Hyperbolic Paraboloid
The quadratic and doubly
ruled surface given by the Cartesian equation
 |
(1)
|
(left figure). An alternative form is
 |
(2)
|
(right figure; Fischer 1986), which has parametric
equations
(Gray 1997, pp. 297-298).
The coefficients of the first fundamental form
are
and the second fundamental form coefficients
are
giving surface area element
 |
(12)
|
The Gaussian curvature is
 |
(13)
|
and the mean curvature is
 |
(14)
|
The Gaussian curvature can be given implicitly as
 |
(15)
|
Three skew lines always define a one-sheeted hyperboloid, except in the case where they are all parallel to a single plane
but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert
and Cohn-Vossen 1999, p. 15).
SEE ALSO: Doubly Ruled Surface,
Elliptic Paraboloid,
Paraboloid,
Ruled Surface,
Saddle,
Skew Quadrilateral
REFERENCES:
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 227,
1987.
Fischer, G. (Ed.). Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Kommentarband.
Braunschweig, Germany: Vieweg, pp. 3-4, 1986.
Fischer, G. (Ed.). Plates 7-9 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig,
Germany: Vieweg, pp. 8-10, 1986.
Gray, A. "The Hyperbolic Paraboloid." Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 297-298 and 449, 1997.
Hilbert, D. and Cohn-Vossen, S. Geometry
and the Imagination. New York: Chelsea, 1999.
JavaView. "Classic Surfaces from Differential Geometry: Hyperbolic Paraboloid."
http://www-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_HyperbolicParaboloid.html.
McCrea, W. H. Analytical
Geometry of Three Dimensions. Edinburgh: Oliver and Boyd, 1947.
Meyer, W. "Spezielle algebraische Flächen." Encylopädie der
Math. Wiss. III, 22B, 1439-1779.
Salmon, G. Analytic
Geometry of Three Dimensions. New York: Chelsea, 1979.
Steinhaus, H. Mathematical
Snapshots, 3rd ed. New York: Dover, p. 245, 1999.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 110-112, 1991.
CITE THIS AS:
Weisstein, Eric W. "Hyperbolic Paraboloid."
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