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In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (
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,
parabola
parabola
s, and
hyperbola
hyperbola
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 Lev
Quadrics
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 ...

University of Minnesota
: \sum_^ x_i Q_ x_j + \sum_^ P_i x_i + R = 0 which may be compactly written in vector and matrix notation as: : x Q x^\mathrm + P x^\mathrm + R = 0\, where is a row vector, ''x''T is the
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: : + +\varepsilon_1 + \varepsilon_2=0, : - + \varepsilon_3=0 : + \varepsilon_4 =0, :z= +\varepsilon_5 , 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 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: \varepsilon_1=\varepsilon_2=1 (''imaginary ellipsoid''), \varepsilon_1=0, \varepsilon_2=1 (''imaginary elliptic cylinder''), and \varepsilon_4=1 (pair of complex conjugate parallel planes, a reducible quadric). In one case, the ''imaginary 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, 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, \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, \ldots, X_n)=X_0^2\,p\left(\frac , \ldots,\frac \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,\ldots,x_n)=0, where the polynomial has the form :p(x_1,\ldots,x_n)=\sum_^n a_x_i^2+\sum_ (a_+a_)x_ix_j +\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 :\mathbf x = \begin 1&x_1&\cdots&x_n\end^, then the equation may be shortened in the matrix equation :\mathbf x^A\mathbf x=0. The equation of the projective completion of this quadric is :\sum_^n a_X_i^2+\sum_ (a_+a_)X_iX_j, or :\mathbf X^A\mathbf X=0, with :\mathbf X = \begin X_0&X_1&\cdots&X_n\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 R''D''+1 are :(x_1,\dots,x_) one introduces new coordinates on R''D''+2 :[X_0,\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)=\sum_ a_X_iX_j=0\, 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) = \pm X_0^2 \pm X_1^2 \pm\cdots\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) = \begin 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 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. \, 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.


Projective quadrics over fields

The definition of a projective quadric in a real projective space (see above) can be formally adopted defining a projective quadric in an n-dimensional projective space over a Field (mathematics), field. In order to omit dealing with coordinates a projective quadric is usually defined starting with a quadratic form on a vector space


Quadratic 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) \;q(\lambda\vec x)=\lambda^2q(\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 a bilinear form. is called quadratic form. The bilinear form f is symmetric''.'' In case of \operatornameK\ne2 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 \operatornameK=2 (that means: 1+1=0) the bilinear form has the property f(\vec x,\vec x)=0, i.e. f is ''Symplectic vector space, symplectic''. For V=K^n\ and \ \vec x=\sum_^x_i\vec e_i\quad (\ is a base of V) \ q has the familiar form : q(\vec x)=\sum_^ a_x_ix_k\ \text\ a_:= f(\vec e_i,\vec e_k)\ \text\ i\ne k\ \text\ a_:= q(\vec e_i)\ and : f(\vec x,\vec y)=\sum_^ a_(x_iy_k+x_ky_i). For example: : 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\; .


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

Let K be a field, 2\le n\in\N, :V_ an -dimension (vector space), dimensional vector space over the field K, :\langle\vec x\rangle the 1-dimensional linear span, subspace generated by \vec 0\ne \vec x\in V_, : =\,\ the ''set of points'' , : =\,\ the ''set of lines''. :P_n(K)=(,)\ 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 \;n>3\; a (n-1)-dimensional subspace is called ''hyperplane''.


Projective quadric

For a quadratic form q on a vector space V_ a point \langle\vec x\rangle \in is called ''singular'' if q(\vec x)=0. The set : \mathcal Q=\ of singular points of q is called quadric (with respect to the quadratic form q). Examples in P_2(K).:
(E1): For \;q(\vec x)=x_1x_2-x^2_3\; one gets a Conic section, conic.
(E2): For \;q(\vec x)=x_1x_2\; one gets the pair of lines with the equations x_1=0 and x_2=0, respectively. They intersect at point \langle(0,0,1)^\text\rangle; For the considerations below it is assumed that \mathcal Q\ne \emptyset.


Polar space

For point P=\langle\vec p\rangle \in the set : P^\perp:=\ is called Duality (mathematics)#Polarities of general projective spaces, polar space of P (with respect to q). If \;f(\vec p,\vec x)=0\; for any \vec x , one gets P^\perp=\mathcal P. If \;f(\vec p,\vec x)\ne 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 a hyperplane or .


Intersection with a line

For the intersection of a line with a quadric \mathcal Q the familiar statement is true: :For an arbitrary line g the following cases occur: :a) g\cap \mathcal Q=\emptyset\; and g is called ''exterior line'' or :b) g \subset \mathcal Q\; and g is called ''tangent line'' or :b′) , g\cap \mathcal Q, =1\; and g is called ''tangent line'' or :c) , g\cap \mathcal Q, =2\; and g is called ''secant 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 gets
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\; for ''any'' x\in K or \;q(x\vec u+\vec v)\ne 0\; for ''any'' x\in K, which proves b) and b').
II) In case of g\not\subset U^\perp one gets f(\vec u,\vec v)\ne 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 a ''tangent'' line if and only if g\subset P^\perp.


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

In the classical cases K=\R or \C there exists only one radical, because of \operatornameK\ne2 and f and q are closely connected. In case of \operatornameK=2 the quadric \mathcal Q is not determined by f (see above) and so one has to deal with two radicals: :a) \mathcal R:=\ is a projective subspace. \mathcal R is called ''f''-radical of quadric \mathcal Q. :b) \mathcal S:=\mathcal R\cap\mathcal Q is called singular radical or ''q-radical'' of \mathcal Q. :c) In case of \operatornameK\ne2 one has \mathcal R=\mathcal S. A quadric is called non-degenerate if \mathcal S=\emptyset. Examples in P_2(K) (see above):
(E1): For \;q(\vec x)=x_1x_2-x^2_3\; (conic) the bilinear form is f(\vec x,\vec y)=x_1y_2+x_2y_1-2x_3y_3\; .
In case of \operatornameK\ne2 the polar spaces are never \mathcal P. Hence \mathcal R=\mathcal S=\empty.
In case of \operatornameK=2 the bilinear form is reduced to f(\vec x,\vec y)=x_1y_2+x_2y_1\; and \mathcal R=\langle(0,0,1)^\text\rangle\notin \mathcal Q. Hence \mathcal R\ne \mathcal S=\empty \; . In this case the ''f''-radical is the common point of all tangents, the so called ''knot''.
In both cases S=\empty and the quadric (conic) ist ''non-degenerate''.
(E2): For \;q(\vec x)=x_1x_2\; (pair of lines) the bilinear form is f(\vec x,\vec y)=x_1y_2+x_2y_1\; and \mathcal R=\langle(0,0,1)^\text\rangle=\mathcal S\; , the intersection point.
In this example the quadric is ''degenerate''.


Symmetries

A quadric is a rather homogeneous object: :For any point P\notin \mathcal Q\cup \; there exists an Involution (mathematics), involutorial central collineation \sigma_P with center P and \sigma_P(\mathcal Q)=\mathcal Q. Proof: Due to P\notin \mathcal Q\cup the polar space P^\perp is a hyperplane. The linear mapping : \varphi: \vec x \rightarrow \vec x-\frac\vec p induces an ''involutorial central collineation'' \sigma_P with axis P^\perp and centre P which leaves \mathcal Q invariant.
In case of \operatornameK\ne2 mapping \varphi gets the Reflection (mathematics), familiar shape \; \varphi: \vec x \rightarrow \vec x-2\frac\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) 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 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 \mathcal Q then :The integer \;i:=m+1\; is called index of \mathcal Q. Theorem: (BUEKENHOUT) :For the index i of a non-degenerate quadric \mathcal Q in P_n(K) the following is true: ::i\le \frac. Let be \mathcal Q a non-degenerate quadric in P_n(K), n\ge 2, and i its index.
: In case of i=1 quadric \mathcal Q is called ''sphere'' (or oval (projective plane), oval conic if n=2). : In case of i=2 quadric \mathcal Q is called ''hyperboloid'' (of one sheet). Examples: :a) Quadric \mathcal Q in P_2(K) with form \;q(\vec x)=x_1x_2-x^2_3\; is non-degenerate with index 1. :b) If polynomial \;p(\xi)=\xi^2+a_0\xi+b_0\; is Irreducible polynomial, irreducible over K the quadratic form \;q(\vec x)=x^2_1+a_0x_1x_2+b_0x^2_2-x_3x_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 \R (but not over \C !). :c) In P_3(K) the quadratic form \;q(\vec x)=x_1x_2+x_3x_4\; 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, \ a,b \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 axes


References


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=Quadric


External links


Interactive Java 3D models of all quadric surfacesLecture Note ''Planar Circle Geometries'', an Introduction to Moebius, Laguerre and Minkowski Planes
p. 117 Quadrics, Projective geometry ru:Поверхность второго порядка