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 .
Total curvature:  where  is the distance from O to the tangent plane at the considered point.
Case of the rectangular hyperboloid  .
Cylindrical equation : .
First fundamental quadratic form : .
Gauss curvature :

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.