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 homography 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 i ...
of
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, induced by an isomorphism of the
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 ...
s from which the projective spaces derive. It is 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 ...
that maps
lines to lines, and thus a
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 ...
. In general, some collineations are not homographies, but the
fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
Historically, homographies (and projective spaces) have been introduced to study
perspective and
projections in
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
, and the term ''homography'', which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which differed from extending
Euclidean or
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 related ...
s by 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 of parallel lines of the plane. Ad ...
. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines 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 ...
over a given
field (the above definition is based on this version); this construction facilitates the definition of
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 coordinate system, Cartesian coordinates are u ...
and allows using the tools 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 ...
for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (
incidence geometry
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''inciden ...
, see also
synthetic geometry
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...
); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative)
field. Equivalently
Pappus's hexagon theorem
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that
*given one set of collinear points A, B, C, and another set of collinear points a,b,c, then the intersection points X,Y,Z of line pairs Ab and aB, Ac and ...
and
Desargues's theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ...
are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.
Geometric motivation
Historically, the concept of homography had been introduced to understand, explain and study
visual perspective
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
, and, specifically, the difference in appearance of two plane objects viewed from different points of view.
In three-dimensional Euclidean space, a
central projection
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition, i.e., which is idempotent. The restriction to a subspace of a projec ...
from a point ''O'' (the center) onto a plane ''P'' that does not contain ''O'' is the mapping that sends a point ''A'' to the intersection (if it exists) of the line ''OA'' and the plane ''P''. The projection is not defined if the point ''A'' belongs to the plane passing through ''O'' and parallel to ''P''. The notion of
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 ...
was originally introduced by extending the Euclidean space, that is, by 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 of parallel lines of the plane. Ad ...
to it, in order to define the projection for every point except ''O''.
Given another plane ''Q'', which does not contain ''O'', the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
to ''Q'' of the above projection is called a
perspectivity
In geometry and in its applications to drawing, a perspectivity is the formation of an image in a picture plane of a scene viewed from a fixed point.
Graphics
The science of graphical perspective uses perspectivities to make realistic images ...
.
With these definitions, a perspectivity is only a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
, but it becomes 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 ...
if extended to projective spaces. Therefore, this notion is normally defined for projective spaces. The notion is also easily generalized to projective spaces of any dimension, over any
field, in the following way:
If ''f'' is a perspectivity from ''P'' to ''Q'', and ''g'' a perspectivity from ''Q'' to ''P'', with a different center, then is a homography from ''P'' to itself, which is called a ''central collineation'', when the dimension of ''P'' is at least two. (See below and .)
Originally, a homography was defined as 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 a finite number of perspectivities. It is a part of the fundamental theorem of projective geometry (see below) that this definition coincides with the more algebraic definition sketched in the introduction and detailed below.
Definition and expression in homogeneous coordinates
A
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 ...
P(''V'') of dimension ''n'' over a
field ''K'' may be defined as the set of the lines through the origin in a ''K''-vector space ''V'' of dimension . If a basis of ''V'' has been fixed, a point of ''V'' may be represented by a point
of ''K''
''n''+1. A point of P(''V''), being a line in ''V'', may thus be represented by the coordinates of any nonzero point of this line, which are thus called
homogeneous 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 geometr ...
of the projective point.
Given two projective spaces P(''V'') and P(''W'') of the same dimension, a homography is a mapping from P(''V'') to P(''W''), which is induced by 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 ...
of vector spaces
. Such an isomorphism induces 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 P(''V'') to P(''W''), because of the linearity of ''f''. Two such isomorphisms, ''f'' and ''g'', define the same homography if and only if there is a nonzero element ''a'' of ''K'' such that .
This may be written in terms of homogeneous coordinates in the following way: A homography ''φ'' may be defined by a
nonsingular matrix ''i'',''j''">'a''''i'',''j'' called the ''matrix of the homography''. This matrix is defined
up to Two 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'' with respect to ''R'' ...
the multiplication by a nonzero element of ''K''. The homogeneous coordinates