Quadric

From Wikipedia, the free encyclopedia

For the computing company, see Quadrics.

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of zeros of a quadratic polynomial. In coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): \{x_0, x_1, x_2, \ldots, x_D\} , the general quadric is defined by the algebraic equation ^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{i,j=0}^D Q_{i,j} x_i x_j + \sum_{i=0}^D P_i x_i + R = 0

where Q is a (D + 1)-dimensional 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. Every projective variety can be shown to be isomorphic to the intersection of a set of quadrics. 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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. ^[2]

ellipsoid	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,	Image:Quadric Ellipsoid.jpg
spheroid (special case of ellipsoid)	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,
sphere (special case of spheroid)	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,
elliptic paraboloid	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,	Image:Quadric Elliptic Paraboloid.jpg
circular paraboloid	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,
hyperbolic paraboloid	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,	Image:Quadric Hyperbolic Paraboloid.jpg
hyperboloid of one sheet	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,	Image:Quadric Hyperboloid 1.jpg
hyperboloid of two sheets	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,	Image:Quadric Hyperboloid 2.jpg
cone	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,	Image:Quadric Cone.jpg
elliptic cylinder	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over b^2} = 1 \,	Image:Quadric Elliptic Cylinder.jpg
circular cylinder	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} + {y^2 \over a^2} = 1 \,
hyperbolic cylinder	Failed to parse (Missing texvc executable; please see math/README to configure.): {x^2 \over a^2} - {y^2 \over b^2} = 1 \,	Image:Quadric Hyperbolic Cylinder.jpg
parabolic cylinder	Failed to parse (Missing texvc executable; please see math/README to configure.): x^2 + 2ay = 0 \,	Image:Quadric Parabolic Cylinder.jpg

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.

[edit] See also

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

[edit] References

^ [1], Quadrics in Geometry Formulas and Facts by Silvio Levy, excerpted from 30th Edition of the CRC Standard Mathematical Tables and Formulas (CRC Press).
^ Stewart Venit and Wayne Bishop, Elementary Linear Algebra (fourth edition), International Thompson Publishing, 1996.