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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 , , .

Contrary to the plane case, the equation , where F(a, 0, 0) is a point and H the projection of M on a line (D) = Oz does not give all the quadrics. The reduced equation being , we obtain an ellipsoid of revolution for k < 1, a parabolic cylinder for k = 1, and a hyperboloid of revolution for k > 1 (opposite). 
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© Robert FERRÉOL 2022