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{{for|the computing company|Quadrics}}
{{distinguish|Quadratic (disambiguation){{!}}Quadratic|Quartic (disambiguation){{!}}Quartic}}
In mathematics, a '''quadric''' or '''quadric surface''' ('''quadric hypersurface''' in higher [[dimension]]s), is a [[generalization]] of [[conic section]]s ([[ellipse]]s, [[parabola]]s, and [[hyperbola]]s). It is a [[hypersurface]] (of dimension ''D'') in a {{nowrap|(''D'' + 1)}}-dimensional space, and it is defined as  the [[zero set]] of an [[irreducible polynomial]] of [[degree of a polynomial|degree]] two in ''D'' + 1 variables ({{nowrap|1=''D'' = 1}} in the case of conic sections). When the defining polynomial is not [[absolutely irreducible]], the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''.

In coordinates {{nowrap|''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''D''+1</sub>}}, the general quadric is thus defined by the [[algebraic equation]]<ref name="geom">Silvio Levy [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node61.html Quadrics] in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', [[CRC Press]], from [[The Geometry Center]] at [[University of Minnesota]]</ref> 
:<math>
\sum_{i,j=1}^{D+1} x_i Q_{ij} x_j + \sum_{i=1}^{D+1} P_i  x_i + R = 0
</math>

which may be compactly written in vector and matrix notation as:

:<math>
x Q x^\mathrm{T} + P x^\mathrm{T} + R = 0\,
</math>

where {{nowrap|1=''x'' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''D''+1</sub>)}} is a row [[vector (geometry)|vector]], ''x''<sup>T</sup> is the [[transpose]] of ''x'' (a column vector), ''Q'' is a {{nowrap|(''D'' + 1)&thinsp;×&thinsp;(''D'' + 1)}} [[matrix (mathematics)|matrix]] and ''P'' is a {{nowrap|(''D'' + 1)}}-dimensional row vector and ''R'' a scalar constant.  The values ''Q'', ''P'' and ''R'' are often taken to be over [[real number]]s or [[complex number]]s, but a quadric may be defined over any [[field (mathematics)|field]].  

A quadric is an [[affine algebraic variety]], or, if it is reducible, an [[affine algebraic set]]. Quadrics may also be defined in [[projective space]]s; see {{slink||Projective geometry}}, below.

== Euclidean plane ==
{{main|conic section}}
Quadrics in the [[Euclidean plane]] are those of dimension ''D''&nbsp;=&nbsp;1, which is to say that they are [[plane curve]]s. In this case, one talks of ''conic sections'', or ''conics''.
[[Image:Eccentricity.svg|center|thumb|280px|Circle (''e''&nbsp;=&nbsp;0), ellipse (''e''&nbsp;=&nbsp;0.5), parabola (''e''&nbsp;=&nbsp;1), and hyperbola (''e''&nbsp;=&nbsp;2) with fixed focus ''F'' and directrix.]]

== Euclidean space ==
In three-dimensional [[Euclidean space]], quadrics have dimension ''D''&nbsp;=&nbsp;2, and are known as '''quadric surfaces'''.  They are classified and named by their [[orbit (group theory)|orbits]] under [[affine transformation]]s. More precisely, if an affine transformation maps a quadric onto another one, they belong to the same class, and share the same name and many properties.

The [[principal axis theorem]] shows that for any (possibly reducible) quadric, a suitable [[Euclidean transformation]] or a change of [[Cartesian coordinates]] allows putting the [[quadratic equation]] of the quadric into one of the following normal forms:
:<math> {x^2 \over a^2} + {y^2 \over b^2} +\varepsilon_1  {z^2 \over c^2} + \varepsilon_2=0,</math>
:<math> {x^2 \over a^2} - {y^2 \over b^2}  + \varepsilon_3=0</math>
:<math>{x^2 \over a^2}  + \varepsilon_4 =0,</math>
:<math>z={x^2 \over a^2} +\varepsilon_5 {y^2 \over b^2}, </math>
where the <math>\varepsilon_i</math> are either 1, –1 or 0, except <math> \varepsilon_3 </math> which takes only the value 0 or 1. 

Each of these 17 normal forms<ref name="ela">Stewart Venit and Wayne Bishop, ''Elementary Linear Algebra (fourth edition)'', International Thompson Publishing, 1996.</ref><ref>[http://inspirehep.net/author/S.A.Khan.5/ Sameen Ahmed Khan],[http://indapt.org/images/stories/bulletin2010/bulletin_november_2010.pdf Quadratic Surfaces in Science and Engineering], Bulletin of the IAPT, 2(11), 327–330 (November 2010). (Publication of the [[Indian Association of Physics Teachers]]).<BR> 
[https://arxiv.org/a/khan_s_1 Sameen Ahmed Khan], [https://arxiv.org/abs/1311.3602/ Coordinate Geometric Generalization of the Spherometer and Cylindrometer], [https://arxiv.org/abs/1311.3602/ arXiv:1311.3602]
<br>
(Apparently, these [[WP:reliable source|unreliable sources]] forget one case)</ref> 
corresponds to a single orbit under affine transformations. In three cases there are no real points: <math>\varepsilon_1=\varepsilon_2=1</math> (''imaginary ellipsoid''), <math>\varepsilon_1=0, \varepsilon_2=1</math> (''imaginary elliptic cylinder''), and <math>\varepsilon_4=1</math> (pair of [[complex conjugate]] parallel planes, a reducible quadric). In one case, the ''imaginary cone'', there is a single point (<math>\varepsilon_1=1, \varepsilon_2=0</math>). If <math>\varepsilon_1=\varepsilon_2=0,</math> one has a line (in fact two complex conjugate intersecting planes). For <math>\varepsilon_3=0,</math> one has two intersecting planes (reducible quadric).  For <math>\varepsilon_4=0,</math> one has a double plane. For <math>\varepsilon_4=-1,</math> one has two parallel planes (reducible quadric).

Thus, among the 17 normal forms, there are nine true quadrics: a cone, three cylinders (often called degenerate quadrics) and five non-degenerate quadrics ([[ellipsoid]], [[paraboloid]]s and [[hyperboloid]]s), which are detailed in the following tables. The eight remaining quadrics are the imaginary ellipsoid (no real point), the imaginary cylinder (no real point), the imaginary cone (a single real point), and the reducible quadrics, which are decomposed in two planes; there are five such decomposed quadrics, depending whether the planes are distinct or not, parallel or not, real or complex conjugate.

{| class="wikitable" style="background-color: white; margin: 1em auto 1em auto"
! colspan="3" style="background-color: white;" | Non-degenerate real quadric surfaces
|-
| &nbsp; &nbsp; [[Ellipsoid]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1 \,</math>
|[[Image:Ellipsoid Quadric.png|150px]]
|-
| &nbsp; &nbsp; Elliptic [[paraboloid]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} - z = 0 \,</math>
|[[Image:Paraboloid Quadric.Png|150px]]
|-
| &nbsp; &nbsp; Hyperbolic [[paraboloid]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} - z = 0  \,</math>
|[[Image:Hyperbolic Paraboloid Quadric.png|150px]]
|-
| &nbsp; &nbsp;Elliptic [[hyperboloid]] of one sheet
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 1 \,</math>
|[[Image:Hyperboloid Of One Sheet Quadric.png|150px]]
|-
| &nbsp; &nbsp;Elliptic [[hyperboloid]] of two sheets
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = - 1 \,</math>
|[[Image:Hyperboloid Of Two Sheets Quadric.png|150px]]
|}

{| class="wikitable" style="background-color: white; margin: 1em auto 1em auto"
! colspan="3" style="background-color: white;" | Degenerate real quadric surfaces
|-
| &nbsp; &nbsp; Elliptic [[Conical surface|cone]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} - {z^2 \over c^2} = 0 \,</math>
|[[Image:Elliptical Cone Quadric.Png|150px]]
|-
| &nbsp; &nbsp; Elliptic [[Cylinder (geometry)|cylinder]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} = 1 \,</math>
|[[Image:Elliptic Cylinder Quadric.png|150px]]
|-
| &nbsp; &nbsp; Hyperbolic [[Cylinder (geometry)|cylinder]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} = 1 \,</math>
|[[Image:Hyperbolic Cylinder Quadric.png|150px]]
|-
| &nbsp; &nbsp; Parabolic [[Cylinder (geometry)|cylinder]]
| <math>x^2 + 2ay = 0 \,</math>
|[[Image:Parabolic Cylinder Quadric.png|150px]]
|}

When two or more of the parameters of the canonical equation are equal, one gets a quadric [[surface of revolution|of revolution]], which remains invariant when rotated around an axis (or infinitely many axes, in the case of the sphere). 

{| class="wikitable" style="background-color: white; margin: 1em auto 1em auto"
! colspan="3" style="background-color: white;" | Quadrics of revolution
|-
| &nbsp; &nbsp; Oblate and prolate [[spheroid]]s (special cases of ellipsoid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} + {z^2 \over b^2} = 1 \,</math>
|[[Image:Oblate Spheroid Quadric.png|75px]][[Image:Prolate Spheroid Quadric.png|75px]]
|-
| &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>
|[[Image:Sphere Quadric.png|150px]]
|-
| &nbsp; &nbsp; Circular [[paraboloid]] (special case of elliptic paraboloid)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - z = 0  \,</math>
|[[Image:Circular Paraboloid Quadric.png|150px]]
|-
| &nbsp; &nbsp;Circular [[hyperboloid]] of one sheet (special case of elliptic hyperboloid of one sheet)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 1 \,</math>
|[[Image:Circular Hyperboloid Of One Sheet Quadric.png|150px]]
|-
| &nbsp; &nbsp;Circular [[hyperboloid]] of two sheets (special case of elliptic hyperboloid of two sheets)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = -1 \,</math>
|[[Image:Circular Hyperboloid of Two Sheets Quadric.png|150px]]
|-
| &nbsp; &nbsp; Circular [[Conical surface|cone]] (special case of elliptic cone)
| <math>{x^2 \over a^2} + {y^2 \over a^2} - {z^2 \over b^2} = 0 \,</math>
|[[Image:Circular Cone Quadric.png|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>
|[[Image:Circular Cylinder Quadric.png|150px]]
|}

==Definition and basic properties==
A quadric is the set of [[zero of a function|zeros]] of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have [[real number|real]] coefficients, and the zeros are points in a [[Euclidean space]]. However, most properties remain true when the coefficients belong to any [[field (mathematics)|field]] and the points belong in an [[affine space]]. As usually in [[algebraic geometry]], it is often useful to consider points over an [[algebraically closed field]] containing the polynomial coefficients, generally the [[complex number]]s, when the coefficients are real.

This define ''affine quadrics''. Many properties becomes easier to state (and to prove) by extending the quadric to the [[projective space]] by [[projective completion]] consisting of adding [[points at infinity]]. Technically, if 
:<math>p(x_1, \ldots,x_n)</math> 
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by [[homogenization of a polynomial|homogenizing]] {{mvar|p}} into
:<math>P(X_0, \ldots, X_n)=X_0^2\,p(\frac {X_1}{X_0}, \ldots,\frac {X_n}{X_0})</math>
(this is a polynomial, because the degree of {{mvar|p}} is two). The points of the projective completion are the points of the projective space whose [[projective coordinates]] are zeros of {{mvar|P}}.

===Equation===
A quadric in an [[affine space]] of dimension {{mvar|n}} is the set of zeros of a polynomial of degree 2, that is the set of the points whose coordinates satisfy an equation
:<math>p(x_1,\ldots,x_n)=0,</math>
where the polynomial {{mvar|p}} has the form
:<math>p(x_1,\ldots,x_n)=\sum_{i=1}^n a_{i,i}x_i^2+\sum_{1\le i<j\le n} (a_{i,j}+a_{j,i})x_ix_j +\sum_{i=1}^n (a_{i,0}+a_{0,i})x_i +a_{0,0},</math>
where <math>a_{i,j}=a_{j,i}</math> if the [[characteristic (algebra)|characteristic]] of the [[field (mathematics)|field]] of the coefficients is not two and <math>a_{j,i}=0</math> otherwise.

If {{mvar|A}} is the {{math|(''n'' + 1)×(''n'' + 1)}} matrix that has the <math>a_{i,j}</math> as entries, and
:<math>\mathbf x = \begin {pmatrix}1&x_1&\cdots&x_n\end{pmatrix}^{\mathsf T},</math>
then the equation may be shortened in the matrix equation
:<math>\mathbf x^{\mathsf T}A\mathbf x=0.</math>
The equation of the projective completion of this quadric is 
:<math>\sum_{i=0}^n a_{i,i}X_i^2+\sum_{0\le i<j\le n} (a_{i,j}+a_{j,i})X_iX_j,</math>
or 
:<math>\mathbf X^{\mathsf T}A\mathbf X=0,</math>
with
:<math>\mathbf X = \begin {pmatrix}X_0&X_1&\cdots&X_n\end{pmatrix}^{\mathsf T}.</math>

These equations define a quadric as an [[hypersurface|algebraic hypersurface]] of ''dimension'' {{math|''n'' – 1}}  and degree  two in a space of dimension {{mvar|n}}.

==Normal form of projective quadrics==
The quadrics can be treated in a uniform manner by introducing [[homogeneous coordinates]] on a Euclidean space, thus effectively regarding it as a [[projective space]].  Thus if the original (affine) coordinates on '''R'''<sup>''D''+1</sup> are 
:<math>(x_1,\dots,x_{D+1})</math>
one introduces new coordinates on '''R'''<sup>''D''+2</sup>
:<math>[X_0,\dots,X_{D+1}]</math>
related to the original coordinates by <math>x_i=X_i/X_0</math>.  In the new variables, every quadric is defined by an equation of the form
:<math>Q(X)=\sum_{ij} a_{ij}X_iX_j=0\,</math>
where the coefficients ''a''<sub>''ij''</sub> are symmetric in ''i'' and ''j''.  Regarding ''Q''(''X'')&nbsp;=&nbsp;0 as an equation in [[projective space]] exhibits the quadric as a projective [[algebraic variety]].  The quadric is said to be '''non-degenerate''' if the quadratic form is non-singular; equivalently, if the [[matrix (mathematics)|matrix]] (''a''<sub>''ij''</sub>) is [[invertible matrix|invertible]].

In [[real projective space]], by [[Sylvester's law of inertia]], a non-singular [[quadratic form]] ''Q''(''X'') may be put into the normal form

:<math>Q(X) = \pm X_0^2 \pm X_1^2 \pm\cdots\pm X_{D+1}^2</math> 

by means of a suitable [[projective transformation]] (normal forms for singular quadrics can have zeros as well as ±1 as coefficients). For surfaces in space (dimension ''D''&nbsp;=&nbsp;2) there are exactly three nondegenerate cases:

:<math>Q(X) = \begin{cases}
X_0^2+X_1^2+X_2^2+X_3^2\\
X_0^2+X_1^2+X_2^2-X_3^2\\
X_0^2+X_1^2-X_2^2-X_3^2
\end{cases}
</math>

The first case is the empty set.

The second case generates the ellipsoid, the elliptic paraboloid or the hyperboloid of two sheets, depending on whether the chosen plane at infinity cuts the quadric in the empty set, in a point, or in a nondegenerate conic respectively. These all have positive [[Gaussian curvature]].

The third case generates the hyperbolic paraboloid or the hyperboloid of one sheet, depending on whether the plane at infinity cuts it in two lines, or in a nondegenerate conic respectively. These are doubly [[ruled surface]]s of negative Gaussian curvature.

The degenerate form
:<math>X_0^2-X_1^2-X_2^2=0. \, </math>
generates the elliptic cylinder, the parabolic cylinder, the hyperbolic cylinder, or the cone, depending on whether the plane at infinity cuts it in a point, a line, two lines, or a nondegenerate conic respectively. These are singly ruled surfaces of zero Gaussian curvature.

We see that projective transformations don't mix Gaussian curvatures of different sign. This is true for general surfaces. <ref>S. Lazebnik and J. Ponce, {{cite web|url=http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/ijcv04b.pdf|title=The Local Projective Shape of Smooth Surfaces and Their Outlines}}, Proposition 1</ref>

In [[complex projective space]] all of the nondegenerate quadrics become indistinguishable from each other.

== Projective quadrics over fields ==

The definition of a projective quadric in a real projective space (see above) can be formally adopted defining a projective quadric in an n-dimensional projective space over a [[Field (mathematics)|field]]. In order to omit dealing with coordinates a projective quadric is usually defined starting with a quadratic form on a vector space <ref> Beutelspacher/Rosenbaum: p. 158</ref>

===Quadratic form===
Let <math>K</math> be a [[Field (algebra)|field]] and <math>V</math> a [[vector space]] over <math>K</math>. A mapping <math>q</math> from <math>V</math> to <math>K</math> such that<br />

: '''(Q1)''' <math>\;q(\lambda\vec x)=\lambda^2q(\vec x )\;</math>  for any <math>\lambda\in K</math>  and  <math>\vec x \in V</math>.

: '''(Q2)''' <math>\;f(\vec x,\vec y ):=q(\vec x+\vec y)-q(\vec x)-q(\vec y)\;</math> is a [[bilinear form]].
is called '''[[quadratic form]]'''. The bilinear form <math>f</math> is symmetric''.''

In case of <math>\operatorname{char}K\ne2</math> the bilinear form is <math>f(\vec x,\vec x)=2q(\vec x)</math>, i.e. <math>f</math> and <math>q</math> are mutually determined in a unique way.<br />
In case of <math>\operatorname{char}K=2</math> (that means: <math>1+1=0</math>) the bilinear form has the property <math>f(\vec x,\vec x)=0</math>, i.e. <math>f</math> is 
''[[Symplectic vector space|symplectic]]''.

For <math>V=K^n\ </math> and <math>\ \vec x=\sum_{i=1}^{n}x_i\vec e_i\quad </math> 
(<math>\{\vec e_1,\ldots,\vec e_n\} </math> is a base of <math>V</math>) <math>\ q</math> has the familiar form
: <math> 
q(\vec x)=\sum_{1=i\le k}^{n} a_{ik}x_ix_k\ \text{ with }\ a_{ik}:= f(\vec e_i,\vec e_k)\ \text{ for }\ i\ne k\ \text{ and }\ a_{ii}:= q(\vec e_i)\ </math> and

: <math> f(\vec x,\vec y)=\sum_{1=i\le k}^{n} a_{ik}(x_iy_k+x_ky_i)</math>.

For example:

: <math>n=3,\quad q(\vec x)=x_1x_2-x^2_3, \quad f(\vec x,\vec y)=x_1y_2+x_2y_1-2x_3y_3\; . </math>

=== ''n''-dimensional projective space over a field ===
Let <math>K</math> be a field, <math>2\le n\in\N</math>, 
:<math>V_{n+1}</math>  an {{math|(''n'' + 1)}}-[[dimension (vector space)|dimensional]] [[vector space]] over the field <math>K,</math>
:<math>\langle\vec x\rangle</math> the 1-dimensional [[linear span|subspace generated by <math>\vec 0\ne \vec x\in V_{n+1}</math>]],
: <math>{\mathcal P}=\{\langle \vec x\rangle \mid \vec x \in V_{n+1}\},\ </math> the ''set of points'' ,
: <math>{\mathcal G}=\{ \text{2-dimensional subspaces of } V_{n+1}\},\ </math> the ''set of lines''.
:<math>P_n(K)=({\mathcal P},{\mathcal G})\ </math> is the {{mvar|n}}-dimensional '''[[projective space]]''' over <math>K</math>.
:The set of points contained in a <math>(k+1)</math>-dimensional subspace of <math> V_{n+1}</math> is a ''<math>k</math>-dimensional subspace'' of  <math>P_n(K)</math>. A 2-dimensional subspace is a ''plane''. 
:In case of <math>\;n>3\;</math> a <math>(n-1)</math>-dimensional subspace is called ''hyperplane''.

=== Projective quadric ===
For a quadratic form <math>q</math>  on a vector space <math>V_{n+1}</math> a point <math>\langle\vec x\rangle \in {\mathcal P}</math> is called ''singular'' if <math>q(\vec x)=0</math>. The set
: <math>\mathcal Q=\{\langle\vec x\rangle \in {\mathcal P} \mid q(\vec x)=0\}</math>
of singular points of <math>q</math> is called '''quadric''' (with respect to the quadratic form <math>q</math>). 

'''Examples in <math> P_2(K)</math>.:'''<br>
'''(E1):''' For <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> one gets a [[Conic section|conic]].<br>
'''(E2):''' For <math>\;q(\vec x)=x_1x_2\;</math> one gets the pair of lines with the equations <math>x_1=0</math> and <math>x_2=0</math>, respectively. They intersect at point <math>\langle(0,0,1)^\text{T}\rangle</math>;

For the considerations below it is assumed that <math>\mathcal Q\ne \emptyset</math>.

=== Polar space ===
For point <math>P=\langle\vec p\rangle \in {\mathcal P}</math> the set
: <math>P^\perp:=\{\langle\vec x\rangle\in {\mathcal P} \mid f(\vec p,\vec x)=0\}</math>
is called [[Duality (mathematics)#Polarities of general projective spaces|'''polar space''']] of <math>P</math> (with respect to <math>q</math>).

If <math>\;f(\vec p,\vec x)=0\;</math> for any <math>\vec x </math>, one gets <math>P^\perp=\mathcal P</math>.

If <math>\;f(\vec p,\vec x)\ne 0\;</math> for at least one <math>\vec x </math>, the equation <math>\;f(\vec p,\vec x)=0\;</math>is a non trivial linear equation which defines a hyperplane. Hence

:<math>P^\perp</math> is either a [[hyperplane]] or <math>{\mathcal P}</math>.

=== Intersection with a line ===
For the intersection of a line with a quadric <math> \mathcal Q</math> the familiar statement is true:

:For an arbitrary  line <math>g</math> the following cases occur:
:a) <math>g\cap \mathcal Q=\emptyset\;</math> and  <math>g</math>  is called ''exterior line''  or
:b) <math> g \subset \mathcal Q\; </math> and  <math>g</math>  is called ''tangent line''  or
:b′) <math>|g\cap \mathcal Q|=1\; </math> and  <math>g</math>  is called ''tangent line''  or
:c) <math>|g\cap \mathcal Q|=2\; </math> and  <math>g</math>  is called  ''secant line''.

'''Proof:'''
Let <math>g</math> be a line, which intersects <math>\mathcal Q </math> at point <math>\;U=\langle\vec u\rangle\;</math> and <math> \;V= \langle\vec v\rangle\;</math> is a second point on <math>g</math>.
From <math>\;q(\vec u)=0\;</math> one gets<br>
<math>q(x\vec u+\vec v)=q(x\vec u)+q(\vec v)+f(x\vec u,\vec v)=q(\vec v)+xf(\vec u,\vec v)\; .</math><br>
I) In case of <math>g\subset U^\perp</math> the equation <math>f(\vec u,\vec v)=0</math> holds and it is
<math>\; q(x\vec u+\vec v)=q(\vec v)\; </math> for any <math>x\in K</math>. Hence either <math>\;q(x\vec u+\vec v)=0\;</math>
for ''any'' <math>x\in K</math> or <math>\;q(x\vec u+\vec v)\ne 0\;</math> for ''any'' <math>x\in K</math>, which proves b) and b').<br>
II) In case of  <math>g\not\subset U^\perp</math> one gets <math>f(\vec u,\vec v)\ne 0</math> and the equation 
<math>\;q(x\vec u+\vec v)=q(\vec v)+xf(\vec u,\vec v)= 0\;</math> has exactly one solution <math>x</math>.
Hence: <math>|g\cap \mathcal Q|=2</math>, which proves c).<br>

Additionally the proof shows:
:A line <math>g</math> through a point <math>P\in \mathcal Q</math> is a ''tangent'' line if and only if <math>g\subset P^\perp</math>.

=== ''f''-radical, ''q''-radical ===
In the classical cases <math>K=\R</math> or <math> \C</math> there exists only one radical, because of <math>\operatorname{char}K\ne2</math> and <math>f</math> and <math>q</math> are closely connected. In case of <math>\operatorname{char}K=2</math> the quadric <math>\mathcal Q</math> is not determined by <math>f</math> (see above) and so one has to deal with two radicals:

:a) <math>\mathcal R:=\{P\in{\mathcal P} \mid P^\perp=\mathcal P\}</math> is a projective subspace. <math>\mathcal R</math> is called '''''f''-radical''' of quadric <math>\mathcal Q</math>.
:b) <math>\mathcal S:=\mathcal R\cap\mathcal Q</math> is called '''singular radical''' or ''<math>q</math>-radical'' of <math>\mathcal Q</math>.
:c) In case of <math>\operatorname{char}K\ne2</math> one has <math>\mathcal R=\mathcal S</math>.

A quadric is called '''non-degenerate''' if <math>\mathcal S=\emptyset</math>.

'''Examples in <math> P_2(K)</math>''' (see above):<br>
'''(E1):''' For <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> (conic) the bilinear form is 
<math>f(\vec x,\vec y)=x_1y_2+x_2y_1-2x_3y_3\; . </math><br>
In case of <math>\operatorname{char}K\ne2</math> the polar spaces are never <math>\mathcal P</math>. Hence <math>\mathcal R=\mathcal S=\empty</math>.<br>
In case of <math>\operatorname{char}K=2</math> the bilinear form is reduced to
<math>f(\vec x,\vec y)=x_1y_2+x_2y_1\; </math> and <math>\mathcal R=\langle(0,0,1)^\text{T}\rangle\notin \mathcal Q</math>. Hence <math>\mathcal R\ne \mathcal S=\empty \; .</math> 
In this case the ''f''-radical is the common point of all tangents, the so called ''knot''.<br>
In both cases <math> S=\empty</math> and the quadric (conic) ist ''non-degenerate''.<br>
'''(E2):''' For <math>\;q(\vec x)=x_1x_2\;</math> (pair of lines) the bilinear form is <math>f(\vec x,\vec y)=x_1y_2+x_2y_1\;  </math> and <math>\mathcal R=\langle(0,0,1)^\text{T}\rangle=\mathcal S\; ,</math> the intersection point. <br>
In this example the quadric is ''degenerate''.

=== Symmetries ===
A quadric is a rather homogeneous object:

:For any point <math>P\notin \mathcal Q\cup {\mathcal R}\;</math> there exists an [[Involution (mathematics)|involutorial]] central [[collineation]] <math>\sigma_P</math> with center <math>P</math> and <math>\sigma_P(\mathcal Q)=\mathcal Q</math>.

'''Proof:'''
Due to <math>P\notin \mathcal Q\cup {\mathcal R}</math> the polar space <math>P^\perp</math> is a hyperplane.

The linear mapping 

: <math>\varphi: \vec x \rightarrow \vec x-\frac{f(\vec p,\vec x)}{q(\vec p)}\vec p</math>

induces an ''involutorial central collineation'' <math>\sigma_P</math> with axis <math>P^\perp</math> and centre <math>P</math> which leaves <math>\mathcal Q</math> invariant.<br />
In case of <math>\operatorname{char}K\ne2</math> mapping <math>\varphi</math> gets the [[Reflection (mathematics)|familiar shape]] <math>\; \varphi: \vec x \rightarrow \vec x-2\frac{f(\vec p,\vec x)}{f(\vec p,\vec p)}\vec p\; </math> with <math>\; \varphi(\vec p)=-\vec p</math> and <math>\; \varphi(\vec x)=\vec x\; </math> for any <math>\langle\vec x\rangle \in P^\perp</math>.

'''Remark:'''
:a) An exterior line, a tangent line or a secant line is mapped by the involution <math>\sigma_P</math> on an exterior, tangent and secant line, respectively.
:b) <math>{\mathcal R}</math> is pointwise fixed by <math>\sigma_P</math>.

===''q''-subspaces and index of a quadric ===
A subspace <math>\;\mathcal U\;</math> of <math>P_n(K)</math> is called <math>q</math>-subspace if <math>\;\mathcal U\subset\mathcal Q\;</math> 

For example: points on a sphere or [[ruled surface|lines on a hyperboloid]] (s. below).

:Any two ''maximal'' <math>q</math>-subspaces have the same dimension <math>m</math> <ref>Beutelpacher/Rosenbaum, p.139</ref>.

Let be <math>m</math> the dimension of the maximal <math>q</math>-subspaces of <math>\mathcal Q</math> then
:The integer <math>\;i:=m+1\;</math> is called '''index''' of <math>\mathcal Q</math>.

'''Theorem: (BUEKENHOUT)<ref>F. Buekenhout: ''Ensembles Quadratiques des Espace Projective'', Math. Teitschr. 110 (1969), p. 306-318.</ref>'''
:For the index <math>i</math> of a non-degenerate quadric <math>\mathcal Q</math> in <math>P_n(K)</math> the following is true: 
::<math>i\le \frac{n+1}{2}</math>.

Let be <math>\mathcal Q</math> a non-degenerate quadric in <math> P_n(K), n\ge 2</math>, and <math>i</math> its index.<br />
: In case of <math>i=1</math> quadric <math>\mathcal Q</math> is called ''sphere'' (or [[oval (projective plane)|oval]] conic if <math>n=2</math>).
: In case of <math>i=2</math> quadric <math>\mathcal Q</math> is called ''hyperboloid'' (of one sheet).

'''Examples:'''
:a) Quadric <math>\mathcal Q</math> in <math>P_2(K)</math> with form <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> is non-degenerate with index 1.
:b) If polynomial <math>\;p(\xi)=\xi^2+a_0\xi+b_0\;</math> is [[Irreducible polynomial|irreducible]] over <math>K</math> the quadratic form <math>\;q(\vec x)=x^2_1+a_0x_1x_2+b_0x^2_2-x_3x_4\;</math> gives rise to a non-degenerate quadric <math>\mathcal Q</math> in <math>P_3(K)</math> of index 1 (sphere). For example: <math>\;p(\xi)=\xi^2+1\;</math> is irreducible over <math>\R</math> (but not over <math>\C</math> !).
:c) In <math>P_3(K)</math> the quadratic form <math>\;q(\vec x)=x_1x_2+x_3x_4\;</math> generates a ''hyperboloid''.

=== Generalization of quadrics: quadratic sets ===
It is not reasonable to formally extend the definition of quadrics to spaces over genuine skew fields (division rings). Because one would get secants bearing more than 2 points of the quadric which is totally different from ''usual'' quadrics.<ref> R. [[Rafael Artzy|Artzy]]: ''The Conic <math>y=x^2</math> in Moufang Planes'', Aequat.Mathem. 6 (1971), p. 31-35</ref><ref>E. Berz: ''Kegelschnitte in Desarguesschen Ebenen'', Math. Zeitschr. 78 (1962), p. 55-8</ref><ref>external link E. Hartmann: ''Planar Circle Geometries'', p. 123</ref> The reason is the following statement.

:A [[division ring]] <math>K</math> is [[commutative ring|commutative]] if and only if any [[quadratic equation|equation]] <math>x^2+ax+b=0, \ a,b \in K</math>, has at most two solutions.

There are ''generalizations'' of quadrics: [[quadratic set]]s.<ref>Beutelspacher/Rosenbaum: p. 135</ref> A quadratic set is a set of points of a projective space with the same geometric properties as a quadric: every line intersects a quadratic set in at most two points or is contained in the set.

== See also ==
*[[Klein quadric]]
*[[Rotation of axes]]
*[[Superquadrics]]
*[[Translation of axes]]

== References ==

{{reflist}}

== Bibliography ==
* M. Audin: ''Geometry'', Springer, Berlin, 2002, {{ISBN|978-3-540-43498-6}},   p. 200.
* M. Berger: ''Problem Books in Mathematics'', ISSN 0941-3502, Springer New York, pp 79-84.
* A. Beutelspacher, U. Rosenbaum: ''Projektive Geometrie'',  Vieweg + Teubner, Braunschweig u. a. 1992, {{ISBN|3-528-07241-5}}, p. 159.
* P. Dembowski: ''Finite Geometries'', Springer, 1968, {{ISBN|978-3-540-61786-0}}, p. 43.
*{{springer|id=q/q076220|title=Quadric|first=V.A.|last=Iskovskikh}}
*{{mathworld|urlname=Quadric|title=Quadric}}

== External links ==
*[https://web.archive.org/web/20070929094100/http://www.professores.uff.br/hjbortol/arquivo/2007.1/qs/quadric-surfaces_en.html Interactive Java 3D models of all quadric surfaces]
*[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note '''''Planar Circle Geometries''''', an Introduction to Moebius, Laguerre and Minkowski Planes], p.&nbsp;117

[[Category:Quadrics| ]]
[[Category:Projective geometry]]

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