Quadric


In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is a hypersurface in a -dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables. 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 equation
which may be compactly written in vector and matrix notation as:
where is a row vector, xT is the transpose of x, 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 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, below.

Euclidean plane

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.

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 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 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
corresponds to a single orbit under affine transformations. In three cases there are no real points: , , and . In one case, the imaginary cone, there is a single point. If one has a line. For one has two intersecting planes. For one has a double plane. For one has two parallel planes.
Thus, among the 17 normal forms, there are nine true quadrics: a cone, three cylinders and five non-degenerate quadrics, which are detailed in the following tables. The eight remaining quadrics are the imaginary ellipsoid, the imaginary cylinder, the imaginary cone, 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 of revolution, which remains invariant when rotated around an axis.

Definition and basic properties

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 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 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 into
. 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
where the polynomial has the form
where if the characteristic of the field of the coefficients is not two and otherwise.
If is the 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 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 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 = 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 is invertible.
In real projective space, by Sylvester's law of inertia, a non-singular quadratic form Q may be put into the normal form
by means of a suitable projective transformation. For surfaces in space 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.
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 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

Quadratic form

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

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 the bilinear form has the property, i.e. is
symplectic.
For and
has the familiar form
For example:

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

Let be a field,,

Projective quadric

For a quadratic form on a vector space a point is called singular if. The set
of singular points of is called quadric.
Examples in.:
: For one gets a conic.
: 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 space

For point the set
is called polar space of .
If for any, one gets.
If for at least one, the equation is a non trivial linear equation which defines a hyperplane. Hence

Intersection with a line

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

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

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 and so one has to deal with two radicals:
A quadric is called non-degenerate if.
Examples in :
: For 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 ist non-degenerate.
: For the bilinear form is and the intersection point.
In this example the quadric is degenerate.

Symmetries

A quadric is a rather homogeneous object:
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:

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

A subspace of is called -subspace if
For example: points on a sphere or lines on a hyperboloid.
Let be the dimension of the maximal -subspaces of then
Theorem:
Let be a non-degenerate quadric in, and its index.

Examples:

Generalization of quadrics: quadratic sets

It is not reasonable to formally extend the definition of quadrics to spaces over genuine skew fields. 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.
There are generalizations of 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.