In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

In coordinates x1, x2, ..., xD+1, the general quadric is thus defined by the algebraic equation[1]

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

where x = (x1, x2, ..., xD+1) is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1) × (D + 1) matrix and P is a (D + 1)-dimensional row vector and R a scalar constant. The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any 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 § Normal form of projective quadrics, below.

Euclidean plane edit

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.

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

Euclidean space edit

In three-dimensional Euclidean space, quadrics have dimension two, and are known as quadric surfaces. Their quadratic equations have the form


where   are real numbers, and at least one of A, B, and C is nonzero.

The quadric surfaces are classified and named by their shape, which corresponds to the orbits under affine 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.

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 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:


where the   are either 1, –1 or 0, except   which takes only the value 0 or 1.

Each of these 17 normal forms[2] corresponds to a single orbit under affine transformations. In three cases there are no real points:   (imaginary ellipsoid),   (imaginary elliptic cylinder), and   (pair of complex conjugate parallel planes, a reducible quadric). In one case, the imaginary cone, there is a single point ( ). If   one has a line (in fact two complex conjugate intersecting planes). For   one has two intersecting planes (reducible quadric). For   one has a double plane. For   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.

Non-degenerate real quadric surfaces
    Elliptic paraboloid    
    Hyperbolic paraboloid    
   Hyperboloid of one sheet
   Hyperbolic hyperboloid
   Hyperboloid of two sheets
   Elliptic hyperboloid
Degenerate real quadric surfaces
    Elliptic cone
   Conical quadric
    Elliptic cylinder    
    Hyperbolic cylinder    
    Parabolic cylinder    

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

Quadrics of revolution
    Oblate and prolate spheroids (special cases of ellipsoid)     
    Sphere (special case of spheroid)    
    Circular paraboloid (special case of elliptic paraboloid)    
    Hyperboloid of revolution of one sheet (special case of hyperboloid of one sheet)    
    Hyperboloid of revolution of two sheets (special case of hyperboloid of two sheets)    
    Circular cone (special case of elliptic cone)    
    Circular cylinder (special case of elliptic cylinder)    

Definition and basic properties edit

An affine quadric is the set of zeros of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have real coefficients, and the zeros are points in a Euclidean space. However, most properties remain true when the coefficients belong to any 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 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


is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing p into


(this is a polynomial, because the degree of p is two). The points of the projective completion are the points of the projective space whose projective coordinates are zeros of 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 X0 = 1:


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   will include points with  , which are not also solutions of   because these points in projective space correspond to points "at infinity" in affine space.

Equation edit

A quadric in an affine space of dimension 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


where the polynomial p has the form


for a matrix   with   and   running from 0 to  . When the characteristic of the field of the coefficients is not two, generally   is assumed; equivalently  . When the characteristic of the field of the coefficients is two, generally   is assumed when  ; equivalently   is upper triangular.

The equation may be shortened, as the matrix equation




The equation of the projective completion is almost identical:




These equations define a quadric as an algebraic hypersurface of dimension n – 1 and degree two in a space of dimension n.

A quadric is said to be non-degenerate if the matrix   is invertible.

A non-degenerate quadric is non-singular in the sense that its projective completion has no 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 A.

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

Normal form of projective quadrics edit

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


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 = 2) in three-dimensional space, there are exactly three non-degenerate 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 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


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.[3]

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

Rational parametrization edit

Given a non-singular point A of a quadric, a line passing through 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 hyperplane). This means that the lines passing through 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 A. Expressing the points of the quadric in terms of the direction of the corresponding line provides parametric equations 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 curves. In higher dimensions, the parametrization defines a birational map, which is a bijection between dense 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 A.

In the affine case, the parametrization is a rational parametrization of the form


where   are the coordinates of a point of the quadric,   are parameters, and   are polynomials of degree at most two.

In the projective case, the parametrization has the form


where   are the projective coordinates of a point of the quadric,   are parameters, and   are homogeneous polynomials of degree two.

One passes from one parametrization to the other by putting   and  


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 q be the quadratic polynomial that defines the quadric, and   be the coordinate vector of the given point of the quadric (so,   Let   be the coordinate vector of the point of the quadric to be parametrized, and   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   passing through   are the points   such that


for some value of the scalar   This is an equation of degree two in   except for the values of   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   otherwise). The coefficients of   and   are respectively of degree at most one and two in   As the constant coefficient is   the equation becomes linear by dividing by   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   one obtains the desired parametrization as fractions of polynomials of degree at most two.

Example: circle and spheres edit

Let consider the quadric of equation


For   this is the unit circle; for   this is the unit sphere; in higher dimensions, this is the unit hypersphere.

The point   belongs to the quadric (the choice of this point among other similar points is only a question of convenience). So, the equation   of the preceding section becomes


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


Substituting this into   and simplifying the expression of the last coordinate, one obtains the parametric equation


By homogenizing, one obtains the projective parametrization


A straightforward verification shows that this induces a bijection between the points of the quadric such that   and the points such that   in the projective space of the parameters. On the other hand, all values of   such that   and   give the point  

In the case of conic sections ( ), there is exactly one point with   and one has a bijection between the circle and the projective line.

For   there are many points with   and thus many parameter values for the point   On the other hand, the other points of the quadric for which   (and thus  ) 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   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 edit

A quadric is defined over a field   if the coefficients of its equation belong to   When   is the field   of the rational numbers, one can suppose that the coefficients are integers by clearing denominators.

A point of a quadric defined over a field   is said rational over   if its coordinates belong to   A rational point over the field   of the real numbers, is called a real point.

A rational point over   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  ) 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 a Diophantine equation.

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

It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if 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   the parametrization takes the form


where the   are homogeneous polynomials of degree two with integer coefficients. Because of the homogeneity, one can consider only parameters that are setwise coprime integers. If   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 edit

This is well illustrated by Pythagorean triples. A Pythagorean triple is a triple   of positive integers such that   A Pythagorean triple is primitive if   are setwise coprime, or, equivalently, if any of the three pairs     and   is coprime.

By choosing   the above method provides the parametrization


for the quadric of equation   (The names of variables and parameters are being changed from the above ones to those that are common when considering Pythagorean triples).

If m and n are coprime integers such that   the resulting triple is a Pythagorean triple. If one of m and n is even and the other is odd, this resulting triple is primitive; otherwise, m and n are both odd, and one obtains a primitive triple by dividing by 2.

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


with m and n coprime integers such that one is even and   (this is Euclid's formula). The primitive Pythagorean triples with   odd are obtained as


with m and n coprime odd integers such that  

As the exchange of a and b transforms 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 edit

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. In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space.[4]

Quadratic form edit

Let   be a field and   a vector space over  . A mapping   from   to   such that

(Q1)   for any   and  .
(Q2)   is a bilinear form.

is called quadratic form. The bilinear form   is symmetric.

In case of   the bilinear form is  , i.e.   and   are mutually determined in a unique way.
In case of   (that means:  ) the bilinear form has the property  , i.e.   is symplectic.

For   and   (  is a base of  )   has the familiar form


For example:


n-dimensional projective space over a field edit

Let   be a field,  ,

  an (n + 1)-dimensional vector space over the field  
  the 1-dimensional subspace generated by  ,
  the set of points ,
  the set of lines.
  is the n-dimensional projective space over  .
The set of points contained in a  -dimensional subspace of   is a  -dimensional subspace of  . A 2-dimensional subspace is a plane.
In case of   a  -dimensional subspace is called hyperplane.

Projective quadric edit

A quadratic form   on a vector space   defines a quadric   in the associated projective space   as the set of the points   such that  . That is,


Examples in  .:
(E1): For   one obtains a conic.
(E2): For   one obtains the pair of lines with the equations   and  , respectively. They intersect at point  ;

For the considerations below it is assumed that  .

Polar space edit

For point   the set


is called polar space of   (with respect to  ).

If   for all  , one obtains  .

If   for at least one  , the equation  is a non trivial linear equation which defines a hyperplane. Hence

  is either a hyperplane or  .

Intersection with a line edit

For the intersection of an arbitrary line   with a quadric  , the following cases may occur:

a)   and   is called exterior line
b)   and   is called a line in the quadric
c)   and   is called tangent line
d)   and   is called secant line.

Proof: Let   be a line, which intersects   at point   and   is a second point on  . From   one obtains
I) In case of   the equation   holds and it is   for any  . Hence either   for any   or   for any  , which proves b) and b').
II) In case of   one obtains   and the equation   has exactly one solution  . Hence:  , which proves c).

Additionally the proof shows:

A line   through a point   is a tangent line if and only if  .

f-radical, q-radical edit

In the classical cases   or   there exists only one radical, because of   and   and   are closely connected. In case of   the quadric   is not determined by   (see above) and so one has to deal with two radicals:

a)   is a projective subspace.   is called f-radical of quadric  .
b)   is called singular radical or  -radical of  .
c) In case of   one has  .

A quadric is called non-degenerate if  .

Examples in   (see above):
(E1): For   (conic) the bilinear form is  
In case of   the polar spaces are never  . Hence  .
In case of   the bilinear form is reduced to   and  . Hence   In this case the f-radical is the common point of all tangents, the so called knot.
In both cases   and the quadric (conic) ist non-degenerate.
(E2): For   (pair of lines) the bilinear form is   and   the intersection point.
In this example the quadric is degenerate.

Symmetries edit

A quadric is a rather homogeneous object:

For any point   there exists an involutorial central collineation   with center   and  .

Proof: Due to   the polar space   is a hyperplane.

The linear mapping


induces an involutorial central collineation   with axis   and centre   which leaves   invariant.
In the case of  , the mapping   produces the familiar shape   with   and   for any  .


a) An exterior line, a tangent line or a secant line is mapped by the involution   on an exterior, tangent and secant line, respectively.
b)   is pointwise fixed by  .

q-subspaces and index of a quadric edit

A subspace   of   is called  -subspace if  

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

Any two maximal  -subspaces have the same dimension  .[5]

Let be   the dimension of the maximal  -subspaces of   then

The integer   is called index of  .

Theorem: (BUEKENHOUT)[6]

For the index   of a non-degenerate quadric   in   the following is true:

Let be   a non-degenerate quadric in  , and   its index.

In case of   quadric   is called sphere (or oval conic if  ).
In case of   quadric   is called hyperboloid (of one sheet).


a) Quadric   in   with form   is non-degenerate with index 1.
b) If polynomial   is irreducible over   the quadratic form   gives rise to a non-degenerate quadric   in   of index 1 (sphere). For example:   is irreducible over   (but not over   !).
c) In   the quadratic form   generates a hyperboloid.

Generalization of quadrics: quadratic sets edit

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.[7][8][9] The reason is the following statement.

A division ring   is commutative if and only if any equation  , has at most two solutions.

There are generalizations of quadrics: quadratic sets.[10] 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 edit

References edit

  1. ^ Silvio Levy 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
  2. ^ Stewart Venit and Wayne Bishop, Elementary Linear Algebra (fourth edition), International Thompson Publishing, 1996.
  3. ^ S. Lazebnik and J. Ponce, "The Local Projective Shape of Smooth Surfaces and Their Outlines" (PDF)., Proposition 1
  4. ^ Beutelspacher/Rosenbaum p.158
  5. ^ Beutelpacher/Rosenbaum, p.139
  6. ^ F. Buekenhout: Ensembles Quadratiques des Espace Projective, Math. Teitschr. 110 (1969), p. 306-318.
  7. ^ R. Artzy: The Conic   in Moufang Planes, Aequat.Mathem. 6 (1971), p. 31-35
  8. ^ E. Berz: Kegelschnitte in Desarguesschen Ebenen, Math. Zeitschr. 78 (1962), p. 55-8
  9. ^ external link E. Hartmann: Planar Circle Geometries, p. 123
  10. ^ Beutelspacher/Rosenbaum: p. 135

Bibliography edit

External links edit