Hyperbolic Paraboloid

A hyperbolic paraboloid is the quadratic and doubly ruled surface given by the Cartesian equation

(1)

(left figure). An alternative form is

(2)

(right figure; Fischer 1986), which has parametric equations

			(3)
			(4)
			(5)

(Gray 1997, pp. 297-298).

The coefficients of the first fundamental form are

			(6)
			(7)
			(8)

and the second fundamental form coefficients are

			(9)
			(10)
			(11)

giving surface area element

(12)

The Gaussian curvature is

(13)

and the mean curvature is

(14)

The Gaussian curvature can be given implicitly as

(15)

Three skew lines always define a one-sheeted hyperboloid, except in the case where they are all parallel to a single plane but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert and Cohn-Vossen 1999, p. 15).