used to construct pressure vessels.

Let be the distance from the center of the torus to the center of the torus tube, let be the radius of the torus tube, and let be the height from the base of the dome to the top. Then the radius of the base is given by . In addition, by elementary geometry, a torispherical dome satisfies

(1)

so

(2)

The transition from sphere to torus occurs at the critical radius

(3)

so the dome has equation

(4)

where

(5)

The torispherical dome has volume

			(6)
			(7)