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TWOSHEETED HYPERBOLOID H_{2}
 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: .
Surface element:

The twosheeted hyperboloid is the only nonconnected quadric.
The twosheeted 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 twosheeted 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 2017