TWO-SHEETED HYPERBOLOID H2 Cartesian equation: , . For a = b: two-sheeted hyperboloid of revolution. For a = b = c: rectangular two-sheeted hyperboloid. Small exercise: of what type is the quadric ? Answer: by a change of orthonormal frame such that OZ is the line x=y=z we get hence H1 for , a cone for , H2 for all of them of revolution around OZ.
Cartesian parametrization
- the coordinate lines of which give a family of hyperbolas and a family of ellipses: , or , or also: .
- the coordinate lines of which are the curvature lines: with, for c < b < a The values u = v = a² give the 4 umbilics, with coordinates Cylindrical equation in the case of the hyperboloid of revolution: Surface element:
Total curvature: where is the distance from O to the tangent plane at the considered point.

The two-sheeted hyperboloid is the only non-connected quadric.

The two-sheeted hyperboloid of revolution can be defined as the surface of revolution generated by the rotation of a hyperbola around its transverse axis. It is the locus of the points M satisfying , where F and F' are the common foci of these hyperbolas. View of the curvature lines of the two-sheeted hyperboloid; they are circles and hyperboloids only in the case of the hyperboloid of revolution, otherwise, they are biquadratics. The 4 singularities are the umbilics. View of one of the two families of circles included in any H2, even if it is not of revolution, with the two corresponding umbilics.