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Elliptic Paraboloid


EllipticParaboloid

A quadratic surface which has elliptical cross section. The elliptic paraboloid of height h, semimajor axis a, and semiminor axis b can be specified parametrically by

x=asqrt(u)cosv
(1)
y=bsqrt(u)sinv
(2)
z=u.
(3)

for v in [0,2pi) and u in [0,h].

This gives first fundamental form coefficients of

E=1+(a^2cos^2v+b^2sin^2v)/(4u)
(4)
F=1/4(b^2-a^2)sin(2v)
(5)
G=u(b^2cos^2v+a^2sin^2v),
(6)

second fundamental form coefficients of

e=(ab)/(2usqrt(a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v)))
(7)
f=0
(8)
g=(2abu)/(sqrt(a^2b^2+2u(a^2+b^2)+2(b^2-a^2)cos(2v))).
(9)

The Gaussian curvature and mean curvature are

K=(4a^2b^2)/([a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v)]^2)
(10)
H=(ab(a^2+b^2+4u))/([a^2b^2+2u(a^2+b^2)+2(b^2-a^2)ucos(2v)]^(3/2)).
(11)

The Gaussian curvature can be expressed implicitly as

 K(x,y,z)=(4a^6b^6)/((a^4b^4+4b^4x^2+4a^4y^2)^2).
(12)

See also

Elliptic Cone, Elliptic Cylinder, Paraboloid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227, 1987.Fischer, G. (Ed.). Plate 66 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, p. 61, 1986.JavaView. "Classic Surfaces from Differential Geometry: Elliptic Paraboloid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_EllipticParaboloid.html.

Cite this as:

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

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