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:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)