Three-dimensional space
Quadric surfaces

In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,

where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero, is called a quadric surface.

There are six types of non-degenerate quadric surfaces:

The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of R through that conic that are normal to π). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.

Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus.
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