Quadric
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2008-07-09T02:39:39Z
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{{for|the computing company|Quadrics}}
In mathematics, a '''quadric''', or '''quadric surface''', is any ''D''-dimensional [[hypersurface]] defined as the [[locus (mathematics)|locus]] of [[root (mathematics)|zeros]] of a [[quadratic polynomial]]. In coordinates <math>\{x_0, x_1, x_2, \ldots, x_D\}</math>, the general quadric is defined by the [[algebraic equation]] <ref name="geom"> [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).</ref>
:<math>
\sum_{i,j=0}^D Q_{ij} x_i x_j + \sum_{i=0}^D P_i x_i + R = 0
</math>
where ''Q'' is a (''D'' + 1)×(''D'' + 1) [[matrix (mathematics)|matrix]] and ''P'' is a (''D'' + 1)-dimensional [[vector (spatial)|vector]] and ''R'' a constant. The values ''Q'', ''P'' and ''R'' are often taken to be [[real number]]s or [[complex number]]s, but in fact, a quadric may be defined over any [[ring (mathematics)|ring]]. In general, the locus of zeros of a set of [[polynomial]]s 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:
:<math>
\pm {x^2 \over a^2} \pm {y^2 \over b^2} \pm {z^2 \over c^2}=1.
</math>
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 ''[[Degeneracy (mathematics)|nondegenerate]]'' forms are given below. The remaining forms are called ''[[Degeneracy (mathematics)|degenerate]]'' forms and include [[plane (mathematics)|plane]]s, [[line (mathematics)|line]]s, [[point (mathematics)|point]]s or even no points at all. <ref name="ela"> Stewart Venit and Wayne Bishop, ''Elementary Linear Algebra (fourth edition)'', International Thompson Publishing, 1996.</ref>
{|
|[[ellipsoid]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,</math>
|[[Image:Quadric Ellipsoid.jpg|150px]]
|-
| [[spheroid]] (special case of ellipsoid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,</math>
|-
| [[sphere]] (special case of spheroid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over a^2} = 1 \,</math>
|-
|elliptic [[paraboloid]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,</math>
|[[Image:Quadric Elliptic Paraboloid.jpg|150px]]
|-
| circular [[paraboloid]] (special case of elliptic paraboloid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - z = 0 \,</math>
|-
|hyperbolic [[paraboloid]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} - z = 0 \,</math>
|[[Image:Quadric Hyperbolic Paraboloid.jpg|150px]]
|-
|[[hyperboloid]] of one sheet
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,</math>
|[[Image:Quadric Hyperboloid 1.jpg|150px]]
|-
|[[hyperboloid]] of two sheets
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,</math>
|[[Image:Quadric Hyperboloid 2.jpg|150px]]
|-
|[[conical surface|cone]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,</math>
|[[Image:Quadric Cone.jpg|150px]]
|-
|elliptic [[Cylinder (geometry)|cylinder]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} = 1 \,</math>
|[[Image:Quadric Elliptic Cylinder.jpg|150px]]
|-
| circular [[Cylinder (geometry)|cylinder]] (special case of elliptic cylinder)
| <math>{x^2 \over a^2} + {y^2 \over a^2} = 1 \,</math>
|-
|hyperbolic [[Cylinder (geometry)|cylinder]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} = 1 \,</math>
|[[Image:Quadric Hyperbolic Cylinder.jpg|150px]]
|-
|parabolic [[Cylinder (geometry)|cylinder]]
| <math>x^2 + 2ay = 0 \,</math>
|[[Image:Quadric Parabolic Cylinder.jpg|150px]]
|}
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 surface]]s); 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==
{{reflist}}
*{{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]
[[Category:Geometry]]
[[Category:Surfaces]]
[[Category:Quadrics| ]]
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