TOPICS
Search

Hyperbolic Paraboloid


HyperbolicParaboloidHyperbolicParaboloid2

A hyperbolic paraboloid is the quadratic and doubly ruled surface given by the Cartesian equation

 z=(y^2)/(b^2)-(x^2)/(a^2)
(1)

(left figure). An alternative form is

 z=xy
(2)

(right figure; Fischer 1986), which has parametric equations

x(u,v)=u
(3)
y(u,v)=v
(4)
z(u,v)=uv
(5)

(Gray 1997, pp. 297-298).

The coefficients of the first fundamental form are

E=1+v^2
(6)
F=uv
(7)
G=1+u^2,
(8)

and the second fundamental form coefficients are

e=0
(9)
f=(1+u^2+v^2)^(-1/2)
(10)
g=0,
(11)

giving surface area element

 dS=sqrt(1+u^2+v^2).
(12)

The Gaussian curvature is

 K=-(1+u^2+v^2)^(-2)
(13)

and the mean curvature is

 H=-(uv)/((1+u^2+v^2)^(3/2)).
(14)

The Gaussian curvature can be given implicitly as

 K(x,y,z)=-(4a^6b^6)/((a^4b^4+4b^4x^2+4a^4y^2)^2).
(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

Explore with Wolfram|Alpha

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

Subject classifications