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.
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.
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.
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.
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.
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:
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.
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
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by homogenizing p into
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.
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
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.
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 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
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.
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
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
In real projective space, by Sylvester's law of inertia, a non-singular quadratic form Q(X) may be put into the normal form
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 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.
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]
Let be a field and
a vector space over
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.
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]
Let 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.
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]
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 In case of For For example:
Let 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
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.
and
(
is a base of
)
has the familiar form
n-dimensional projective space over a field
be a field,
,
an (n + 1)-dimensional vector space over the field
be a field,
,
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 .
For point the set
Examples in .:
(E1): For .:
(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 .
For point the set
is called polar space of (with respect to
).
If for any
, one gets
for any
, one gets
.
If for at least one
, the equation
is a non trivial linear equation which defines a hyperplane. Hence
For the intersection of a line with a quadric the familiar statement is true:
Additionally the proof shows:
In the classical cases or
there exists only one radical, because of
and
and
are closely connected. In case of
the quadric
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:
Examples in Examples in A quadric is a rather homogeneous object:
Proof:
Due to The linear mapping
induces an involutorial central collineation induces an involutorial central collineation Remark:
(see above):
(E1): For (conic) the bilinear form is
In case of the polar spaces are never
. Hence
.
In case of (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.
the polar space
is a hyperplane.
with axis
with axis
and centre
which leaves
invariant.
In case of mapping
gets the familiar shape
with
and
for any
.
on an exterior, tangent and secant line, respectively.
is pointwise fixed by
.
q-subspaces and index of a quadric
For example: points on a sphere or lines on a hyperboloid (s. below).
Let be the dimension of the maximal
-subspaces of lines on a hyperboloid
Let be the dimension of the maximal
-subspaces of
then
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.
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.