# Dictionary Definition

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

# Extensive Definition

In mathematics, a quadric, or quadric surface, is
any D-dimensional hypersurface defined as the
locus
of zeros
of a quadratic
polynomial. In coordinates \, the general quadric is defined by
the algebraic
equation

\sum_^D Q_ x_i x_j + \sum_^D P_i x_i + R =
0

where Q is a
(D + 1)×(D + 1)
matrix
and P is a (D + 1)-dimensional vector
and R a 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).

The normalized equation for a three-dimensional
(D=3) quadric centred at the origin (0,0,0) is:

\pm \pm \pm =1.

Via translations and rotations every quadric can
be transformed to one of several "normalized" forms. In
three-dimensional Euclidean space there are 16 such normalized
forms, and the most interesting, the nondegenerate forms are given
below. The remaining forms are called degenerate forms and include
planes,
lines,
points
or even no points at all.

In real
projective space, the ellipsoid, the elliptic paraboloid and
the hyperboloid of two sheets are equivalent to each other up to a projective
transformation; the hyperbolic paraboloid and the hyperboloid
of one sheet are not different from each other (these are ruled
surfaces); the cone and the cylinder are not different from
each other (these are "degenerate" quadrics, since their Gaussian
curvature is zero).

In complex
projective space all of the nondegenerate quadrics become
indistinguishable from each other.

## See also

- Conic section
- Focus (geometry), an overview of properties of conic sections related to the foci.
- Quadratic function

## References

## External links

quadric in Arabic: سطح ثنائي

quadric in German: Quadrik

quadric in Spanish: Cuádrica

quadric in French: Quadrique

quadric in Italian: Quadrica

quadric in Dutch: Kwadratisch oppervlak

quadric in Japanese: 二次曲面

quadric in Polish: Kwadryka

quadric in Portuguese: Quádrica

quadric in Russian: Поверхность второго
порядка

quadric in Thai: ผิวกำลังสอง

quadric in Vietnamese: Mặt bậc hai

quadric in Chinese: 二次曲面