What is "quadric"

Quadric \Quad"ric\, a. (Math.) Of or pertaining to the second degree.

Quadric \Quad"ric\, n.

1. (Alg.) A quantic of the second degree. See Quantic.

2. (Geom.) A surface whose equation in three variables is of the second degree. Spheres, spheroids, ellipsoids, paraboloids, hyperboloids, also cones and cylinders with circular bases, are quadrics.

a. (context mathematics English) Of or relating to the second degree; quadratic. n. (context mathematics English) A surface whose shape is defined in terms of a quadratic equation

n. a curve or surface whose equation (in Cartesian coordinates) is of the second degree [syn: quadric surface]

In mathematics, a quadric or quadric surface (quadric hypersurface if D > 2), is a generalization of conic sections ( ellipses, parabolas, and hyperbolas) to any number of dimensions. It is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial (D=1 in the case of conic sections). In coordinates , the general quadric is defined by the algebraic equation

∑xQx + ∑Px + R = 0

which may be compactly written in vector and matrix notation as:

xQx + Px + R = 0 

where is a row vector, x is the transpose of x (a column vector), Q is a matrix and P is a -dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any ring.

In general, the locus of zeros of a set of polynomials is known as an algebraic set, and is studied in the branch of algebraic geometry. A quadric is thus an example of an algebraic set. For the projective theory see Quadric (projective geometry).

In projective geometry, a quadric is the set of points of a projective space where a certain quadratic form on the homogeneous coordinates becomes zero. We shall restrict ourself to the case of finite-dimensional projective spaces.