Quadric (projective Geometry) - Formal Definition

Formal Definition

More formally, let be an -dimensional vector space with scalar field, and let be a quadratic form on . Let be the -dimensional projective space corresponding to, that is the set, where denotes the set of all nonzero multiples of . The (projective) quadric defined by is the set of all points of such that . (This definition is consistent because implies for some, and by definition of a quadratic form.)

When is the real or complex projective plane, the quadric is also called a (projective) quadratic curve, conic section, or just conic.

When is the real or complex projective space, the quadric is also called a (projective) quadratic surface.

In general, if is the field of real numbers, a quadric is an -dimensional sub-manifold of the projective space . The exceptions are certain degenerate quadrics that are associated to quadratic forms with special properties. For instance, if is the trivial or null form (that yields 0 for any vector ), the quadric consists of all points of ; if is a definite form (everywhere positive, or everywhere negative), the quadric is empty; if factors into the product of two non-trivial linear forms, the quadric is the union of two hyperplanes; and so on. Some authors may define "quadric" so as to exclude some or all of these special cases.