In
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, a collineation is a
one-to-one and
onto map (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 ...
) from one
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 ...
to another, or from a projective space to itself, such that the
images
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
points are themselves collinear. A collineation is thus 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 i ...
'' between projective spaces, or an
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism. The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all collineations of a space to itself form a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
, called the collineation group.
Definition
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently.
Linear algebra
For a projective space defined in terms of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
(as the projectivization of 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 ...
), a collineation is a map between the projective spaces that is
order-preserving with respect to
inclusion of subspaces.
Formally, let ''V'' be a vector space over a
field ''K'' and ''W'' a vector space over a field ''L''. Consider the projective spaces ''PG''(''V'') and ''PG''(''W''), consisting of the
vector lines of ''V'' and ''W''.
Call ''D''(''V'') and ''D''(''W'') the set of subspaces of ''V'' and ''W'' respectively. A collineation from ''PG''(''V'') to ''PG''(''W'') is a map α : ''D''(''V'') → ''D''(''W''), such that:
* α is a bijection.
* ''A'' ⊆ ''B'' ⇔ α(''A'') ⊆ α(''B'') for all ''A'', ''B'' in ''D''(''V'').
Axiomatically
Given a
projective space defined axiomatically in terms of an
incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
(a set of points ''P,'' lines ''L,'' and an
incidence relation ''I'' specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function ''f'' between the sets of points and a bijective function ''g'' between the set of lines, preserving the incidence relation.
Every projective space of dimension greater than or equal to three is isomorphic to the
projectivization
In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multi ...
of a linear space over 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 ...
, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the
non-Desarguesian plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...
s, and this definition allows one to define collineations in such projective planes.
For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.
Collineations of the projective line
For a projective space of dimension one (a projective line; the projectivization of a vector space of
dimension
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 coord ...
two), all points are collinear, so the collineation group is exactly the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the
fundamental theorem of projective geometry holds.
In this definition, when ''V'' has dimension two, a collineation from ''PG''(''V'') to ''PG''(''W'') is a map , such that:
* The
zero subspace of ''V'' is mapped to the zero subspace of ''W''.
* ''V'' is mapped to ''W''.
* There is a nonsingular
semilinear map ''β'' from ''V'' to ''W'' such that, for all ''v'' in ''V'',
This last requirement ensures that collineations are all semilinear maps.
Types
The main examples of collineations are projective linear transformations (also known as
homographies) and
automorphic collineations
Automorphic may refer to
*Automorphic number, in mathematics
*Automorphic form, in mathematics
* Automorphic representation, in mathematics
* Automorphic L-function, in mathematics
*Automorphism
In mathematics, an automorphism is an isomorphi ...
. For projective spaces coming from a linear space, the
fundamental theorem of projective geometry states that all collineations are a combination of these, as described below.
Projective linear transformations
Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are projective linear transformations. The group of projective linear transformations (
PGL) is in general a proper
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of the collineation group.
Automorphic collineations
An is a map that, in coordinates, is a
field automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
applied to the coordinates.
Fundamental theorem of projective geometry
If the geometric dimension of a
pappian projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the
projective semilinear group In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
, which is the
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in wh ...
of homographies by automorphic collineations.
In particular, the collineations of are exactly the homographies, as R has no non-trivial automorphisms (that is, Gal(R/Q) is trivial).
Suppose ''φ'' is a nonsingular semilinear map from ''V'' to ''W'', with the dimension of ''V'' at least three. Define by saying that for all ''Z'' in ''D''(''V''). As ''φ'' is semilinear, one easily checks that this map is properly defined, and furthermore, as ''φ'' is not singular, it is bijective. It is obvious now that ''α'' is a collineation. We say that ''α'' is induced by ''φ''.
The fundamental theorem of projective geometry states the converse:
Suppose ''V'' is a vector space over a field ''K'' with dimension at least three, ''W'' is a vector space over a field ''L'', and ''α'' is a collineation from PG(''V'') to PG(''W''). This implies ''K'' and ''L'' are isomorphic fields, ''V'' and ''W'' have the same dimension, and there is a semilinear map ''φ'' such that ''φ'' induces ''α''.
For , the collineation group is the
projective semilinear group In linear algebra, particularly projective geometry, a semilinear map between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear map "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K' ...
, PΓL – this is PGL, twisted by
field automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s; formally, the
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in wh ...
, where ''k'' is the
prime field
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
for ''K''.
Linear structure
Thus for ''K'' a prime field (
or
), we have , but for ''K'' not a prime field (such as
or
for ), the projective linear group is in general a proper subgroup of the collineation group, which can be thought of as "transformations preserving a projective ''semi''-linear structure". Correspondingly, the quotient group corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure. Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and PΓL, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup , these choices forming a
torsor over Gal(''K''/''k'').
History
The idea of a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Art ...
was abstracted to a
ternary relation
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.
Just as a binary relati ...
determined by
collinearity (points lying on a single line). According to
Wilhelm Blaschke
Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry.
Education and career
Blaschke was the son of mathematician Josef Blaschke, who taugh ...
it was
August Möbius that first abstracted this essence of geometrical transformation:
:What do our geometric transformations mean now? Möbius threw out and fielded this question already in his ''Barycentric Calculus'' (1827). There he spoke not of ''transformations'' but of ''permutations''
erwandtschaften when he said two elements drawn from a domain were ''permuted'' when they were interchanged by an arbitrary equation. In our particular case, linear equations between homogeneous point coordinates, Möbius called a permutation
erwandtschaftof both point spaces in particular a ''collineation''. This signification would be changed later by
Chasles to ''homography''. Möbius’ expression is immediately comprehended when we follow Möbius in calling points
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
when they lie on the same line. Möbius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight.
Contemporary mathematicians view geometry as an
incidence structure
In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
with an
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
consisting of mappings of the underlying space that preserve
incidence. Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.
As mentioned by Blaschke and Klein,
Michel Chasles preferred the term ''homography'' to ''collineation''. A distinction between the terms arose when the distinction was clarified between the
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
and the
complex projective line
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers p ...
. Since there are no non-trivial field automorphisms of the
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 ...
field, all the collineations are homographies in the real projective plane, however due to the field automorphism of
complex conjugation, not all collineations of the complex projective line are homographies. In applications such as
computer vision where the underlying field is the real number field, ''homography'' and ''collineation'' can be used interchangeably.
Anti-homography
The operation of taking the
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 - ...
in the
complex plane amounts to a
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
in the
real line. With the notation ''z''
∗ for the conjugate of ''z'', an anti-homography is given by
:
Thus an anti-homography is the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of conjugation with a
homography
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, ...
, and so is an example of a collineation which is not an homography. For example, geometrically, the mapping
amounts to
circle inversion
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
. The transformations of
inversive geometry of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.
p. 43
p. 42
Notes
References
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External links
* {{PlanetMath, urlname=Collineation, title=projectivity
Projective geometry