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| parabolic cylinder |
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In real projective space, the ellipsoid, the elliptic paraboloid and the hyperboloid of two sheets are equivalent to each other up to a projective transformation; the hyperbolic paraboloid and the hyperboloid of one sheet are not different from each other (these are ruled surfaces); the cone and the cylinder are not different from each other (these are "degenerate" quadrics, since their Gaussian curvature is zero).
In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.
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