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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.

See also Poinsot spiral.
 
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© Robert FERRÉOL , Jacques MANDONNET 2017