In mathematics, a quadric or quadric surface (quadric hypersurface in higher
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
s), is a
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
of
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s (
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s,
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
s, and
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
s). It is a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
(of dimension ''D'') in a -dimensional space, and it is defined as the
zero set
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equi ...
of an
irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted ...
of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
two in ''D'' + 1 variables; for example, in the case of conic sections. When the defining polynomial is not
absolutely irreducible In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.. For example, x^2+y^2-1 is absolutely irreducible, but while x^2+y^2 is irreducible over the intege ...
, 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
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
[Silvio Lev]
Quadrics
in "Geometry Formulas and Facts", excerpted from 30th Edition of ''CRC Standard Mathematical Tables and Formulas'', CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information tec ...
, from The Geometry Center
The Geometry Center was a mathematics research and education center at the University of Minnesota. It was established by the National Science Foundation in the late 1980s and closed in 1998. The focus of the center's work was the use of computer ...
at University of Minnesota
The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN Twin Cities, the U of M, or Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Tw ...
:
which may be compactly written in vector and matrix notation as:
:
where is a row
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, ''x''
T is the
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
of ''x'' (a column vector), ''Q'' is a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, but a quadric may be defined over any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
.
A quadric is an
affine algebraic variety
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
, or, if it is reducible, an
affine algebraic set
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
. Quadrics may also be defined in
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
s; see , below.
Euclidean plane
As the dimension of a
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
is two, quadrics in a Euclidean plane have dimension one and are thus
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
s. They are called ''conic sections'', or ''conics''.
Euclidean space
In three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, quadrics have dimension two, and are known as quadric surfaces. Their
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
s have the form
:
where
are real numbers, and at least one of , , and is nonzero.
The quadric surfaces are classified and named by their shape, which corresponds to the
orbits
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
under
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
s. 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
In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis the ...
shows that for any (possibly reducible) quadric, a suitable change of
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
or, equivalently, a
Euclidean transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations ...
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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
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
[Stewart Venit and Wayne Bishop, ''Elementary Linear Algebra (fourth edition)'', International Thompson Publishing, 1996.] 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
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
,
paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plane ...
s and
hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
s), 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.
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).
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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
coefficients, and the zeros are points in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. However, most properties remain true when the coefficients belong to any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and the points belong in an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
. As usually in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, it is often useful to consider points over an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
containing the polynomial coefficients, generally the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, when the coefficients are real.
Many properties becomes easier to state (and to prove) by extending the quadric to the
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
by
projective completion
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
, consisting of adding
points at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each Pencil (mathematics), pencil of parallel l ...
. Technically, if
:
is a polynomial of degree two that defines an affine quadric, then its projective completion is defined by
homogenizing
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, s ...
into
:
(this is a polynomial, because the degree of is two). The points of the projective completion are the points of the projective space whose
projective coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
are zeros of .
So, a ''projective quadric'' is the set of zeros in a projective space of a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
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. 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
A quadric in an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
of dimension 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 has the form
:
for a matrix
with
and
running from 0 to
. When the
characteristic of the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
.
The equation may be shortened, as the matrix equation
:
with
:
The equation of the projective completion is almost identical:
:
with
:
These equations define a quadric as an
algebraic hypersurface of
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
and degree two in a space of dimension .
A quadric is said to be non-degenerate if the matrix
is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
.
A non-degenerate quadric is
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
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
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel ...
of the matrix .
A quadric is reducible if and only if the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
of is one (case of a double hyperplane) or two (case of two hyperplanes).
Normal form of projective quadrics
In
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properties Construction
A ...
, by
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symme ...
, a non-singular
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
''P''(''X'') may be put into the normal form
:
by means of a suitable
projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ...
(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
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
.
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 surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
s 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
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 ...
all of the nondegenerate quadrics become indistinguishable from each other.
Rational parametrization
Given a non-singular point of a quadric, a line passing through 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 and not tangent to the quadric are in
one to one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
with the points of the quadric that do not belong to the tangent hyperplane at . Expressing the points of the quadric in terms of the direction of the corresponding line provides
parametric equation
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
s of the following forms.
In the case of conic sections (quadric curves), this pametrization establishes a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between a projective conic section and a
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
; this bijection is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
of
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s. In higher dimensions, the parametrization defines a
birational map
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...
, which is a bijection between
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
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 algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
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 .
In the affine case, the parametrization is a
rational parametrization In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to
:K(U_1, \dots , U_d),
t ...
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 be the quadratic polynomial that defines the quadric, and
be the
coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
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 geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each Pencil (mathematics), pencil of parallel l ...
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
Let consider the quadric of equation
:
For
this is the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
; for
this is the
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
; in higher dimension, 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 out the constant terms, dividing by
and solving in
one gets
:
Substituting this into
and simplifying the expression of the last coordinate, one gets the parametric equation
:
By homogenizing, one gets 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 getting real points that are not obtained with the resulting parametrization.
Rational points
A quadric is ''defined over'' a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
if the coefficients of its equation belong to
When
is the field
of the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, one can suppose that the coefficients are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s by
clearing denominators In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions.
Example
Co ...
.
A point of a quadric defined over a field
is said
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
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
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
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
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
.
Given a rational point over a quadric over a field , the parametrization described in the preceding section provides rational points when the parameters are in , and, conversely, every rational point of the quadric can be obtained from parameters in , if the point is not in the tangent hyperplane at .
It follows that, if a quadric has a rational point, it has many other rational points (infinitely many if 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
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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 Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
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
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
for getting the associated primitive solution.
This is well illustrated by
Pythagorean triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
s. A Pythagorean triple is a
triple
Triple is used in several contexts to mean "threefold" or a " treble":
Sports
* Triple (baseball), a three-base hit
* A basketball three-point field goal
* A figure skating jump with three rotations
* In bowling terms, three strikes in a row
* ...
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 and are coprime integers such that
the resulting triple is a Pythagorean triple. If one of and is even and the other is even, this resulting triple is primitive; otherwise, and are both odd, and one gets a primitive triple by dividing by 2.
In summary, the primitive Pythagorean triples with
even are obtained as
:
with and coprime integers such that one is even and
(this is
Euclid's formula
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
). The primitive Pythagorean triples with
odd are obtained as
:
with and coprime odd integers such that
As the exchange of and transforms a Pythagorean triple into another Pythagorean triple, only one of the two cases is sufficient for getting all primitive Pythagorean triples.
Projective quadrics over fields
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space.
Quadratic form
Let
be a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and
a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over
. A mapping
from
to
such that
: (Q1)
for any
and
.
: (Q2)
is a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
.
is called
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
. 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 field
Let
be a field,
,
:
an -
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over the field
:
the 1-dimensional
subspace generated by ,
:
the ''set of points'' ,
:
the ''set of lines''.
:
is the -dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
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
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
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special ...
.
(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
For point
the set
:
is called
polar space of
(with respect to
).
If
for any
, one obtains
.
If
for at least one
, the equation
is a non trivial linear equation which defines a hyperplane. Hence
:
is either a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
or
.
Intersection with a line
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
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
A quadric is a rather homogeneous object:
:For any point
there exists an
involutorial central
collineation
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thu ...
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 quadric
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
.
Let be
the dimension of the maximal
-subspaces of
then
:The integer
is called index of
.
Theorem: (BUEKENHOUT)
: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
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...
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
In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole.
Emergence ...
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
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. The reason is the following statement.
:A
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
if and only if any
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, has at most two solutions.
There are ''generalizations'' of quadrics:
quadratic set In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
Definition of a qu ...
s.
[Beutelspacher/Rosenbaum: p. 135] 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
*
Klein quadric
In mathematics, the lines of a 3-dimensional projective space, ''S'', can be viewed as points of a 5-dimensional projective space, ''T''. In that 5-space, the points that represent each line in ''S'' lie on a quadric, ''Q'' known as the Klein q ...
*
Rotation of axes
In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x′y′''-Cartesian coordinate system in which the origin is kept fixed and the ''x′'' and ''y′'' axes are ...
*
Superquadrics
In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary ...
*
Translation of axes
In mathematics, a translation of axes in two dimensions is a mapping from an ''xy''-Cartesian coordinate system to an ''x'y-Cartesian coordinate system in which the ''x axis is parallel to the ''x'' axis and ''k'' units away, and the ''y ...
References
Bibliography
* M. Audin: ''Geometry'', Springer, Berlin, 2002, , p. 200.
* M. Berger: ''Problem Books in Mathematics'', ISSN 0941-3502, Springer New York, pp 79–84.
* A. Beutelspacher, U. Rosenbaum: ''Projektive Geometrie'', Vieweg + Teubner, Braunschweig u. a. 1992, , p. 159.
* P. Dembowski: ''Finite Geometries'', Springer, 1968, , p. 43.
*
*{{mathworld, urlname=Quadric, title=Quadric
External links
Interactive Java 3D models of all quadric surfacesLecture Note ''Planar Circle Geometries'', an Introduction to Moebius, Laguerre and Minkowski Planes p. 117
Projective geometry
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