I made these models of curved surfaces: z=x*y, z=x^2+y^2, x^2+y^2-(¾)z^2=¼ and x^2+4y^2-(¾)z^2=¼.
It was hard to make;-)
I think no surfaces present which contain straight lines in other than hyperbolic paraboloid and one sheet of hyperboloid. Is it right?
The gif animations of these curved surfaces: