Quadric

Not to be confused withQuadraticorQuartic.

For the computing company, seeQuadrics (company).For quadrics in algebraic geometry, seeQuadric (algebraic geometry).

In mathematics, aquadricorquadric surface(quadric hypersurfacein higherdimensions), is ageneralizationofconic sections(ellipses,parabolas,andhyperbolas). It is ahypersurface(of dimensionD) in a(D+ 1)-dimensional space, and it is defined as thezero setof anirreducible polynomialofdegreetwo inD+ 1 variables; for example,D= 1in the case of conic sections. When the defining polynomial is notabsolutely irreducible,the zero set is generally not considered a quadric, although it is often called adegenerate quadricor areducible quadric.

In coordinatesx₁,x₂,...,x_D+1,the general quadric is thus defined by thealgebraic equation

$\sum _{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{D+1}P_{i}x_{i}+R=0$

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

$xQx^{\mathrm {T} }+Px^{\mathrm {T} }+R=0\,$

wherex= (x₁,x₂,...,x_D+1)is a rowvector,x^Tis thetransposeofx(a column vector),Qis a(D+ 1) × (D+ 1)matrixandPis a(D+ 1)-dimensional row vector andRa scalar constant. The valuesQ,PandRare often taken to be overreal numbersorcomplex numbers,but a quadric may be defined over anyfield.

A quadric is anaffine algebraic variety,or, if it is reducible, anaffine algebraic set.Quadrics may also be defined inprojective spaces;see§ Normal form of projective quadrics,below.

Euclidean plane

Main article:conic section

As the dimension of aEuclidean planeis two, quadrics in a Euclidean plane have dimension one and are thusplane curves.They are calledconic sections,orconics.

Circle (e= 0), ellipse (e= 0.5), parabola (e= 1), and hyperbola (e= 2) with fixed focusFand directrix.

Euclidean space

In three-dimensionalEuclidean space,quadrics have dimension two, and are known asquadric surfaces.Theirquadratic equationshave the form

$Ax^{2}+By^{2}+Cz^{2}+Dxy+Eyz+Fxz+Gx+Hy+Iz+J=0,$

where$A,B,\ldots,J$are real numbers, and at least one ofA,B,andCis nonzero.

The quadric surfaces are classified and named by their shape, which corresponds to theorbitsunderaffine transformations.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.

Theprincipal axis theoremshows that for any (possibly reducible) quadric, a suitable change ofCartesian coordinatesor, equivalently, aEuclidean transformationallows 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 thenormal formof the equation, since two quadrics have the same normal formif and only ifthere is a Euclidean transformation that maps one quadric to the other. The normal forms are as follows:

${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+\varepsilon _{1}{z^{2} \over c^{2}}+\varepsilon _{2}=0,$

${x^{2} \over a^{2}}-{y^{2} \over b^{2}}+\varepsilon _{3}=0$

${x^{2} \over a^{2}}+\varepsilon _{4}=0,$

$z={x^{2} \over a^{2}}+\varepsilon _{5}{y^{2} \over b^{2}},$

where the$\varepsilon _{i}$are either 1, –1 or 0, except$\varepsilon _{3}$which takes only the value 0 or 1.

Each of these 17 normal formscorresponds to a single orbit under affine transformations. In three cases there are no real points:$\varepsilon _{1}=\varepsilon _{2}=1$(imaginary ellipsoid),$\varepsilon _{1}=0,\varepsilon _{2}=1$(imaginary elliptic cylinder), and$\varepsilon _{4}=1$(pair ofcomplex conjugateparallel planes, a reducible quadric). In one case, theimaginary cone,there is a single point ($\varepsilon _{1}=1,\varepsilon _{2}=0$). If$\varepsilon _{1}=\varepsilon _{2}=0,$one has a line (in fact two complex conjugate intersecting planes). For$\varepsilon _{3}=0,$one has two intersecting planes (reducible quadric). For$\varepsilon _{4}=0,$one has a double plane. For$\varepsilon _{4}=-1,$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,paraboloidsandhyperboloids), 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.

Non-degenerate real quadric surfaces
Ellipsoid	${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1\,$
Elliptic paraboloid	${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-z=0\,$
Hyperbolic paraboloid	${x^{2} \over a^{2}}-{y^{2} \over b^{2}}-z=0\,$
Hyperboloid of one sheet or Hyperbolic hyperboloid	${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=1\,$
Hyperboloid of two sheets or Elliptic hyperboloid	${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=-1\,$

Degenerate real quadric surfaces
Elliptic cone or Conical quadric	${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-{z^{2} \over c^{2}}=0\,$
Elliptic cylinder	${x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\,$
Hyperbolic cylinder	${x^{2} \over a^{2}}-{y^{2} \over b^{2}}=1\,$
Parabolic cylinder	$x^{2}+2ay=0\,$

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

Quadrics of revolution
Oblate and prolatespheroids(special cases of ellipsoid)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}+{z^{2} \over b^{2}}=1\,$
Sphere(special case of spheroid)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}+{z^{2} \over a^{2}}=1\,$
Circular paraboloid(special case of elliptic paraboloid)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-z=0\,$
Hyperboloid of revolutionof one sheet (special case of hyperboloid of one sheet)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=1\,$
Hyperboloid of revolutionof two sheets (special case of hyperboloid of two sheets)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=-1\,$
Circular cone(special case of elliptic cone)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}-{z^{2} \over b^{2}}=0\,$
Circular cylinder(special case of elliptic cylinder)	${x^{2} \over a^{2}}+{y^{2} \over a^{2}}=1\,$

Definition and basic properties

Anaffine quadricis the set ofzerosof a polynomial of degree two. When not specified otherwise, the polynomial is supposed to haverealcoefficients, and the zeros are points in aEuclidean space.However, most properties remain true when the coefficients belong to anyfieldand the points belong in anaffine space.As usual inalgebraic geometry,it is often useful to consider points over analgebraically closed fieldcontaining the polynomial coefficients, generally thecomplex numbers,when the coefficients are real.

Many properties becomes easier to state (and to prove) by extending the quadric to theprojective spacebyprojective completion,consisting of addingpoints at infinity.Technically, if

$p(x_{1},\ldots,x_{n})$

is a polynomial of degree two that defines an affine quadric, then its projective completion is defined byhomogenizingpinto

$P(X_{0},\ldots,X_{n})=X_{0}^{2}\,p\left({\frac {X_{1}}{X_{0}}},\ldots,{\frac {X_{n}}{X_{0}}}\right)$

(this is a polynomial, because the degree ofpis two). The points of the projective completion are the points of the projective space whoseprojective coordinatesare zeros ofP.

So, aprojective quadricis the set of zeros in a projective space of ahomogeneous polynomialof degree two.

As the above process of homogenization can be reverted by settingX₀= 1:

$p(x_{1},\ldots,x_{n})=P(1,x_{1},\ldots,x_{n})\,,$

it is often useful to not distinguish an affine quadric from its projective completion, and to talk of theaffine equationor theprojective equationof a quadric. However, this is not a perfect equivalence; it is generally the case that$P(\mathbf {X} )=0$will include points with$X_{0}=0$,which are not also solutions of$p(\mathbf {x} )=0$because these points in projective space correspond to points "at infinity" in affine space.

Equation

A quadric in anaffine spaceof dimensionnis the set of zeros of a polynomial of degree 2. That is, it is the set of the points whose coordinates satisfy an equation

$p(x_{1},\ldots,x_{n})=0,$

where the polynomialphas the form

$p(x_{1},\ldots,x_{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}x_{i}x_{j}+\sum _{i=1}^{n}(a_{i,0}+a_{0,i})x_{i}+a_{0,0}\,,$

for a matrix$A=(a_{i,j})$with$i$and$j$running from 0 to$n$.When thecharacteristicof thefieldof the coefficients is not two, generally$a_{i,j}=a_{j,i}$is assumed; equivalently$A=A^{\mathsf {T}}$.When the characteristic of the field of the coefficients is two, generally$a_{i,j}=0$is assumed when$j<i$;equivalently$A$isupper triangular.

The equation may be shortened, as the matrix equation

$\mathbf {x} ^{\mathsf {T}}A\mathbf {x} =0\,,$

with

$\mathbf {x} ={\begin{pmatrix}1&x_{1}&\cdots &x_{n}\end{pmatrix}}^{\mathsf {T}}\,.$

The equation of the projective completion is almost identical:

$\mathbf {X} ^{\mathsf {T}}A\mathbf {X} =0,$

with

$\mathbf {X} ={\begin{pmatrix}X_{0}&X_{1}&\cdots &X_{n}\end{pmatrix}}^{\mathsf {T}}.$

These equations define a quadric as analgebraic hypersurfaceofdimensionn– 1and degree two in a space of dimensionn.

A quadric is said to benon-degenerateif the matrix$A$isinvertible.

A non-degenerate quadric is non-singular in the sense that its projective completion has nosingular 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 thenull spaceof the matrixA.

A quadric is reducible if and only if therankofAis one (case of a double hyperplane) or two (case of two hyperplanes).

Normal form of projective quadrics

Inreal projective space,bySylvester's law of inertia,a non-singularquadratic formP(X) may be put into the normal form

$P(X)=\pm X_{0}^{2}\pm X_{1}^{2}\pm \cdots \pm X_{D+1}^{2}$

by means of a suitableprojective transformation(normal forms for singular quadrics can have zeros as well as ±1 as coefficients). For two-dimensional surfaces (dimensionD= 2) in three-dimensional space, there are exactly three non-degenerate cases:

$P(X)={\begin{cases}X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}\\X_{0}^{2}+X_{1}^{2}+X_{2}^{2}-X_{3}^{2}\\X_{0}^{2}+X_{1}^{2}-X_{2}^{2}-X_{3}^{2}\end{cases}}$

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 positiveGaussian 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 doublyruled surfacesof negative Gaussian curvature.

The degenerate form

$X_{0}^{2}-X_{1}^{2}-X_{2}^{2}=0.\,$

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.

Incomplex projective spaceall of the nondegenerate quadrics become indistinguishable from each other.

Rational parametrization

Given a non-singular pointAof a quadric, a line passing throughAis 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 thetangent hyperplane). This means that the lines passing throughAand not tangent to the quadric are inone to one correspondencewith the points of the quadric that do not belong to the tangent hyperplane atA.Expressing the points of the quadric in terms of the direction of the corresponding line providesparametric equationsof the following forms.

In the case of conic sections (quadric curves), this parametrization establishes abijectionbetween a projective conic section and aprojective line;this bijection is anisomorphismofalgebraic curves.In higher dimensions, the parametrization defines abirational map,which is a bijection betweendenseopensubsets 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 theZariski topologyin 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 atA.

In the affine case, the parametrization is arational parametrizationof the form

$x_{i}={\frac {f_{i}(t_{1},\ldots,t_{n-1})}{f_{0}(t_{1},\ldots,t_{n-1})}}\quad {\text{for }}i=1,\ldots,n,$

where$x_{1},\ldots,x_{n}$are the coordinates of a point of the quadric,$t_{1},\ldots,t_{n-1}$are parameters, and$f_{0},f_{1},\ldots,f_{n}$are polynomials of degree at most two.

In the projective case, the parametrization has the form

$X_{i}=F_{i}(T_{1},\ldots,T_{n})\quad {\text{for }}i=0,\ldots,n,$

where$X_{0},\ldots,X_{n}$are the projective coordinates of a point of the quadric,$T_{1},\ldots,T_{n}$are parameters, and$F_{0},\ldots,F_{n}$are homogeneous polynomials of degree two.

One passes from one parametrization to the other by putting$x_{i}=X_{i}/X_{0},$and$t_{i}=T_{i}/T_{n}\,:$

$F_{i}(T_{1},\ldots,T_{n})=T_{n}^{2}\,f_{i}\!{\left({\frac {T_{1}}{T_{n}}},\ldots,{\frac {T_{n-1}}{T_{n}}}\right)}.$

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.

Letqbe the quadratic polynomial that defines the quadric, and$\mathbf {a} =(a_{1},\ldots a_{n})$be thecoordinate vectorof the given point of the quadric (so,$q(\mathbf {a} )=0).$Let$\mathbf {x} =(x_{1},\ldots x_{n})$be the coordinate vector of the point of the quadric to be parametrized, and$\mathbf {t} =(t_{1},\ldots,t_{n-1},1)$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 bypoints at infinityin the space of parameters). The points of the intersection of the quadric and the line of direction$\mathbf {t}$passing through$\mathbf {a}$are the points$\mathbf {x} =\mathbf {a} +\lambda \mathbf {t}$such that

$q(\mathbf {a} +\lambda \mathbf {t} )=0$

for some value of the scalar$\lambda.$This is an equation of degree two in$\lambda,$except for the values of$\mathbf {t}$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$0=0$otherwise). The coefficients of$\lambda$and$\lambda ^{2}$are respectively of degree at most one and two in$\mathbf {t}.$As the constant coefficient is$q(\mathbf {a} )=0,$the equation becomes linear by dividing by$\lambda,$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$\mathbf {x},$one obtains the desired parametrization as fractions of polynomials of degree at most two.

Example: circle and spheres

Let consider the quadric of equation

$x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}-1=0.$

For$n=2,$this is theunit circle;for$n=3$this is theunit sphere;in higher dimensions, this is theunit hypersphere.

The point$\mathbf {a} =(0,\ldots,0,-1)$belongs to the quadric (the choice of this point among other similar points is only a question of convenience). So, the equation$q(\mathbf {a} +\lambda \mathbf {t} )=0$of the preceding section becomes

$(\lambda t_{1}^{2})+\cdots +(\lambda t_{n-1})^{2}+(1-\lambda )^{2}-1=0.$

By expanding the squares, simplifying the constant terms, dividing by$\lambda,$and solving in$\lambda,$one obtains

$\lambda ={\frac {2}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}.$

Substituting this into$\mathbf {x} =\mathbf {a} +\lambda \mathbf {t}$and simplifying the expression of the last coordinate, one obtains the parametric equation

${\begin{cases}x_{1}={\frac {2t_{1}}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}\\\vdots \\x_{n-1}={\frac {2t_{n-1}}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}\\x_{n}={\frac {1-t_{1}^{2}-\cdots -t_{n-1}^{2}}{1+t_{1}^{2}+\cdots +t_{n-1}^{2}}}.\end{cases}}$

By homogenizing, one obtains the projective parametrization

${\begin{cases}X_{0}=T_{1}^{2}+\cdots +T_{n}^{2}\\X_{1}=2T_{1}T_{n}\\\vdots \\X_{n-1}=2T_{n-1}T_{n}\\X_{n}=T_{n}^{2}-T_{1}^{2}-\cdots -T_{n-1}^{2}.\end{cases}}$

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

In the case of conic sections ($n=2$), there is exactly one point with$T_{n}=0.$and one has a bijection between the circle and the projective line.

For$n>2,$there are many points with$T_{n}=0,$and thus many parameter values for the point$A.$On the other hand, the other points of the quadric for which$X_{n}=-X_{0}$(and thus$x_{n}=-1$) 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$A.$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 isdefined overafield$F$if the coefficients of its equation belong to$F.$When$F$is the field$\mathbb {Q}$of therational numbers,one can suppose that the coefficients areintegersbyclearing denominators.

A point of a quadric defined over a field$F$is saidrationalover$F$if its coordinates belong to$F.$A rational point over the field$\mathbb {R}$of the real numbers, is called a real point.

A rational point over$\mathbb {Q}$is called simply arational point.By clearing denominators, one can suppose and one supposes generally that theprojective coordinatesof a rational point (in a quadric defined over$\mathbb {Q}$) 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 solve aDiophantine equation.

Given a rational pointAover a quadric over a fieldF,the parametrization described in the preceding section provides rational points when the parameters are inF,and, conversely, every rational point of the quadric can be obtained from parameters inF,if the point is not in the tangent hyperplane atA.

It follows that, if a quadric has a rational point, it has many other rational points (infinitely many ifFis 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$\mathbb {Q},$the parametrization takes the form

$X_{i}=F_{i}(T_{1},\ldots,T_{n})\quad {\text{for }}i=0,\ldots,n,$

where the$F_{i}$are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that aresetwise coprimeintegers. If$Q(X_{0},\ldots,X_{n})=0$is the equation of the quadric, a solution of this equation is saidprimitiveif its components are setwise coprime integers. The primitive solutions are in one to one correspondence with the rational points of the quadric (up toa 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 agreatest common divisorto arrive at the associated primitive solution.

Pythagorean triples

This is well illustrated byPythagorean triples.A Pythagorean triple is atriple$(a,b,c)$of positive integers such that$a^{2}+b^{2}=c^{2}.$A Pythagorean triple isprimitiveif$a,b,c$are setwise coprime, or, equivalently, if any of the three pairs$(a,b),$$(b,c)$and$(a,c)$is coprime.

By choosing$A=(-1,0,1),$the above method provides the parametrization

${\begin{cases}a=m^{2}-n^{2}\\b=2mn\\c=m^{2}+n^{2}\end{cases}}$

for the quadric of equation$a^{2}+b^{2}-c^{2}=0.$(The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).

Ifmandnare coprime integers such that$m>n>0,$the resulting triple is a Pythagorean triple. If one ofmandnis even and the other is odd, this resulting triple is primitive; otherwise,mandnare both odd, and one obtains a primitive triple by dividing by 2.

In summary, the primitive Pythagorean triples with$b$even are obtained as

$a=m^{2}-n^{2},\quad b=2mn,\quad c=m^{2}+n^{2},$

withmandncoprime integers such that one is even and$m>n>0$(this isEuclid's formula). The primitive Pythagorean triples with$b$odd are obtained as

$a={\frac {m^{2}-n^{2}}{2}},\quad b=mn,\quad c={\frac {m^{2}+n^{2}}{2}},$

withmandncoprime odd integers such that$m>n>0.$

As the exchange ofaandbtransforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for producing all primitive Pythagorean triples.

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 ann-dimensional projective space over afield.In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space.

Quadratic form

Let$K$be afieldand$V$avector spaceover$K$.A mapping$q$from$V$to$K$such that

(Q1)$\;q(\lambda {\vec {x}})=\lambda ^{2}q({\vec {x}})\;$for any$\lambda \in K$and${\vec {x}}\in V$.

(Q2)$\;f({\vec {x}},{\vec {y}}):=q({\vec {x}}+{\vec {y}})-q({\vec {x}})-q({\vec {y}})\;$is abilinear form.

is calledquadratic form.The bilinear form$f$is symmetric.

In case of$\operatorname {char} K\neq 2$the bilinear form is$f({\vec {x}},{\vec {x}})=2q({\vec {x}})$,i.e.$f$and$q$are mutually determined in a unique way.
In case of$\operatorname {char} K=2$(that means:$1+1=0$) the bilinear form has the property$f({\vec {x}},{\vec {x}})=0$,i.e.$f$is symplectic.

For$V=K^{n}\$and$\ {\vec {x}}=\sum _{i=1}^{n}x_{i}{\vec {e}}_{i}\quad$ ($\{{\vec {e}}_{1},\ldots,{\vec {e}}_{n}\}$is a base of$V$)$\ q$has the familiar form

$q({\vec {x}})=\sum _{1=i\leq k}^{n}a_{ik}x_{i}x_{k}\ {\text{ with }}\ a_{ik}:=f({\vec {e}}_{i},{\vec {e}}_{k})\ {\text{ for }}\ i\neq k\ {\text{ and }}\ a_{ii}:=q({\vec {e}}_{i})\$and

$f({\vec {x}},{\vec {y}})=\sum _{1=i\leq k}^{n}a_{ik}(x_{i}y_{k}+x_{k}y_{i})$.

For example:

$n=3,\quad q({\vec {x}})=x_{1}x_{2}-x_{3}^{2},\quad f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}\;.$

n-dimensional projective space over a field

Let$K$be a field,$2\leq n\in \mathbb {N}$,

$V_{n+1}$an(n+ 1)-dimensionalvector spaceover the field$K,$

$\langle {\vec {x}}\rangle$the 1-dimensionalsubspace generated by${\vec {0}}\neq {\vec {x}}\in V_{n+1}$,

${\mathcal {P}}=\{\langle {\vec {x}}\rangle \mid {\vec {x}}\in V_{n+1}\},\$theset of points,

${\mathcal {G}}=\{{\text{2-dimensional subspaces of }}V_{n+1}\},\$theset of lines.

$P_{n}(K)=({\mathcal {P}},{\mathcal {G}})\$is then-dimensionalprojective spaceover$K$.

The set of points contained in a$(k+1)$-dimensional subspace of$V_{n+1}$is a$k$-dimensional subspaceof$P_{n}(K)$.A 2-dimensional subspace is aplane.

In case of$\;n>3\;$a$(n-1)$-dimensional subspace is calledhyperplane.

Projective quadric

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

${\mathcal {Q}}=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid q({\vec {x}})=0\}.$

Examples in$P_{2}(K)$.:
(E1):For$\;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;$one obtains aconic.
(E2):For$\;q({\vec {x}})=x_{1}x_{2}\;$one obtains the pair of lines with the equations$x_{1}=0$and$x_{2}=0$,respectively. They intersect at point$\langle (0,0,1)^{\text{T}}\rangle$;

For the considerations below it is assumed that${\mathcal {Q}}\neq \emptyset$.

Polar space

For point$P=\langle {\vec {p}}\rangle \in {\mathcal {P}}$the set

$P^{\perp }:=\{\langle {\vec {x}}\rangle \in {\mathcal {P}}\mid f({\vec {p}},{\vec {x}})=0\}$

is calledpolar spaceof$P$(with respect to$q$).

If$\;f({\vec {p}},{\vec {x}})=0\;$for all${\vec {x}}$,one obtains$P^{\perp }={\mathcal {P}}$.

If$\;f({\vec {p}},{\vec {x}})\neq 0\;$for at least one${\vec {x}}$,the equation$\;f({\vec {p}},{\vec {x}})=0\;$is a non trivial linear equation which defines a hyperplane. Hence

$P^{\perp }$is either ahyperplaneor${\mathcal {P}}$.

Intersection with a line

For the intersection of an arbitrary line$g$with a quadric${\mathcal {Q}}$,the following cases may occur:

a)$g\cap {\mathcal {Q}}=\emptyset \;$and$g$is calledexterior line

b)$g\subset {\mathcal {Q}}\;$and$g$is called aline in the quadric

c)$|g\cap {\mathcal {Q}}|=1\;$and$g$is calledtangent line

d)$|g\cap {\mathcal {Q}}|=2\;$and$g$is calledsecant line.

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

Additionally the proof shows:

A line$g$through a point$P\in {\mathcal {Q}}$is atangentline if and only if$g\subset P^{\perp }$.

f-radical,q-radical

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

a)${\mathcal {R}}:=\{P\in {\mathcal {P}}\mid P^{\perp }={\mathcal {P}}\}$is a projective subspace.${\mathcal {R}}$is calledf-radicalof quadric${\mathcal {Q}}$.

b)${\mathcal {S}}:={\mathcal {R}}\cap {\mathcal {Q}}$is calledsingular radicalor$q$-radicalof${\mathcal {Q}}$.

c) In case of$\operatorname {char} K\neq 2$one has${\mathcal {R}}={\mathcal {S}}$.

A quadric is callednon-degenerateif${\mathcal {S}}=\emptyset$.

Examples in$P_{2}(K)$(see above):
(E1):For$\;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;$(conic) the bilinear form is $f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}\;.$
In case of$\operatorname {char} K\neq 2$the polar spaces are never${\mathcal {P}}$.Hence${\mathcal {R}}={\mathcal {S}}=\emptyset$.
In case of$\operatorname {char} K=2$the bilinear form is reduced to $f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}\;$and${\mathcal {R}}=\langle (0,0,1)^{\text{T}}\rangle \notin {\mathcal {Q}}$.Hence${\mathcal {R}}\neq {\mathcal {S}}=\emptyset \;.$ In this case thef-radical is the common point of all tangents, the so calledknot.
In both cases$S=\emptyset$and the quadric (conic) istnon-degenerate.
(E2):For$\;q({\vec {x}})=x_{1}x_{2}\;$(pair of lines) the bilinear form is$f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}\;$and${\mathcal {R}}=\langle (0,0,1)^{\text{T}}\rangle ={\mathcal {S}}\;,$the intersection point.
In this example the quadric isdegenerate.

Symmetries

A quadric is a rather homogeneous object:

For any point$P\notin {\mathcal {Q}}\cup {\mathcal {R}}\;$there exists aninvolutorialcentralcollineation$\sigma _{P}$with center$P$and$\sigma _{P}({\mathcal {Q}})={\mathcal {Q}}$.

Proof: Due to$P\notin {\mathcal {Q}}\cup {\mathcal {R}}$the polar space$P^{\perp }$is a hyperplane.

The linear mapping

$\varphi:{\vec {x}}\rightarrow {\vec {x}}-{\frac {f({\vec {p}},{\vec {x}})}{q({\vec {p}})}}{\vec {p}}$

induces aninvolutorial central collineation$\sigma _{P}$with axis$P^{\perp }$and centre$P$which leaves${\mathcal {Q}}$invariant.
In the case of$\operatorname {char} K\neq 2$,the mapping$\varphi$produces thefamiliar shape$\;\varphi:{\vec {x}}\rightarrow {\vec {x}}-2{\frac {f({\vec {p}},{\vec {x}})}{f({\vec {p}},{\vec {p}})}}{\vec {p}}\;$with$\;\varphi ({\vec {p}})=-{\vec {p}}$and$\;\varphi ({\vec {x}})={\vec {x}}\;$for any$\langle {\vec {x}}\rangle \in P^{\perp }$.

Remark:

a) An exterior line, a tangent line or a secant line is mapped by the involution$\sigma _{P}$on an exterior, tangent and secant line, respectively.

b)${\mathcal {R}}$is pointwise fixed by$\sigma _{P}$.

q-subspaces and index of a quadric

A subspace$\;{\mathcal {U}}\;$of$P_{n}(K)$is called$q$-subspace if$\;{\mathcal {U}}\subset {\mathcal {Q}}\;$

For example: points on a sphere orlines on a hyperboloid(s. below).

Any twomaximal$q$-subspaces have the same dimension$m$.

Let be$m$the dimension of the maximal$q$-subspaces of${\mathcal {Q}}$then

The integer$\;i:=m+1\;$is calledindexof${\mathcal {Q}}$.

Theorem: (BUEKENHOUT)

For the index$i$of a non-degenerate quadric${\mathcal {Q}}$in$P_{n}(K)$the following is true:

$i\leq {\frac {n+1}{2}}$.

Let be${\mathcal {Q}}$a non-degenerate quadric in$P_{n}(K),n\geq 2$,and$i$its index.

In case of$i=1$quadric${\mathcal {Q}}$is calledsphere(orovalconic if$n=2$).

In case of$i=2$quadric${\mathcal {Q}}$is calledhyperboloid(of one sheet).

Examples:

a) Quadric${\mathcal {Q}}$in$P_{2}(K)$with form$\;q({\vec {x}})=x_{1}x_{2}-x_{3}^{2}\;$is non-degenerate with index 1.

b) If polynomial$\;p(\xi )=\xi ^{2}+a_{0}\xi +b_{0}\;$isirreducibleover$K$the quadratic form$\;q({\vec {x}})=x_{1}^{2}+a_{0}x_{1}x_{2}+b_{0}x_{2}^{2}-x_{3}x_{4}\;$gives rise to a non-degenerate quadric${\mathcal {Q}}$in$P_{3}(K)$of index 1 (sphere). For example:$\;p(\xi )=\xi ^{2}+1\;$is irreducible over$\mathbb {R}$(but not over$\mathbb {C}$!).

c) In$P_{3}(K)$the quadratic form$\;q({\vec {x}})=x_{1}x_{2}+x_{3}x_{4}\;$generates ahyperboloid.

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 fromusualquadrics.The reason is the following statement.

Adivision ring$K$iscommutativeif and only if anyequation$x^{2}+ax+b=0,\ a,b\in K$,has at most two solutions.

There aregeneralizationsof quadrics:quadratic sets.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

Bibliography

• M. Audin:Geometry,Springer, Berlin, 2002,ISBN978-3-540-43498-6,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,ISBN3-528-07241-5,p. 159.
• P. Dembowski:Finite Geometries,Springer, 1968,ISBN978-3-540-61786-0,p. 43.
• Iskovskikh, V.A. (2001) [1994],"Quadric",Encyclopedia of Mathematics,EMS Press
• Weisstein, Eric W."Quadric".MathWorld.

External links

• Interactive Java 3D models of all quadric surfaces
• Lecture NotePlanar Circle Geometries,an Introduction to Moebius, Laguerre and Minkowski Planes,p. 117