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

(20)

(Beyer 1987).

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.

Gray, A. "The Paraboloid." §13.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 307-308, 1997.

Harris, J. W. and Stocker, H. "Paraboloid of Revolution." §4.10.2 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998.

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, pp. 10-11, 1999.

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.

CITE THIS AS:

Weisstein, Eric W. "Paraboloid." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Paraboloid.html