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, 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, 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 extended Euclidean and affine spaces by the addition of new points called points at infinity. 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 over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra 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, see also synthetic geometry); 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 and Desargues's theorem 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, and, specifically, the difference in appearance of two plane objects viewed from different points of view. In three-dimensional Euclidean space, a central projection 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 was originally introduced by extending the Euclidean space, that is, by adding points at infinity to it, in order to define the projection for every point except O. Given another plane Q, which does not contain O, the restriction to Q of the above projection is called a perspectivity. With these definitions, a perspectivity is only a partial function, but it becomes a bijection 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:
Given two projective spaces P and Q of dimension n, a perspectivity is a bijection from P to Q that may be obtained by embedding P and Q in a projective space R of dimension n + 1 and restricting to P a central projection onto Q. If f is a perspectivity from P to Q, and g a perspectivity from Q to P, with a different center, then g ⋅ f is a homography from P to itself, which is called a central collineation, when the dimension of P is at least two. (See § Central collineations below and Perspectivity § Perspective collineations.) Originally, a homography was defined as the composition 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 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 n + 1. If a basis of V has been fixed, a point of V may be represented by a point (x0, ..., xn) of Kn+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 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 of vector spaces f : V → W. Such an isomorphism induces a bijection 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 g = af. This may be written in terms of homogeneous coordinates in the following way: A homography φ may be defined by a nonsingular (n+1) × (n+1) matrix [ai,j], called the matrix of the homography. This matrix is defined up to the multiplication by a nonzero element of K. The homogeneous coordinates [x0 : ... : xn] of a point and the coordinates [y0 : ... : yn] of its image by φ are related by
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