A hyperboloid is a quadratic surface which may be one- or two-sheeted. The two-sheeted hyperboloid is a surface
of revolution obtained by rotating a hyperbola
about the line joining the foci (Hilbert and Cohn-Vossen
1991, p. 11).
A two-sheeted circular hyperboloid oriented along the z -axis
has Cartesian coordinates equation
(1)
The parametric equations of the top sheet
are
for
and
(Gray 1997, p. 406). The Gaussian curvature
of this surface can be given implicitly as
(5)
The volume of a two-sheeted hyperboloid of half-separation ,
height ,
and radius
is
where
(8)
(Harris and Stocket 1998). An obvious generalization gives the two-sheeted elliptic
hyperboloid .
See also Hyperboloid ,
One-Sheeted
Hyperboloid
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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.). Plates 67 and 69 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig,
Germany: Vieweg, pp. 62 and 64, 1986. Gray, A. "The Hyperboloid
of Revolution." §20.5 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, p. 470, 1997. Harris, J. W. and Stocker,
H. "Hyperboloid of Revolution." §4.10.3 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. JavaView.
"Classic Surfaces from Differential Geometry: Hyperboloid." http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_Hyperboloid.html . Steinhaus,
H. Mathematical
Snapshots, 3rd ed. New York: Dover, 1999. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 112-113, 1991. Referenced on Wolfram|Alpha Two-Sheeted Hyperboloid
Cite this as:
Weisstein, Eric W. "Two-Sheeted Hyperboloid."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Two-SheetedHyperboloid.html
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