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)

,

,

(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).