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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 (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 § Projective geometry, below.

Euclidean planeEdit

Quadrics in the Euclidean plane are those of dimension D = 1, which is to say that they are plane curves. In this case, one talks of 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 spaceEdit

In three-dimensional Euclidean space, quadrics have dimension D = 2, and are known as quadric surfaces. They are classified and named by their 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:

 
 
 
 

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

Each of these 17 normal forms[2][3] 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
    Ellipsoid    
    Elliptic paraboloid    
    Hyperbolic paraboloid    
   Elliptic hyperboloid of one sheet    
   Elliptic hyperboloid of two sheets    
Degenerate real quadric surfaces
    Elliptic cone    
    Elliptic cylinder    
    Hyperbolic cylinder    
    Parabolic cylinder    

When two or more of the parameters of the canonical equation are equal, one gets 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)    
   Circular hyperboloid of one sheet (special case of elliptic hyperboloid of one sheet)    
   Circular hyperboloid of two sheets (special case of elliptic hyperboloid of two sheets)    
    Circular cone (special case of elliptic cone)    
    Circular cylinder (special case of elliptic cylinder)    

Definition and basic propertiesEdit

A 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 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.

This define affine quadrics. 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.

EquationEdit

A quadric in an affine space of dimension n is the set of zeros of a polynomial of degree 2, that is the set of the points whose coordinates satisfy an equation

 

where the polynomial p has the form

 

where   if the characteristic of the field of the coefficients is not two and   otherwise.

If A is the (n + 1)×(n + 1) matrix that has the   as entries, and

 

then the equation may be shortened in the matrix equation

 

The equation of the projective completion of this quadric is

 

or

 

with

 

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

Normal form of projective quadricsEdit

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 RD+1 are

 

one introduces new coordinates on RD+2

 

related to the original coordinates by  . In the new variables, every quadric is defined by an equation of the form

 

where the coefficients aij 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 (aij) is 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

 

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:

 

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. [4]

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

Projective quadrics over fieldsEdit

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. In order to omit dealing with coordinates a projective quadric is usually defined starting with a quadratic form on a vector space [5]

Quadratic formEdit

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

  and
 .

For example:

 

n-dimensional projective space over a fieldEdit

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 quadricEdit

For a quadratic form   on a vector space   a point   is called singular if  . The set

 

of singular points of   is called quadric (with respect to the quadratic form  ).

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

For the considerations below it is assumed that  .

Polar spaceEdit

For point   the set

 

is called polar space of   (with respect to  ).

If   for any  , one gets  .

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 lineEdit

For the intersection of a line with a quadric   the familiar statement is true:

For an arbitrary line   the following cases occur:
a)   and   is called exterior line or
b)   and   is called tangent line or
b′)   and   is called tangent line or
c)   and   is called secant line.

Proof: Let   be a line, which intersects   at point   and   is a second point on  . From   one gets
 
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 gets   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-radicalEdit

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.

SymmetriesEdit

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 case of   mapping   gets the familiar shape   with   and   for any  .

Remark:

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 quadricEdit

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   [6].

Let be   the dimension of the maximal  -subspaces of   then

The integer   is called index of  .

Theorem: (BUEKENHOUT)[7]

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).

Examples:

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 setsEdit

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.[8][9][10] 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.[11] 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 alsoEdit

ReferencesEdit

  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. ^ Sameen Ahmed Khan,Quadratic Surfaces in Science and Engineering, Bulletin of the IAPT, 2(11), 327–330 (November 2010). (Publication of the Indian Association of Physics Teachers).
    Sameen Ahmed Khan, Coordinate Geometric Generalization of the Spherometer and Cylindrometer, arXiv:1311.3602
    (Apparently, these unreliable sources forget one case)
  4. ^ S. Lazebnik and J. Ponce, "The Local Projective Shape of Smooth Surfaces and Their Outlines" (PDF)., Proposition 1
  5. ^ Beutelspacher/Rosenbaum: p. 158
  6. ^ Beutelpacher/Rosenbaum, p.139
  7. ^ F. Buekenhout: Ensembles Quadratiques des Espace Projective, Math. Teitschr. 110 (1969), p. 306-318.
  8. ^ R. Artzy: The Conic   in Moufang Planes, Aequat.Mathem. 6 (1971), p. 31-35
  9. ^ E. Berz: Kegelschnitte in Desarguesschen Ebenen, Math. Zeitschr. 78 (1962), p. 55-8
  10. ^ external link E. Hartmann: Planar Circle Geometries, p. 123
  11. ^ Beutelspacher/Rosenbaum: p. 135

BibliographyEdit

External linksEdit