Rembs' Surface

A surface of constant Gaussian curvature that can be given parametrically by

			(1)
			(2)
			(3)

where

			(4)
			(5)
			(6)

and , and . The value of is restricted to

(7)

(Reckziegel 1986), and the values correspond to the ends of the cleft in the surface. The surface illustrated above corresponds to .

The coefficients of the first fundamental form are given by

			(8)
			(9)
			(10)

with coefficients of the second fundamental form given by similar, but rather complicated, expressions. The Gaussian curvature is

(11)

with the mean curvature given by a rather complicated expression.