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Paraboloid


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

 z=b(x^2+y^2).
(1)

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

x(u,v)=asqrt(u/h)cosv
(2)
y(u,v)=asqrt(u/h)sinv
(3)
z(u,v)=u,
(4)

where u>=0, v in [0,2pi).

The coefficients of the first fundamental form are given by

E=1+(a^2)/(4hu)
(5)
F=0
(6)
G=(a^2u)/h
(7)

and the second fundamental form coefficients are

e=(a^2)/(2usqrt(a^4+4a^2hu))
(8)
f=0
(9)
g=(2a^2u)/(sqrt(a^4+4a^2hu))
(10)

The area element is then

 dS=(sqrt(a^4+4a^2hu))/(2h)du ^ dv,
(11)

giving surface area

S=int_0^(2pi)int_0^hdS
(12)
=(pia)/(6h^2)[(a^2+4h^2)^(3/2)-a^3].
(13)

The Gaussian curvature is given by

 K=(4h^2)/((a^2+4hu)^2),
(14)

and the mean curvature

 H=(2h(a^2+2hu))/((a^2+4hu)sqrt(a^4+4a^2hu)).
(15)

The volume of the paraboloid of height h is then

V=piint_0^h(a^2z)/hdz
(16)
=1/2pia^2h.
(17)

The weighted mean of z over the paraboloid is

<z>=piint_0^h(a^2z)/hzdz
(18)
=1/3pia^2h^2.
(19)

The geometric centroid is then given by

 z^_=(<z>)/V=2/3h
(20)

(Beyer 1987).


See also

Elliptic Paraboloid, Hyperbolic Paraboloid, Parabola, Poweroid

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References

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. https://mathworld.wolfram.com/Paraboloid.html

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