Quadric (projective geometry)
Encyclopedia
In projective geometry
a quadric is the set of points of a projective space where a certain quadratic form
on the homogeneous coordinates
becomes zero. It may also be defined as the set of all points that lie on their dual hyperplane
s, under some projective duality of the space.
be an 
-dimensional vector space
with scalar
field
, and let 
be a quadratic form on 
. Let 
be the 
-dimensional projective space corresponding to 
, that is the set 
, where 
denotes the set of all nonzero multiples of 
. The (projective) quadric defined by 
is the set of all points 
of 
such that 
. (This definition is consistent because 
implies 
for some 
, and 
by definition of a quadratic form.)
When
is the real
or complex projective plane, the quadric is also called a (projective) quadratic curve, conic section, or just conic.
When
is the real
or complex projective space
, the quadric is also called a (projective) quadratic surface.
In general, if
is the field of real numbers, a quadric is an 
-dimensional sub-manifold
of the projective space
. The exceptions are certain degenerate quadrics that are associated to quadratic forms with special properties. For instance, if 
is the trivial or null form (that yields 0 for any vector 
), the quadric consists of all points of 
; if 
is a definite form
(everywhere positive, or everywhere negative), the quadric is empty; if
factors into the product of two non-trivial linear forms, the quadric is the union of two hyperplanes; and so on. Some authors may define "quadric" so as to exclude some or all of these special cases.
can be expressed as
where
are the coordinates of 
with respect to some chosen basis, and 
is a certain 
symmetric matrix
with entries
in 
, that depends on 
and on the basis.
This formula can also be written as
where 
is the standard inner product of 
, and 
is the vector of 
defined by
The quadratic form
is trivial if and only if all the entries 
are 0. If 
is the real numbers, there is always a basis such that 
is a diagonal matrix
. In this case, the signs of the diagonal elements
determine whether the quadric is degenerate or not.
defines a projective polarity: a mapping that takes any point 
of 
to a hyperplane 
of 
, and vice-versa, while preserving the incidence relation between points and hyperplanes. The coefficient vector of the polar hyperplane 
, relative to the chosen basis of 
, is 
.
If
is not on the quadric, the hyperplane 
is well-defined (that is, not identically zero) and does not contain 
.
If
is on the quadric and the hyperplane 
is well-defined, and contains 
(which is said to be a regular point). In fact, it is the hyperplane that is tangent
to the quadric at
.
If
is on the quadric, it may happen that all coefficients 
are zero. In that case the polar 
is not defined, and 
is said to be a singular point or singularity of the quadric.
The tangent hyperplane turns out to be the union of all lines that are either entirely contained in
, or intersect 
at only one point.
The condition for a point
to be in the hyperplane that is tangent to 
at 
is 
, which is equivalent to 
The condition for a point
to be singular is 
. The quadric has singular points if and only the matrix 
, in diagonal form, has one or more zeros in its diagonal. It follows that the set of all singular points on the quadric is a projective subspace.
at zero, one, or two points, or may be entirely contained in it. The line defined by two distinct points 
and 
is the set of points of the form 
where 
are arbitrary scalars from 
, not both zero. This generic point lies on 
if and only if 
, which is equivalent to
The number of intersections depends on the three coefficients
, 
, and 
. If all three of 
are zero, any pair 
satisfies the equation, so the line is entirely contained in 
. Otherwise, the line has zero, one, or two distinct intersections depending on whether 
is negative, zero, or positive, respectively.
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
a quadric is the set of points of a projective space where a certain quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
on the homogeneous coordinates
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
becomes zero. It may also be defined as the set of all points that lie on their dual hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s, under some projective duality of the space.
Formal definition
More formally, let be an -dimensional vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
with scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and let be a quadratic form on . Let be the -dimensional projective space corresponding to , that is the set , where denotes the set of all nonzero multiples of . The (projective) quadric defined by is the set of all points of such that . (This definition is consistent because implies for some , and by definition of a quadratic form.)When
is the realReal projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
or complex projective plane, the quadric is also called a (projective) quadratic curve, conic section, or just conic.
When
is the realReal projective space
In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
or complex projective space
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...
, the quadric is also called a (projective) quadratic surface.
In general, if
is the field of real numbers, a quadric is an -dimensional sub-manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
of the projective space
. The exceptions are certain degenerate quadrics that are associated to quadratic forms with special properties. For instance, if is the trivial or null form (that yields 0 for any vector ), the quadric consists of all points of ; if is a definite formDefinite quadratic form
In mathematics, a definite quadratic form is a real-valued quadratic form over some vector space V that has the same sign for every nonzero vector of V...
(everywhere positive, or everywhere negative), the quadric is empty; if
factors into the product of two non-trivial linear forms, the quadric is the union of two hyperplanes; and so on. Some authors may define "quadric" so as to exclude some or all of these special cases.Matrix representation
Any quadratic form can be expressed aswhere
are the coordinates of with respect to some chosen basis, and is a certain symmetric matrixMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
with entries
in , that depends on and on the basis.This formula can also be written as
where is the standard inner product of , and is the vector of defined byThe quadratic form
is trivial if and only if all the entries are 0. If is the real numbers, there is always a basis such that is a diagonal matrixDiagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
. In this case, the signs of the diagonal elements
determine whether the quadric is degenerate or not.Polarity, tangent hyperplane, and singular points
In general, a projective quadric defines a projective polarity: a mapping that takes any point of to a hyperplane of , and vice-versa, while preserving the incidence relation between points and hyperplanes. The coefficient vector of the polar hyperplane , relative to the chosen basis of , is .If
is not on the quadric, the hyperplane is well-defined (that is, not identically zero) and does not contain .If
is on the quadric and the hyperplane is well-defined, and contains (which is said to be a regular point). In fact, it is the hyperplane that is tangentTangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
to the quadric at
.If
is on the quadric, it may happen that all coefficients are zero. In that case the polar is not defined, and is said to be a singular point or singularity of the quadric.The tangent hyperplane turns out to be the union of all lines that are either entirely contained in
, or intersect at only one point.The condition for a point
to be in the hyperplane that is tangent to at is , which is equivalent to The condition for a point
to be singular is . The quadric has singular points if and only the matrix , in diagonal form, has one or more zeros in its diagonal. It follows that the set of all singular points on the quadric is a projective subspace.Intersection of lines with quadrics
In projective space, a straight line may intersect a quadric at zero, one, or two points, or may be entirely contained in it. The line defined by two distinct points and is the set of points of the form where are arbitrary scalars from , not both zero. This generic point lies on if and only if , which is equivalent toThe number of intersections depends on the three coefficients
, , and . If all three of are zero, any pair satisfies the equation, so the line is entirely contained in . Otherwise, the line has zero, one, or two distinct intersections depending on whether is negative, zero, or positive, respectively.