Quadric 145570 224494415 2008-07-09T02:39:39Z 99.235.122.139 {{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''&nbsp;+&nbsp;1)&times;(''D''&nbsp;+&nbsp;1) [[matrix (mathematics)|matrix]] and ''P'' is a (''D''&nbsp;+&nbsp;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]] |- | &nbsp; &nbsp; [[spheroid]] (special case of ellipsoid) &nbsp; | <math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,</math> |- | &nbsp; &nbsp; &nbsp;&nbsp; [[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]] |- | &nbsp; &nbsp; 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]] |- | &nbsp; &nbsp; 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| ]] [[ar:سطح ثنائي]] [[de:Quadrik]] [[eo:Kvadriko]] [[es:Cuádrica]] [[fr:Quadrique]] [[it:Quadrica]] [[nl:Kwadratisch oppervlak]] [[ja:二次曲面]] [[pl:Kwadryka]] [[pt:Quádrica]] [[ru:Поверхность второго порядка]] [[th:ผิวกำลังสอง]] [[vi:Mặt bậc hai]] [[zh:二次曲面]]