Paraboloid

A paraboloid is a quadratic surface of either elliptic or hyperbolic type. A circular paraboloid is the surface of revolution of a parabola about its axis and is the shape used in the reflectors of automobile headlights (Steinhaus 1999, p. 242; Hilbert and Cohn-Vossen 1999). It can be specified by the Cartesian equation

(1)

where . The circular paraboloid which has radius at height has and the parametric equations

			(2)
			(3)
			(4)

where , .

For , the coefficients of the first fundamental form are given by

			(5)
			(6)
			(7)

and, with the upward-pointing unit normal vector, the second fundamental form coefficients are

			(8)
			(9)
			(10)

The area element is then

(11)

giving surface area

			(12)
			(13)

The Gaussian curvature is given by

(14)

and the mean curvature

(15)

The volume enclosed by the circular paraboloid and the plane is then

			(16)
			(17)

The first moment of over this solid is

			(18)
			(19)

The geometric centroid is then given by

(20)

(Beyer 1987).