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.

ParametricPlot3D[{u, v, u*v}, {u, -1, 1}, {v, -1, 1}]

Other algebaric surfaces that has cross-sections of conic sections are: ellipsoid, paraboloid, hyperbolic paraboloid, hyperboloid of one sheet, hyperboloid of two sheets.

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