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QUADRIC
From the Latin quadrus: square. 
A quadric is an algebraic surface of degree 2; see the classification below.
Cartesian equation: ,
where P is a polynomial of degree 2.
The quadric is said to be non degenerate if the homogeneous quadratic form is non degenerate, i.e. of rank 4 (which amounts to saying that the surface is smooth). When the rank of this form is equal to 3, we get the cones and cylinders of second degree, and when it is less than or equal to 2, the quadric is the union of two planes. Reduced Cartesian equation (up to isometry) of the quadrics: with . I) Real affine classification.
II) Real projective classification. There remain only two nonempty types:
and one nonempty type of rank 3: . III) Complex projective classification:

Homofocal quadrics and triple orthogonal system of nondegenerate quadrics.
If a > b > c, then the quadrics with equation are nondegenerate quadrics the sections by the symmetry planes of which are confocal conics (i.e. with the same focus); the 6 foci are , , .

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© Robert FERRÉOL 2017