Editing Quadric

{{short description|Locus of the zeros of a polynomial of degree two}}
{{distinguish|Quadratic (disambiguation){{!}}Quadratic|Quartic (disambiguation){{!}}Quartic}}
{{for-multi|the computing company|Quadrics (company)|quadrics in algebraic geometry|Quadric (algebraic geometry)}}

In mathematics, a '''quadric''' or '''quadric surface''' is a [[generalization]] of [[conic section]]s ([[ellipse]]s, [[parabola]]s, and [[hyperbola]]s). In [[three-dimensional space]], quadrics include [[ellipsoid]]s, [[paraboloid]]s, and [[hyperboloid]]s. 

More generally, a quadric [[hypersurface]] (of dimension ''D'') embedded in a [[higher dimensional]] space (of dimension {{nowrap|''D'' + 1}}) is defined as the [[zero set]] of an [[irreducible polynomial]] of [[degree of a polynomial|degree]] two in {{nowrap|''D'' + 1}} variables; for example, ''D''{{=}}1 is the case of conic sections ([[plane curve]]s). 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''.

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||Normal form of projective quadrics}}, below.

==Formulation==
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]].

== Euclidean plane ==
{{main|Conic section}}

As the dimension of a [[Euclidean plane]] is two, quadrics in a Euclidean plane have dimension one and are thus [[plane curve]]s. They are called ''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 two, and are known as '''quadric surfaces'''. Their [[quadratic equation]]s have the form
:<math>A x^2 + B y^2 + C z^2 + D xy + E yz + F xz + G x + H y + I z + J = 0,</math>
where <math>A, B, \ldots, J</math> are real numbers, and at least one of {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} is nonzero.

The quadric surfaces are classified and named by their shape, which corresponds to the [[orbit (group theory)|orbits]] under [[affine transformation]]s. That is, 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 change of [[Cartesian coordinates]] or, equivalently, a [[Euclidean transformation]] allows putting the equation of the quadric into a unique simple form on which the class of the quadric is immediately visible. This form is called the [[Canonical form|normal form]] of the equation, since two quadrics have the same normal form [[if and only if]] there is a Euclidean transformation that maps one quadric to the other. The normal forms are as follows:
:<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> 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;[[Hyperboloid of one sheet]] <br />&nbsp; &nbsp;&nbsp; &nbsp;or<br />&nbsp; &nbsp;Hyperbolic hyperboloid
| <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;[[Hyperboloid of two sheets]] <br />&nbsp; &nbsp;&nbsp; &nbsp;or<br />&nbsp; &nbsp;Elliptic hyperboloid
| <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 cone]]<br />&nbsp; &nbsp;&nbsp; &nbsp;or<br />&nbsp; &nbsp;Conical quadric
| <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]]
| <math>{x^2 \over a^2} + {y^2 \over b^2} = 1 \,</math>
|[[Image:Elliptic Cylinder Quadric.png|150px]]
|-
| &nbsp; &nbsp; [[Hyperbolic cylinder]]
| <math>{x^2 \over a^2} - {y^2 \over b^2} = 1 \,</math>
|[[Image:Hyperbolic Cylinder Quadric.png|150px]]
|-
| &nbsp; &nbsp;  [[Parabolic 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 obtains 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; [[Hyperboloid of revolution]] of one sheet (special case of 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; [[Hyperboloid of revolution]] of two sheets (special case of 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 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]] (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==
An ''affine 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 usual 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.

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\left(\frac {X_1}{X_0}, \ldots,\frac {X_n}{X_0}\right)</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}}.

So, a ''projective quadric'' is the set of zeros in a projective space of a [[homogeneous polynomial]] of degree two.

As the above process of homogenization can be reverted by setting {{math|1=''X''{{sub|0}} = 1}}:
:<math>p(x_1, \ldots, x_n)=P(1, x_1, \ldots, x_n)\,,</math>
it is often useful to not distinguish an affine quadric from its projective completion, and to talk of the ''affine equation'' or the ''projective equation'' of a quadric.  However, this is not a perfect equivalence; it is generally the case that <math>P(\mathbf{X}) = 0</math> will include points with <math>X_0 = 0</math>, which are not also solutions of <math>p(\mathbf{x}) = 0</math> because these points in projective space correspond to points "at infinity" in affine space.

===Equation===
A quadric in an [[affine space]] of dimension {{mvar|n}} is the set of zeros of a polynomial of degree 2. That is, it 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 \sum_{j=1}^n a_{i,j}x_i x_j + \sum_{i=1}^n (a_{i,0}+a_{0,i})x_i +  a_{0,0}\,,</math>
for a matrix <math>A = (a_{i,j})</math> with <math>i</math> and  <math>j</math> running from 0 to <math>n</math>.  When the [[characteristic (algebra)|characteristic]] of the [[field (mathematics)|field]] of the coefficients is not two, generally <math>a_{i,j} = a_{j,i}</math> is assumed; equivalently <math>A = A^{\mathsf T}</math>.  When the characteristic of the field of the coefficients is two, generally <math>a_{i,j} = 0</math> is assumed when <math>j < i</math>; equivalently <math>A</math> is [[upper triangular]].

The equation may be shortened, as the matrix equation
:<math>\mathbf x^{\mathsf T}A\mathbf x=0\,,</math>
with
:<math>\mathbf x = \begin {pmatrix}1&x_1&\cdots&x_n\end{pmatrix}^{\mathsf T}\,.</math>
The equation of the projective completion is almost identical:
:<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}}.

A quadric is said to be '''non-degenerate''' if the matrix <math>A</math> is [[invertible matrix|invertible]].

A non-degenerate quadric is non-singular in the sense that its projective completion has no [[singular point of an algebraic variety|singular point]] (a cylinder is non-singular in the affine space, but it is a degenerate quadric that has a singular point at infinity).

The singular points of a degenerate quadric are the points whose projective coordinates belong to the [[null space]] of the matrix {{mvar|A}}.

A quadric is reducible if and only if the [[rank (linear algebra)|rank]] of {{mvar|A}}  is one (case of a double hyperplane) or two (case of two hyperplanes).

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

:<math>P(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 two-dimensional surfaces (dimension ''D''&nbsp;=&nbsp;2) in three-dimensional space, there are exactly three non-degenerate cases:

:<math>P(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=https://slazebni.cs.illinois.edu/publications/iccv03b.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.

==Rational parametrization==
Given a non-singular point {{mvar|A}} of a quadric, a line passing through {{mvar|A}} is either tangent to the quadric, or intersects the quadric in exactly one other point (as usual, a line contained in the quadric is considered as a tangent, since it is contained in the [[tangent space|tangent hyperplane]]). This means that the lines passing through {{mvar|A}} and not tangent to the quadric are in [[one to one correspondence]] with the points of the quadric that do not belong to the tangent hyperplane at {{mvar|A}}. Expressing the points of the quadric in terms of the direction of the corresponding line provides [[parametric equation]]s of the following forms.

In the case of conic sections (quadric curves), this parametrization establishes a [[bijection]] between a projective conic section and a [[projective line]]; this bijection is an [[isomorphism]] of [[algebraic curve]]s. In higher dimensions, the parametrization defines a [[birational map]], which is a bijection between [[dense set|dense]] [[open set|open]] subsets of the quadric and a projective space of the same dimension (the topology that is considered is the usual one in the case of a real or complex quadric, or the [[Zariski topology]] in all cases). The points of the quadric that are not in the image of this bijection are the points of intersection of the quadric and its tangent hyperplane at {{mvar|A}}.

In the affine case, the parametrization is a [[rational parametrization]] of the form
:<math>x_i=\frac{f_i(t_1,\ldots, t_{n-1})}{f_0(t_1,\ldots, t_{n-1})}\quad\text{for }i=1, \ldots, n,</math>
where <math>x_1, \ldots, x_n</math> are the coordinates of a point of the quadric, <math>t_1,\ldots,t_{n-1}</math> are parameters, and <math>f_0, f_1, \ldots, f_n</math> are polynomials of degree at most two.

In the projective case, the parametrization has the form 
:<math>X_i=F_i(T_1,\ldots, T_n)\quad\text{for }i=0, \ldots, n,</math>
where <math>X_0, \ldots, X_n</math> are the projective coordinates of a point of the quadric, <math>T_1,\ldots,T_n</math> are parameters, and <math>F_0, \ldots, F_n</math> are homogeneous polynomials of degree two.

One passes from one parametrization to the other by putting <math>x_i=X_i/X_0,</math> and <math>t_i=T_i/T_n\,:</math> 
:<math>F_i(T_1,\ldots, T_n)=T_n^2 \,f_i\!{\left(\frac{T_1}{T_n},\ldots,\frac{T_{n-1}}{T_n}\right)}.</math>

For computing the parametrization and proving that the degrees are as asserted, one may proceed as follows in the affine case. One can proceed similarly in the projective case.

Let {{mvar|q}} be the quadratic polynomial that defines the quadric, and <math>\mathbf a=(a_1,\ldots a_n)</math> be the [[coordinate vector]] of the given point of the quadric (so, <math>q(\mathbf a)=0).</math> Let <math>\mathbf x=(x_1,\ldots x_n)</math> be the coordinate vector of the point of the quadric to be parametrized, and <math>\mathbf t=(t_1,\ldots, t_{n-1},1)</math> be a vector defining the direction used for the parametrization (directions whose last coordinate is zero are not taken into account here; this means that some points of the affine quadric are not parametrized; one says often that they are parametrized by [[points at infinity]] in the space of parameters) . The points of the intersection of the quadric and the line of direction <math>\mathbf t</math> passing through <math>\mathbf a</math> are the points <math>\mathbf x=\mathbf a +\lambda \mathbf t</math> such that
:<math>q(\mathbf a +\lambda \mathbf t)=0</math>
for some value of the scalar <math>\lambda.</math> This is an equation of degree two in <math>\lambda,</math> except for the values of <math>\mathbf t</math> such that the line is tangent to the quadric (in this case, the degree is one if the line is not included in the quadric, or the equation becomes <math>0=0</math> otherwise). The coefficients of <math>\lambda</math> and <math>\lambda^2</math> are respectively of degree at most one and two in <math>\mathbf t.</math> As the constant coefficient is <math>q(\mathbf a)=0,</math> the equation becomes linear by dividing by <math>\lambda,</math> and its unique solution is the quotient of a polynomial of degree at most one by a polynomial of degree at most two. Substituting this solution into the expression of <math>\mathbf x,</math> one obtains the desired parametrization as fractions of polynomials of degree at most two.

===Example: circle and spheres===

Let consider the quadric of equation
:<math>x_1^2+ x_2^2+\cdots x_n^2 -1=0.</math>
For <math>n=2,</math> this is the [[unit circle]]; for <math>n=3</math> this is the [[unit sphere]]; in higher dimensions, this is the [[unit hypersphere]].

The point <math>\mathbf a=(0, \ldots, 0, -1)</math> belongs to the quadric (the choice of this point among other similar points is only a question of convenience). So, the equation <math>q(\mathbf a +\lambda \mathbf t)=0</math> of the preceding section becomes
:<math>(\lambda t_1^2)+\cdots +(\lambda t_{n-1})^2+ (1-\lambda)^2-1=0.</math>
By expanding the squares, simplifying the constant terms, dividing by <math>\lambda,</math> and solving in <math>\lambda,</math> one obtains
:<math>\lambda = \frac{2}{1+ t_1^2+ \cdots +t_{n-1}^2}.</math>
Substituting this into <math>\mathbf x=\mathbf a +\lambda \mathbf t</math> and simplifying the expression of the last coordinate, one obtains the parametric equation
:<math>\begin{cases}
x_1=\frac{2t_1}{1+ t_1^2+ \cdots +t_{n-1}^2}\\
\vdots\\
x_{n-1}=\frac{2 t_{n-1}}{1+ t_1^2+ \cdots +t_{n-1}^2}\\
x_n =\frac{1- t_1^2- \cdots -t_{n-1}^2}{1+ t_1^2+ \cdots +t_{n-1}^2}.
\end{cases}</math>

By homogenizing, one obtains the projective parametrization
:<math>\begin{cases}
X_0=T_1^2+ \cdots +T_n^2\\
X_1=2T_1 T_n\\
\vdots\\
X_{n-1}=2T_{n-1}T_n\\
X_n =T_n^2- T_1^2- \cdots -T_{n-1}^2.
\end{cases}</math>

A straightforward verification shows that this induces a bijection between the points of the quadric such that <math>X_n\neq -X_0</math> and the points such that <math>T_n\neq 0</math> in the projective space of the parameters. On the other hand, all values of <math>(T_1,\ldots, T_n)</math> such that <math>T_n=0</math> and <math>T_1^2+ \cdots +T_{n-1}^2\neq 0</math> give the point <math>A.</math>

In the case of conic sections (<math>n=2</math>), there is exactly one point with <math>T_n=0.</math> and one has a bijection between the circle and the projective line.

For <math>n>2,</math> there are many points with <math>T_n=0,</math> and thus many parameter values for the point <math>A.</math> On the other hand, the other points of the quadric for which <math>X_n=-X_0</math> (and thus <math>x_n=-1</math>) cannot be obtained for any value of the parameters. These points are the points of the intersection of the quadric and its tangent plane at <math>A.</math> In this specific case, these points have nonreal complex coordinates, but it suffices to change one sign in the equation of the quadric for producing real points that are not obtained with the resulting parametrization.

==Rational points==
A quadric is ''defined over'' a [[field (mathematics)|field]] <math>F</math> if the coefficients of its equation belong to <math>F.</math> When <math>F</math> is the field <math>\Q</math> of the [[rational number]]s, one can suppose that the coefficients are [[integer]]s by [[clearing denominators]].

A point of a quadric defined over a field <math>F</math> is said [[rational point|rational]] over <math>F</math> if its coordinates belong to <math>F.</math> A rational point over the field <math>\R</math> of the real numbers, is called a real point.

A rational point over <math>\Q</math> is called simply a ''rational point''. By clearing denominators, one can suppose and one supposes generally that the [[projective coordinates]] of a rational point (in a quadric defined over <math>\Q</math>) are integers. Also, by clearing denominators of the coefficients, one supposes generally that all the coefficients of the equation of the quadric and the polynomials occurring in the parametrization are integers.

Finding the rational points of a projective quadric amounts thus to solving a [[Diophantine equation]].

Given a rational point {{mvar|A}} over a quadric over a field {{mvar|F}}, the parametrization described in the preceding section provides rational points when the parameters are in {{mvar|F}}, and, conversely, every rational point of the quadric can be obtained from parameters in {{mvar|F}}, if the point is not in the tangent hyperplane at {{mvar|A}}.

It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if {{mvar|F}} is infinite), and these points can be algorithmically generated as soon one knows one of them.

As said above, in the case of projective quadrics defined over <math>\Q,</math> the parametrization takes the form
:<math>X_i=F_i(T_1, \ldots, T_n)\quad \text{for } i=0,\ldots,n,</math>
where the <math>F_i</math> are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that are [[setwise coprime]] integers. If <math>Q(X_0,\ldots, X_n)=0</math> is the equation of the quadric, a solution of this equation is said ''primitive'' if its components are setwise coprime integers. The primitive solutions are in one to one correspondence with the rational points of the quadric ([[up to]] a change of sign of all components of the solution). The non-primitive integer solutions are obtained by multiplying primitive solutions by arbitrary integers; so they do not deserve a specific study. However, setwise coprime parameters can produce non-primitive solutions, and one may have to divide by a [[greatest common divisor]] to arrive at the associated primitive solution.

===Pythagorean triples===
This is well illustrated by [[Pythagorean triple]]s. A Pythagorean triple is a [[triple (mathematics)|triple]] <math>(a,b,c)</math> of positive integers such that <math>a^2+b^2=c^2.</math> A Pythagorean triple is ''primitive'' if <math>a, b, c</math> are setwise coprime, or, equivalently, if any of the three pairs <math>(a,b),</math> <math>(b,c)</math> and <math>(a,c)</math> is coprime.

By choosing <math>A=(-1, 0, 1),</math> the above method provides the parametrization
:<math>\begin{cases}
a=m^2-n^2\\b=2mn\\c=m^2+n^2 
\end{cases}</math> 
for the quadric of equation <math>a^2+b^2-c^2=0.</math> (The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).

If {{mvar|m}} and {{mvar|n}} are coprime integers such that <math>m>n>0,</math>  the resulting triple is a Pythagorean triple. If one of {{mvar|m}} and {{mvar|n}} is even and the other is odd, this resulting triple is primitive; otherwise, {{mvar|m}} and {{mvar|n}} are both odd, and one obtains a primitive triple by dividing by 2.

In summary, the primitive Pythagorean triples with <math>b</math> even are obtained as
:<math>a=m^2-n^2,\quad b=2mn,\quad c= m^2+n^2,</math> 
with {{mvar|m}} and {{mvar|n}} coprime integers such that one is even and <math>m>n>0</math> (this is [[Euclid's formula]]). The primitive Pythagorean triples with <math>b</math> odd are obtained as
:<math>a=\frac{m^2-n^2}{2},\quad b=mn, \quad c= \frac{m^2+n^2}2,</math>
with {{mvar|m}} and {{mvar|n}} coprime odd integers such that <math>m>n>0.</math>

As the exchange of {{mvar|a}} and {{mvar|b}} transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples [[up to]] the order of {{mvar|a}} and {{mvar|b}}.

== Projective quadrics over fields ==

The definition of a projective quadric in a real projective space (see above) can be formally adapted by 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 by 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

: '''(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 ===
A quadratic form <math>q</math>  on a vector space <math>V_{n+1}</math> defines a ''quadric'' <math>\mathcal Q</math> in the associated projective space <math>\mathcal P,</math> as the set of the points <math>\langle\vec x\rangle \in {\mathcal P}</math> such that <math>q(\vec x)=0</math>. That is,
: <math>\mathcal Q=\{\langle\vec x\rangle \in {\mathcal P} \mid q(\vec x)=0\}.</math>

'''Examples in <math> P_2(K)</math>.:'''<br />
'''(E1):''' For <math>\;q(\vec x)=x_1x_2-x^2_3\;</math> one obtains a [[Conic section|conic]].<br />
'''(E2):''' For <math>\;q(\vec x)=x_1x_2\;</math> one obtains 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 all <math>\vec x </math>, one obtains <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 an arbitrary line <math>g</math> with a quadric <math> \mathcal Q</math>, the following cases may occur:
:a) <math>g\cap \mathcal Q=\emptyset\;</math> and <math>g</math> is called ''exterior line''
:b) <math> g \subset \mathcal Q\; </math> and <math>g</math> is called a ''line in the quadric''
:c) <math>|g\cap \mathcal Q|=1\; </math> and <math>g</math> is called ''tangent line''
:d) <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 obtains<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 obtains <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).

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 the case of <math>\operatorname{char}K\ne2</math>, the mapping <math>\varphi</math> produces 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 obtain 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.&nbsp;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.&nbsp;159.
* P. Dembowski: ''Finite Geometries'', Springer, 1968, {{ISBN|978-3-540-61786-0}}, p.&nbsp;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

{{Authority control}}

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

[[ru:Поверхность второго порядка]]