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# Two-Sheeted Hyperboloid

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

 (2) (3) (4)

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

 (6) (7)

where

 (8)

(Harris and Stocket 1998). An obvious generalization gives the two-sheeted elliptic hyperboloid.

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