Hyperbolic Paraboloid

hyperbolic paraboloid 4fCHP
hyperbolic paraboloid
ParametricPlot3D[{u, v, u*v}, {u, -1, 1}, {v, -1, 1}, Axes -> False,
 Boxed -> False,
 PlotStyle ->
  Directive[RandomColor[], Opacity[0.7], Specularity[White, 30]]]
hyperbolic paraboloid
A hyperbolic-paraboloid in a virtual world Second Life.

This is a ruled surface. That is, it can be formed by tracing a line in space. We can also say that, for any point P on the surface, there is a line on the surface passing P. Hyperbolic paraboloid is in fact doubly-ruled surface, meaning that for each point P, there are two such lines on the surface passing P.

The diagonal, vertical cross sections of the surface are parabolas. The horizontal cross sections are hyperbolas

Conic Section Surfaces