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 -axisCartesian coordinates equation
(1)
The parametric equations of the top sheet
are
for Gaussian curvature
of this surface can be given implicitly as
(5)
The volume of a two-sheeted hyperboloid of half-separation
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. Fischer, G. (Ed.). Plates 67 and 69 in Mathematische
Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Gray, A. "The Hyperboloid
of Revolution." §20.5 in Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Harris, J. W. and Stocker,
H. "Hyperboloid of Revolution." §4.10.3 in Handbook
of Mathematics and Computational Science. Hilbert, D. and Cohn-Vossen, S. Geometry
and the Imagination. 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. Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Two-Sheeted Hyperboloid
Cite this as:
Weisstein, Eric W. "Two-Sheeted Hyperboloid."
From MathWorld https://mathworld.wolfram.com/Two-SheetedHyperboloid.html
Subject classifications
and 
(Gray 1997, p. 406). The Gaussian curvature
of this surface can be given implicitly as
![K(x,y,z)=(c^6)/([c^4-(a^2+c^2)z^2]^2).](/images/equations/Two-SheetedHyperboloid/NumberedEquation2.svg)
,
height 
,
and radius 
is
