Hyperbolic Paraboloid

ParametricPlot3D[{u, v, u*v}, {u, -1, 1}, {v, -1, 1}, Axes -> True,
 Boxed -> True,
 PlotStyle ->
  Directive[RandomColor[], Opacity[0.7], Specularity[White, 30]]]

• This is a ruled surface.
• That is, it can be formed by trails of a moving 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 horizontal cross sections are hyperbolas
• Some vertical cross sections of the surface are parabolas.