Hyperbolic Paraboloid

A Quadratic Surface given by the Cartesian equation

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

(left figure). This form has parametric equations

$$x(u,v) = a(u+v)$$ (2)

$$y(u,v) = \pm bv$$ (3)

$$z(u,v) = u^2+2uv$$ (4)

(Gray 1993, p. 336). An alternative form is

$$z=xy$$ (5)

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

$$x(u,v) = u$$ (6)

$$y(u,v) = v$$ (7)

$$z(u,v) = uv.$$ (8)

See also Elliptic Paraboloid, Paraboloid, Ruled Surface

References

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.