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# Hyperbolic Paraboloid

A hyperbolic paraboloid is 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

 (3) (4) (5)

(Gray 1997, pp. 297-298).

The coefficients of the first fundamental form are

 (6) (7) (8)

and the second fundamental form coefficients are

 (9) (10) (11)

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).

Doubly Ruled Surface, Elliptic Paraboloid, Paraboloid, Ruled Surface, Saddle, Skew Quadrilateral

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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.). 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." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicParaboloid.html