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One sheet (usually the positive sheet) of a hyperboloid of two sheets.
(Note that if one replaces with in the above definition, one obtains a hyperboloid of two sheets.
(hyperboloid of two sheets).
In spite of its positive curvature, the hyperboloid of two sheets with another suitably chosen metric can also be used as a model for hyperbolic geometry.
The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively.
Alternatively, a hyperboloid of two sheets of axis AB is obtained as the set of points P such that AP BP is a constant, AP being the distance between A and P. Points A and B are then called the foci of the hyperboloid.
The two-sheet hyperboloid has one positive eigenvalue and two negative eigenvalues.
Whereas the Gaussian curvature of a hyperboloid of one sheet is negative, that of a two-sheet hyperboloid is positive.
The orthogonal group O(1, n) acts by norm-preserving transformations on Minkowski space R, and it acts transitively on the two-sheet hyperboloid of norm 1 vectors.
