Quadric

In mathematics, a quadric, or quadric surface, is any "D"-dimensional hypersurface defined as the locus of zeros of a quadratic polynomial. In coordinates $\{x\_0,\; x\_1,\; x\_2,\; ldots,\; x\_D\}$, the general quadric is defined by the algebraic equation [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html] , "Quadrics" in "Geometry Formulas and Facts" by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).]

:$sum\_\{i,j=0\}^D\; Q\_\{ij\}\; x\_i\; x\_j\; +\; sum\_\{i=0\}^D\; P\_i\; x\_i\; +\; R\; =\; 0$

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 two-dimensional (D=2) quadric in three-dimensional space centred at the origin (0,0,0) is:

:$pm\; \{x^2\; over\; a^2\}\; pm\; \{y^2\; over\; b^2\}\; pm\; \{z^2\; over\; c^2\}=1.$

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. Stewart Venit and Wayne Bishop, "Elementary Linear Algebra (fourth edition)", International Thompson Publishing, 1996.]

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

*mathworld|urlname=Quadric|title=Quadric

External links

* [http://www.professores.uff.br/hjbortol/arquivo/2007.1/qs/quadric-surfaces_en.html Interactive Java 3D models of all quadric surfaces]