Bour's Minimal Surface

Gray (1997) defines Bour's minimal curve over complex by

			(1)
			(2)
			(3)

and then derives a family of minimal surfaces.

The order-three Bour surface resembles a cross-cap and is given using Enneper-Weierstrass parameterization by

			(4)
			(5)

or explicitly by the parametric equations

			(6)
			(7)
			(8)

(Maeder 1997). It is an algebraic surface of order 16.

The coefficients of the first fundamental form are given by

			(9)
			(10)
			(11)

and the coefficients of the second fundamental form by

			(12)
			(13)
			(14)

The area element is

(15)

The Gaussian and mean curvatures are given by

			(16)
			(17)