In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (

Quadrics

in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'',^{T} is the

^{''D''+1} are
:$(x\_1,\backslash dots,x\_)$
one introduces new coordinates on R^{''D''+2}
:$[X\_0,\backslash dots,X\_]$
related to the original coordinates by $x\_i=X\_i/X\_0$. In the new variables, every quadric is defined by an equation of the form
:$Q(X)=\backslash sum\_\; a\_X\_iX\_j=0\backslash ,$
where the coefficients ''a''_{''ij''} are symmetric in ''i'' and ''j''. Regarding ''Q''(''X'') = 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 (''a''_{''ij''}) 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
:$Q(X)\; =\; \backslash pm\; X\_0^2\; \backslash pm\; X\_1^2\; \backslash pm\backslash cdots\backslash pm\; X\_^2$
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'' = 2) there are exactly three nondegenerate cases:
:$Q(X)\; =\; \backslash begin\; X\_0^2+X\_1^2+X\_2^2+X\_3^2\backslash \backslash \; X\_0^2+X\_1^2+X\_2^2-X\_3^2\backslash \backslash \; X\_0^2+X\_1^2-X\_2^2-X\_3^2\; \backslash end$
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 surfaces of negative Gaussian curvature.
The degenerate form
:$X\_0^2-X\_1^2-X\_2^2=0.\; \backslash ,$
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.
In complex projective space all of the nondegenerate quadrics become indistinguishable from each other.

In case of $\backslash operatornameK=2$ (that means: $1+1=0$) the bilinear form has the property $f(\backslash vec\; x,\backslash vec\; x)=0$, i.e. $f$ is ''Symplectic vector space, symplectic''. For $V=K^n\backslash $ and $\backslash \; \backslash vec\; x=\backslash sum\_^x\_i\backslash vec\; e\_i\backslash quad$ ($\backslash $ is a base of $V$) $\backslash \; q$ has the familiar form : $q(\backslash vec\; x)=\backslash sum\_^\; a\_x\_ix\_k\backslash \; \backslash text\backslash \; a\_:=\; f(\backslash vec\; e\_i,\backslash vec\; e\_k)\backslash \; \backslash text\backslash \; i\backslash ne\; k\backslash \; \backslash text\backslash \; a\_:=\; q(\backslash vec\; e\_i)\backslash $ and : $f(\backslash vec\; x,\backslash vec\; y)=\backslash sum\_^\; a\_(x\_iy\_k+x\_ky\_i)$. For example: : $n=3,\backslash quad\; q(\backslash vec\; x)=x\_1x\_2-x^2\_3,\; \backslash quad\; f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1-2x\_3y\_3\backslash ;\; .$

(E1): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2-x^2\_3\backslash ;$ one gets a Conic section, conic.

(E2): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2\backslash ;$ one gets the pair of lines with the equations $x\_1=0$ and $x\_2=0$, respectively. They intersect at point $\backslash langle(0,0,1)^\backslash text\backslash rangle$; For the considerations below it is assumed that $\backslash mathcal\; Q\backslash ne\; \backslash emptyset$.

$q(x\backslash vec\; u+\backslash vec\; v)=q(x\backslash vec\; u)+q(\backslash vec\; v)+f(x\backslash vec\; u,\backslash vec\; v)=q(\backslash vec\; v)+xf(\backslash vec\; u,\backslash vec\; v)\backslash ;\; .$

I) In case of $g\backslash subset\; U^\backslash perp$ the equation $f(\backslash vec\; u,\backslash vec\; v)=0$ holds and it is $\backslash ;\; q(x\backslash vec\; u+\backslash vec\; v)=q(\backslash vec\; v)\backslash ;$ for any $x\backslash in\; K$. Hence either $\backslash ;q(x\backslash vec\; u+\backslash vec\; v)=0\backslash ;$ for ''any'' $x\backslash in\; K$ or $\backslash ;q(x\backslash vec\; u+\backslash vec\; v)\backslash ne\; 0\backslash ;$ for ''any'' $x\backslash in\; K$, which proves b) and b').

II) In case of $g\backslash not\backslash subset\; U^\backslash perp$ one gets $f(\backslash vec\; u,\backslash vec\; v)\backslash ne\; 0$ and the equation $\backslash ;q(x\backslash vec\; u+\backslash vec\; v)=q(\backslash vec\; v)+xf(\backslash vec\; u,\backslash vec\; v)=\; 0\backslash ;$ has exactly one solution $x$. Hence: $,\; g\backslash cap\; \backslash mathcal\; Q,\; =2$, which proves c). Additionally the proof shows: :A line $g$ through a point $P\backslash in\; \backslash mathcal\; Q$ is a ''tangent'' line if and only if $g\backslash subset\; P^\backslash perp$.

(E1): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2-x^2\_3\backslash ;$ (conic) the bilinear form is $f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1-2x\_3y\_3\backslash ;\; .$

In case of $\backslash operatornameK\backslash ne2$ the polar spaces are never $\backslash mathcal\; P$. Hence $\backslash mathcal\; R=\backslash mathcal\; S=\backslash empty$.

In case of $\backslash operatornameK=2$ the bilinear form is reduced to $f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1\backslash ;$ and $\backslash mathcal\; R=\backslash langle(0,0,1)^\backslash text\backslash rangle\backslash notin\; \backslash mathcal\; Q$. Hence $\backslash mathcal\; R\backslash ne\; \backslash mathcal\; S=\backslash empty\; \backslash ;\; .$ In this case the ''f''-radical is the common point of all tangents, the so called ''knot''.

In both cases $S=\backslash empty$ and the quadric (conic) ist ''non-degenerate''.

(E2): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2\backslash ;$ (pair of lines) the bilinear form is $f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1\backslash ;$ and $\backslash mathcal\; R=\backslash langle(0,0,1)^\backslash text\backslash rangle=\backslash mathcal\; S\backslash ;\; ,$ the intersection point.

In this example the quadric is ''degenerate''.

In case of $\backslash operatornameK\backslash ne2$ mapping $\backslash varphi$ gets the Reflection (mathematics), familiar shape $\backslash ;\; \backslash varphi:\; \backslash vec\; x\; \backslash rightarrow\; \backslash vec\; x-2\backslash frac\backslash vec\; p\backslash ;$ with $\backslash ;\; \backslash varphi(\backslash vec\; p)=-\backslash vec\; p$ and $\backslash ;\; \backslash varphi(\backslash vec\; x)=\backslash vec\; x\backslash ;$ for any $\backslash langle\backslash vec\; x\backslash rangle\; \backslash in\; P^\backslash perp$. Remark: :a) An exterior line, a tangent line or a secant line is mapped by the involution $\backslash sigma\_P$ on an exterior, tangent and secant line, respectively. :b) $$ is pointwise fixed by $\backslash sigma\_P$.

: In case of $i=1$ quadric $\backslash mathcal\; Q$ is called ''sphere'' (or oval (projective plane), oval conic if $n=2$). : In case of $i=2$ quadric $\backslash mathcal\; Q$ is called ''hyperboloid'' (of one sheet). Examples: :a) Quadric $\backslash mathcal\; Q$ in $P\_2(K)$ with form $\backslash ;q(\backslash vec\; x)=x\_1x\_2-x^2\_3\backslash ;$ is non-degenerate with index 1. :b) If polynomial $\backslash ;p(\backslash xi)=\backslash xi^2+a\_0\backslash xi+b\_0\backslash ;$ is Irreducible polynomial, irreducible over $K$ the quadratic form $\backslash ;q(\backslash vec\; x)=x^2\_1+a\_0x\_1x\_2+b\_0x^2\_2-x\_3x\_4\backslash ;$ gives rise to a non-degenerate quadric $\backslash mathcal\; Q$ in $P\_3(K)$ of index 1 (sphere). For example: $\backslash ;p(\backslash xi)=\backslash xi^2+1\backslash ;$ is irreducible over $\backslash R$ (but not over $\backslash C$ !). :c) In $P\_3(K)$ the quadratic form $\backslash ;q(\backslash vec\; x)=x\_1x\_2+x\_3x\_4\backslash ;$ generates a ''hyperboloid''.

Interactive Java 3D models of all quadric surfaces

Lecture Note ''Planar Circle Geometries'', an Introduction to Moebius, Laguerre and Minkowski Planes

p. 117 Quadrics, Projective geometry ru:Поверхность второго порядка

ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...

s, s, and s). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in ''D'' + 1 variables ( 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 , the general quadric is thus defined by the algebraic equationSilvio LevQuadrics

in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'',

CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technology ...

, from The Geometry Center at University of Minnesota
The University of Minnesota, Twin Cities (the U of M or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Twin Cities of Minneapolis and Saint Paul, Minnesota. The Tw ...

:$\backslash sum\_^\; x\_i\; Q\_\; x\_j\; +\; \backslash sum\_^\; P\_i\; x\_i\; +\; R\; =\; 0$
which may be compactly written in vector and matrix notation as:
:$x\; Q\; x^\backslash mathrm\; +\; P\; x^\backslash mathrm\; +\; R\; =\; 0\backslash ,$
where is a row vector, ''x''transpose
In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...

of ''x'' (a column vector), ''Q'' is a matrix and ''P'' is a -dimensional row vector and ''R'' a scalar constant. The values ''Q'', ''P'' and ''R'' are often taken to be over real number
Real may refer to:
* Reality, the state of things as they exist, rather than as they may appear or may be thought to be
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish col ...

s or complex numbers, 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 spaces; see , below.
Euclidean plane

As the dimension of a Euclidean plane is two, quadrics in a Euclidean plane have dimension one and are thus plane curves. They are called ''conic sections'', or ''conics''. Image:Eccentricity.svg, center, 280px, Circle (''e'' = 0), ellipse (''e'' = 0.5), parabola (''e'' = 1), and hyperbola (''e'' = 2) with fixed focus ''F'' and directrix.Euclidean space

In three-dimensional Euclidean space, quadrics have dimension ''D'' = 2, and are known as quadric surfaces. They are classified and named by their orbit (group theory), orbits under affine transformations. 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: :$+\; +\backslash varepsilon\_1\; +\; \backslash varepsilon\_2=0,$ :$-\; +\; \backslash varepsilon\_3=0$ :$+\; \backslash varepsilon\_4\; =0,$ :$z=\; +\backslash varepsilon\_5\; ,$ where the $\backslash varepsilon\_i$ are either 1, –1 or 0, except $\backslash varepsilon\_3$ which takes only the value 0 or 1. Each of these 17 normal formsStewart Venit and Wayne Bishop, ''Elementary Linear Algebra (fourth edition)'', International Thompson Publishing, 1996. corresponds to a single orbit under affine transformations. In three cases there are no real points: $\backslash varepsilon\_1=\backslash varepsilon\_2=1$ (''imaginary ellipsoid''), $\backslash varepsilon\_1=0,\; \backslash varepsilon\_2=1$ (''imaginary elliptic cylinder''), and $\backslash varepsilon\_4=1$ (pair of complex conjugate parallel planes, a reducible quadric). In one case, the ''imaginary cone'', there is a single point ($\backslash varepsilon\_1=1,\; \backslash varepsilon\_2=0$). If $\backslash varepsilon\_1=\backslash varepsilon\_2=0,$ one has a line (in fact two complex conjugate intersecting planes). For $\backslash varepsilon\_3=0,$ one has two intersecting planes (reducible quadric). For $\backslash varepsilon\_4=0,$ one has a double plane. For $\backslash varepsilon\_4=-1,$ 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, paraboloids and hyperboloids), 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. 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).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 usually in algebraic geometry, it is often useful to consider points over an algebraically closed field containing the polynomial coefficients, generally the complex numbers, 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 :$p(x\_1,\; \backslash ldots,x\_n)$ is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenization of a polynomial, homogenizing into :$P(X\_0,\; \backslash ldots,\; X\_n)=X\_0^2\backslash ,p\backslash left(\backslash frac\; ,\; \backslash ldots,\backslash frac\; \backslash right)$ (this is a polynomial, because the degree of is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of . 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 , 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.Equation

A quadric in an affine space of dimension is the set of zeros of a polynomial of degree 2, that is the set of the points whose coordinates satisfy an equation :$p(x\_1,\backslash ldots,x\_n)=0,$ where the polynomial has the form :$p(x\_1,\backslash ldots,x\_n)=\backslash sum\_^n\; a\_x\_i^2+\backslash sum\_\; (a\_+a\_)x\_ix\_j\; +\backslash sum\_^n\; (a\_+a\_)x\_i\; +a\_,$ where $a\_=a\_$ if the characteristic (algebra), characteristic of the field (mathematics), field of the coefficients is not two and $a\_=0$ otherwise. If is the matrix that has the $a\_$ as entries, and :$\backslash mathbf\; x\; =\; \backslash begin\; 1\&x\_1\&\backslash cdots\&x\_n\backslash end^,$ then the equation may be shortened in the matrix equation :$\backslash mathbf\; x^A\backslash mathbf\; x=0.$ The equation of the projective completion of this quadric is :$\backslash sum\_^n\; a\_X\_i^2+\backslash sum\_\; (a\_+a\_)X\_iX\_j,$ or :$\backslash mathbf\; X^A\backslash mathbf\; X=0,$ with :$\backslash mathbf\; X\; =\; \backslash begin\; X\_0\&X\_1\&\backslash cdots\&X\_n\backslash end^.$ These equations define a quadric as an hypersurface, algebraic hypersurface of ''dimension'' and degree two in a space of dimension .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 RProjective 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 spaceQuadratic form

Let $K$ be a Field (algebra), field and $V$ a vector space over $K$. A mapping $q$ from $V$ to $K$ such that : (Q1) $\backslash ;q(\backslash lambda\backslash vec\; x)=\backslash lambda^2q(\backslash vec\; x\; )\backslash ;$ for any $\backslash lambda\backslash in\; K$ and $\backslash vec\; x\; \backslash in\; V$. : (Q2) $\backslash ;f(\backslash vec\; x,\backslash vec\; y\; ):=q(\backslash vec\; x+\backslash vec\; y)-q(\backslash vec\; x)-q(\backslash vec\; y)\backslash ;$ is a bilinear form. is called quadratic form. The bilinear form $f$ is symmetric''.'' In case of $\backslash operatornameK\backslash ne2$ the bilinear form is $f(\backslash vec\; x,\backslash vec\; x)=2q(\backslash vec\; x)$, i.e. $f$ and $q$ are mutually determined in a unique way.In case of $\backslash operatornameK=2$ (that means: $1+1=0$) the bilinear form has the property $f(\backslash vec\; x,\backslash vec\; x)=0$, i.e. $f$ is ''Symplectic vector space, symplectic''. For $V=K^n\backslash $ and $\backslash \; \backslash vec\; x=\backslash sum\_^x\_i\backslash vec\; e\_i\backslash quad$ ($\backslash $ is a base of $V$) $\backslash \; q$ has the familiar form : $q(\backslash vec\; x)=\backslash sum\_^\; a\_x\_ix\_k\backslash \; \backslash text\backslash \; a\_:=\; f(\backslash vec\; e\_i,\backslash vec\; e\_k)\backslash \; \backslash text\backslash \; i\backslash ne\; k\backslash \; \backslash text\backslash \; a\_:=\; q(\backslash vec\; e\_i)\backslash $ and : $f(\backslash vec\; x,\backslash vec\; y)=\backslash sum\_^\; a\_(x\_iy\_k+x\_ky\_i)$. For example: : $n=3,\backslash quad\; q(\backslash vec\; x)=x\_1x\_2-x^2\_3,\; \backslash quad\; f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1-2x\_3y\_3\backslash ;\; .$

''n''-dimensional projective space over a field

Let $K$ be a field, $2\backslash le\; n\backslash in\backslash N$, :$V\_$ an -dimension (vector space), dimensional vector space over the field $K,$ :$\backslash langle\backslash vec\; x\backslash rangle$ the 1-dimensional linear span, subspace generated by $\backslash vec\; 0\backslash ne\; \backslash vec\; x\backslash in\; V\_$, : $=\backslash ,\backslash $ the ''set of points'' , : $=\backslash ,\backslash $ the ''set of lines''. :$P\_n(K)=(,)\backslash $ is the -dimensional projective space over $K$. :The set of points contained in a $(k+1)$-dimensional subspace of $V\_$ is a ''$k$-dimensional subspace'' of $P\_n(K)$. A 2-dimensional subspace is a ''plane''. :In case of $\backslash ;n>3\backslash ;$ a $(n-1)$-dimensional subspace is called ''hyperplane''.Projective quadric

For a quadratic form $q$ on a vector space $V\_$ a point $\backslash langle\backslash vec\; x\backslash rangle\; \backslash in$ is called ''singular'' if $q(\backslash vec\; x)=0$. The set : $\backslash mathcal\; Q=\backslash $ of singular points of $q$ is called quadric (with respect to the quadratic form $q$). Examples in $P\_2(K)$.:(E1): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2-x^2\_3\backslash ;$ one gets a Conic section, conic.

(E2): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2\backslash ;$ one gets the pair of lines with the equations $x\_1=0$ and $x\_2=0$, respectively. They intersect at point $\backslash langle(0,0,1)^\backslash text\backslash rangle$; For the considerations below it is assumed that $\backslash mathcal\; Q\backslash ne\; \backslash emptyset$.

Polar space

For point $P=\backslash langle\backslash vec\; p\backslash rangle\; \backslash in$ the set : $P^\backslash perp:=\backslash $ is called Duality (mathematics)#Polarities of general projective spaces, polar space of $P$ (with respect to $q$). If $\backslash ;f(\backslash vec\; p,\backslash vec\; x)=0\backslash ;$ for any $\backslash vec\; x$, one gets $P^\backslash perp=\backslash mathcal\; P$. If $\backslash ;f(\backslash vec\; p,\backslash vec\; x)\backslash ne\; 0\backslash ;$ for at least one $\backslash vec\; x$, the equation $\backslash ;f(\backslash vec\; p,\backslash vec\; x)=0\backslash ;$is a non trivial linear equation which defines a hyperplane. Hence :$P^\backslash perp$ is either a hyperplane or $$.Intersection with a line

For the intersection of a line with a quadric $\backslash mathcal\; Q$ the familiar statement is true: :For an arbitrary line $g$ the following cases occur: :a) $g\backslash cap\; \backslash mathcal\; Q=\backslash emptyset\backslash ;$ and $g$ is called ''exterior line'' or :b) $g\; \backslash subset\; \backslash mathcal\; Q\backslash ;$ and $g$ is called ''tangent line'' or :b′) $,\; g\backslash cap\; \backslash mathcal\; Q,\; =1\backslash ;$ and $g$ is called ''tangent line'' or :c) $,\; g\backslash cap\; \backslash mathcal\; Q,\; =2\backslash ;$ and $g$ is called ''secant line''. Proof: Let $g$ be a line, which intersects $\backslash mathcal\; Q$ at point $\backslash ;U=\backslash langle\backslash vec\; u\backslash rangle\backslash ;$ and $\backslash ;V=\; \backslash langle\backslash vec\; v\backslash rangle\backslash ;$ is a second point on $g$. From $\backslash ;q(\backslash vec\; u)=0\backslash ;$ one gets$q(x\backslash vec\; u+\backslash vec\; v)=q(x\backslash vec\; u)+q(\backslash vec\; v)+f(x\backslash vec\; u,\backslash vec\; v)=q(\backslash vec\; v)+xf(\backslash vec\; u,\backslash vec\; v)\backslash ;\; .$

I) In case of $g\backslash subset\; U^\backslash perp$ the equation $f(\backslash vec\; u,\backslash vec\; v)=0$ holds and it is $\backslash ;\; q(x\backslash vec\; u+\backslash vec\; v)=q(\backslash vec\; v)\backslash ;$ for any $x\backslash in\; K$. Hence either $\backslash ;q(x\backslash vec\; u+\backslash vec\; v)=0\backslash ;$ for ''any'' $x\backslash in\; K$ or $\backslash ;q(x\backslash vec\; u+\backslash vec\; v)\backslash ne\; 0\backslash ;$ for ''any'' $x\backslash in\; K$, which proves b) and b').

II) In case of $g\backslash not\backslash subset\; U^\backslash perp$ one gets $f(\backslash vec\; u,\backslash vec\; v)\backslash ne\; 0$ and the equation $\backslash ;q(x\backslash vec\; u+\backslash vec\; v)=q(\backslash vec\; v)+xf(\backslash vec\; u,\backslash vec\; v)=\; 0\backslash ;$ has exactly one solution $x$. Hence: $,\; g\backslash cap\; \backslash mathcal\; Q,\; =2$, which proves c). Additionally the proof shows: :A line $g$ through a point $P\backslash in\; \backslash mathcal\; Q$ is a ''tangent'' line if and only if $g\backslash subset\; P^\backslash perp$.

''f''-radical, ''q''-radical

In the classical cases $K=\backslash R$ or $\backslash C$ there exists only one radical, because of $\backslash operatornameK\backslash ne2$ and $f$ and $q$ are closely connected. In case of $\backslash operatornameK=2$ the quadric $\backslash mathcal\; Q$ is not determined by $f$ (see above) and so one has to deal with two radicals: :a) $\backslash mathcal\; R:=\backslash $ is a projective subspace. $\backslash mathcal\; R$ is called ''f''-radical of quadric $\backslash mathcal\; Q$. :b) $\backslash mathcal\; S:=\backslash mathcal\; R\backslash cap\backslash mathcal\; Q$ is called singular radical or ''$q$-radical'' of $\backslash mathcal\; Q$. :c) In case of $\backslash operatornameK\backslash ne2$ one has $\backslash mathcal\; R=\backslash mathcal\; S$. A quadric is called non-degenerate if $\backslash mathcal\; S=\backslash emptyset$. Examples in $P\_2(K)$ (see above):(E1): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2-x^2\_3\backslash ;$ (conic) the bilinear form is $f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1-2x\_3y\_3\backslash ;\; .$

In case of $\backslash operatornameK\backslash ne2$ the polar spaces are never $\backslash mathcal\; P$. Hence $\backslash mathcal\; R=\backslash mathcal\; S=\backslash empty$.

In case of $\backslash operatornameK=2$ the bilinear form is reduced to $f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1\backslash ;$ and $\backslash mathcal\; R=\backslash langle(0,0,1)^\backslash text\backslash rangle\backslash notin\; \backslash mathcal\; Q$. Hence $\backslash mathcal\; R\backslash ne\; \backslash mathcal\; S=\backslash empty\; \backslash ;\; .$ In this case the ''f''-radical is the common point of all tangents, the so called ''knot''.

In both cases $S=\backslash empty$ and the quadric (conic) ist ''non-degenerate''.

(E2): For $\backslash ;q(\backslash vec\; x)=x\_1x\_2\backslash ;$ (pair of lines) the bilinear form is $f(\backslash vec\; x,\backslash vec\; y)=x\_1y\_2+x\_2y\_1\backslash ;$ and $\backslash mathcal\; R=\backslash langle(0,0,1)^\backslash text\backslash rangle=\backslash mathcal\; S\backslash ;\; ,$ the intersection point.

In this example the quadric is ''degenerate''.

Symmetries

A quadric is a rather homogeneous object: :For any point $P\backslash notin\; \backslash mathcal\; Q\backslash cup\; \backslash ;$ there exists an Involution (mathematics), involutorial central collineation $\backslash sigma\_P$ with center $P$ and $\backslash sigma\_P(\backslash mathcal\; Q)=\backslash mathcal\; Q$. Proof: Due to $P\backslash notin\; \backslash mathcal\; Q\backslash cup$ the polar space $P^\backslash perp$ is a hyperplane. The linear mapping : $\backslash varphi:\; \backslash vec\; x\; \backslash rightarrow\; \backslash vec\; x-\backslash frac\backslash vec\; p$ induces an ''involutorial central collineation'' $\backslash sigma\_P$ with axis $P^\backslash perp$ and centre $P$ which leaves $\backslash mathcal\; Q$ invariant.In case of $\backslash operatornameK\backslash ne2$ mapping $\backslash varphi$ gets the Reflection (mathematics), familiar shape $\backslash ;\; \backslash varphi:\; \backslash vec\; x\; \backslash rightarrow\; \backslash vec\; x-2\backslash frac\backslash vec\; p\backslash ;$ with $\backslash ;\; \backslash varphi(\backslash vec\; p)=-\backslash vec\; p$ and $\backslash ;\; \backslash varphi(\backslash vec\; x)=\backslash vec\; x\backslash ;$ for any $\backslash langle\backslash vec\; x\backslash rangle\; \backslash in\; P^\backslash perp$. Remark: :a) An exterior line, a tangent line or a secant line is mapped by the involution $\backslash sigma\_P$ on an exterior, tangent and secant line, respectively. :b) $$ is pointwise fixed by $\backslash sigma\_P$.

''q''-subspaces and index of a quadric

A subspace $\backslash ;\backslash mathcal\; U\backslash ;$ of $P\_n(K)$ is called $q$-subspace if $\backslash ;\backslash mathcal\; U\backslash subset\backslash mathcal\; Q\backslash ;$ For example: points on a sphere or ruled surface, lines on a hyperboloid (s. below). :Any two ''maximal'' $q$-subspaces have the same dimension $m$. Let be $m$ the dimension of the maximal $q$-subspaces of $\backslash mathcal\; Q$ then :The integer $\backslash ;i:=m+1\backslash ;$ is called index of $\backslash mathcal\; Q$. Theorem: (BUEKENHOUT) :For the index $i$ of a non-degenerate quadric $\backslash mathcal\; Q$ in $P\_n(K)$ the following is true: ::$i\backslash le\; \backslash frac$. Let be $\backslash mathcal\; Q$ a non-degenerate quadric in $P\_n(K),\; n\backslash ge\; 2$, and $i$ its index.: In case of $i=1$ quadric $\backslash mathcal\; Q$ is called ''sphere'' (or oval (projective plane), oval conic if $n=2$). : In case of $i=2$ quadric $\backslash mathcal\; Q$ is called ''hyperboloid'' (of one sheet). Examples: :a) Quadric $\backslash mathcal\; Q$ in $P\_2(K)$ with form $\backslash ;q(\backslash vec\; x)=x\_1x\_2-x^2\_3\backslash ;$ is non-degenerate with index 1. :b) If polynomial $\backslash ;p(\backslash xi)=\backslash xi^2+a\_0\backslash xi+b\_0\backslash ;$ is Irreducible polynomial, irreducible over $K$ the quadratic form $\backslash ;q(\backslash vec\; x)=x^2\_1+a\_0x\_1x\_2+b\_0x^2\_2-x\_3x\_4\backslash ;$ gives rise to a non-degenerate quadric $\backslash mathcal\; Q$ in $P\_3(K)$ of index 1 (sphere). For example: $\backslash ;p(\backslash xi)=\backslash xi^2+1\backslash ;$ is irreducible over $\backslash R$ (but not over $\backslash C$ !). :c) In $P\_3(K)$ the quadratic form $\backslash ;q(\backslash vec\; x)=x\_1x\_2+x\_3x\_4\backslash ;$ 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. The reason is the following statement. :A division ring $K$ is commutative ring, commutative if and only if any quadratic equation, equation $x^2+ax+b=0,\; \backslash \; a,b\; \backslash in\; K$, has at most two solutions. There are ''generalizations'' of quadrics: quadratic sets.Beutelspacher/Rosenbaum: p. 135 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 axesReferences

Bibliography

* M. Audin: ''Geometry'', Springer, Berlin, 2002, , 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, , p. 159. * P. Dembowski: ''Finite Geometries'', Springer, 1968, , p. 43. * *{{mathworld, urlname=Quadric, title=QuadricExternal links

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