Quadric surface

This MedLibrary.org supplementary page on Quadric surface is provided directly from the open source Wikipedia as a service to our readers. Please see the note below on authorship of this content, as well as the Wikipedia usage guidelines. To search for other content from our encyclopedia supplement, please use the form below:

Related Sponsors

BM Pharmacy Generic pharmaceuticals at unbeatable prices. Insomnia, men's health, hair health, pain control. bmpharmacy.com
Online Pharmacy Carisoprodol, tadalafil, sildenafil, vardenafil and more... xlpharmacy.com
MedBasket.com Discreet. Inexpensive. Fast shipping. Great selection. What more can we say? medbasket.com
Frugalmed 2000 reasons to shop with us first. frugalmed.com
Ads by Tiva

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 \{x_0, x_1, x_2, \ldots, x_D\}, the general quadric is defined by the algebraic equation 1

\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. 2

ellipsoid {x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,
    spheroid (special case of ellipsoid)   {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,
       sphere (special case of spheroid) {x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,
elliptic paraboloid {x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,
    circular paraboloid (special case of elliptic paraboloid) {x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,
hyperbolic paraboloid {x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,
hyperboloid of one sheet {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,
hyperboloid of two sheets {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,
cone {x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,
elliptic cylinder {x^2 \over a^2} + {y^2 \over b^2} = 1 \,
    circular cylinder (special case of elliptic cylinder) {x^2 \over a^2} + {y^2 \over a^2} = 1 \,
hyperbolic cylinder {x^2 \over a^2} - {y^2 \over b^2} = 1 \,
parabolic cylinder x^2 + 2ay = 0 \,

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

References

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

External links

Wikipedia content modification information:

  • This page was last modified on 3 September 2008, at 15:45.

Wikipedia Authorship and Review

Wikipedia content provided here is not reviewed directly by MedLibrary.org. Wikipedia content is authored by an open community of volunteers and is not produced by or in any way affiliated with MedLibrary.org.

Wikipedia Usage Guidelines

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article on "Quadric surface".

The URL for this specific entry is:

All Wikipedia text is available under the terms of the GNU Free Documentation License. (See Copyrights for details). Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc.