Paraboloid

The surface of revolution of the parabola which is the shape used in the reflectors of automobile headlights (Steinhaus 1999, p. 242; Hilbert and Cohn-Vossen 1999). It is a quadratic surface which can be specified by the Cartesian equation

(1)

The paraboloid which has radius  at height  is then given parametrically by

			(2)
			(3)
			(4)

where , .
The coefficients of the first fundamental form are given by

			(5)
			(6)
			(7)

and 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 of the paraboloid of height  is then

			(16)
			(17)

The weighted mean of  over the paraboloid is

			(18)
			(19)

The geometric centroid is then given by