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
The parametric equations of the top sheet are
for u in (-infty,infty) and v in [0,pi) (Gray 1997, p. 406). The Gaussian curvature of this surface can be given implicitly as
![画像: K(x,y,z)=(c^6)/([c^4-(a^2+c^2)z^2]^2).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Two-SheetedHyperboloid/NumberedEquation2.svg){kind=link}
The volume of a two-sheeted hyperboloid of half-separation a, height h, and radius R is
where
| [画像: R^2=(hb^2)/(a^2)(2a+h) ] |
(8)
|
(Harris and Stocket 1998). An obvious generalization gives the two-sheeted elliptic hyperboloid.
See also
Hyperboloid, One-Sheeted HyperboloidExplore with Wolfram|Alpha
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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.javaview.de/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, England: Penguin, pp. 112-113, 1991.Referenced on Wolfram|Alpha
Two-Sheeted HyperboloidCite this as:
Weisstein, Eric W. "Two-Sheeted Hyperboloid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Two-SheetedHyperboloid.html