A second-order algebraic surface given by the general equation
 |
(1) |
Quadratic surfaces are also called quadrics, and there are 17 standard-form types. A quadratic surface intersects every plane in a (proper or degenerate) conic section. In addition, the cone consisting of all tangents from a fixed point to a quadratic surface cuts every plane in a conic section, and the points of contact of this cone with the surface form a conic section (Hilbert and Cohn-Vossen 1999, p. 12).
Define
 |  |  | (2) |
 | |  | (3) |
 | |  | (4) |
 | |  | (5) |
 | |  | (6) |
and
,
,
are the roots of
 |
(7) |
Also define
 |
(8) |
Then the following table enumerates the 17 quadrics and their properties (Beyer 1987).
| Surface | Equation |  |  |
 | k |
| Coincident planes |  | 1 | 1 | ||
| Ellipsoid (Imaginary) |
 | 3 | 4 |  | 1 |
| ellipsoid (Real) |
 | 3 | 4 |  | 1 |
| Elliptic Cone (Imaginary) |
 | 3 | 3 | 1 | |
| elliptic cone (Real) |
 | 3 | 3 | 0 | |
| Elliptic Cylinder (Imaginary) |
 | 2 | 3 | 1 | |
| elliptic cylinder (Real) |
 | 2 | 3 | 1 | |
| elliptic paraboloid |
 | 2 | 4 | | 1 |
| hyperbolic cylinder |
 | 2 | 3 | 0 | |
| hyperbolic paraboloid |  | 2 | 4 | | 0 |
| hyperboloid of one Sheet |
 | 3 | 4 | | 0 |
| hyperboloid of two Sheets |
 | 3 | 4 | | 0 |
| Intersecting Planes (Imaginary) |
 | 2 | 2 | 1 | |
| Intersecting planes (Real) |
 | 2 | 2 | 0 | |
| parabolic cylinder |  | 1 | 3 | ||
| Parallel Planes (Imaginary) |  | 1 | 2 | ||
| Parallel planes (Real) |  | 1 | 2 |
Of the non-degenerate quadratic surfaces, the elliptic (and usual) cylinder, hyperbolic cylinder, elliptic (and usual) cone are ruled surfaces, while the one-sheeted hyperboloid and hyperbolic paraboloid are doubly ruled surfaces.
A curve in which two arbitrary quadratic surfaces in arbitrary positions intersect cannot meet any plane in more than four points (Hilbert and Cohn-Vossen 1999, p. 24).
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