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.
2.
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.
3.
I can't find it but I saw a picture showing this plane and it looked so much like a hyperboloid of two sheets ( and maybe the two nuclei are the focal points ) If it's true could you show me a proof or something ? thanks .-- talk ) 08 : 50, 18 May 2011 ( UTC)
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