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.
Read more about this topic: Quadric (projective Geometry)
Famous quotes containing the words formal and/or definition:
“Good gentlemen, look fresh and merrily.
Let not our looks put on our purposes,
But bear it as our Roman actors do,
With untired spirits and formal constancy.”
—William Shakespeare (1564–1616)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)