# Quadric

In mathematics, a **quadric** or **quadric surface** (**quadric hypersurface** in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension *D*) in a (*D* + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in *D* + 1 variables (*D* = 1 in the case of conic sections). When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a *degenerate quadric* or a *reducible quadric*.

In coordinates *x*_{1}, *x*_{2}, ..., *x*_{D+1}, the general quadric is thus defined by the algebraic equation[1]

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

where *x* = (*x*_{1}, *x*_{2}, ..., *x*_{D+1}) is a row vector, *x*^{T} 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 over real numbers or complex numbers, but a quadric may be defined over any field.

A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see § Projective geometry, below.