Quadric

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In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates {x1, x2, ..., xD+1}, the general quadric is defined by the algebraic equation[1]

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

where x = {x1, x2, ..., xD+1} is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1)×(D + 1) matrix and P is a (D + 1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be real numbers or complex numbers, but in fact, a quadric may be defined over any ring. In general, the locus of zeros of a set of polynomials is known as an algebraic variety, and is studied in the branch of algebraic geometry.

A quadric is thus an example of an algebraic variety. For the projective theory see quadric (projective geometry).

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Euclidean plane and space

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are curves. Such quadrics are the same as conic sections, and are typically known as conics rather than quadrics.

In Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. By making a suitable Euclidean change of variables, any quadric in Euclidean space can be put into a certain normal form by choosing as the coordinate directions the principal axes of the quadric. In three-dimensional Euclidean space there are 16 such normal forms. Of these 16 forms, five are nondegenerate, and the remaining are degenerate forms. Degenerate forms include planes, lines, points or even no points at all.[2]

Projective geometry

The quadrics can be treated in a uniform manner by introducing homogeneous coordinates on a Euclidean space, thus effectively regarding it as a projective space. Thus if the original (affine) coordinates on RD+1 are

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