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Hyperbolic Paraboloid

A Quadratic Surface given by the Cartesian equation

z={y^2\over b^2}-{x^2\over a^2}
\end{displaymath} (1)

(left figure). This form has parametric equations
$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle a(u+v)$ (2)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle \pm bv$ (3)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle u^2+2uv$ (4)

(Gray 1993, p. 336). An alternative form is
\end{displaymath} (5)

(right figure; Fischer 1986), which has parametric equations
$\displaystyle x(u,v)$ $\textstyle =$ $\displaystyle u$ (6)
$\displaystyle y(u,v)$ $\textstyle =$ $\displaystyle v$ (7)
$\displaystyle z(u,v)$ $\textstyle =$ $\displaystyle uv.$ (8)

See also Elliptic Paraboloid, Paraboloid, Ruled Surface


Fischer, G. (Ed.). Mathematical Models from the Collections of Universities and Museums. Braunschweig, Germany: Vieweg, pp. 3-4, 1986.

Fischer, G. (Ed.). Plates 7-9 in Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume. Braunschweig, Germany: Vieweg, pp. 8-10, 1986.

Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 211-212 and 336, 1993.

Meyer, W. ``Spezielle algebraische Flächen.'' Encylopädie der Math. Wiss. III, 22B, 1439-1779.

Salmon, G. Analytic Geometry of Three Dimensions. New York: Chelsea, 1979.

© 1996-9 Eric W. Weisstein