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, 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 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 ; 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 ; 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 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 onto a plane P that does not contain O is the mapping that sends a point A to the intersection 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:
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.
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 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 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 Kn+1. A point of P, 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 and P of the same dimension, a homography is a mapping from P to P, which is induced by an isomorphism of vector spaces. Such an isomorphism induces a bijection from P to P, 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 , called the matrix of the homography. This matrix is defined up to the multiplication by a nonzero element of K. The homogeneous coordinates of a point and the coordinates of its image by φ are related by
When the projective spaces are defined by adding points at infinity to affine spaces the preceding formulas become, in affine coordinates,
which generalizes the expression of the homographic function of the next section. This defines only a partial function between affine spaces, which is defined only outside the hyperplane where the denominator is zero.

Homographies of a projective line

The projective line over a field K may be identified with the union of K and a point, called the "point at infinity" and denoted by ∞. With this representation of the projective line, the homographies are the mappings
which are called homographic functions or linear fractional transformations.
In the case of the complex projective line, which can be identified with the Riemann sphere, the homographies are called Möbius transformations.
These correspond precisely with those bijections of the Riemann sphere that preserve orientation and are conformal.
In the study of collineations, the case of projective lines is special due to the small dimension. When the line is viewed as a projective space in isolation, any permutation of the points of a projective line is a collineation, since every set of points are collinear. However, if the projective line is embedded in a higher-dimensional projective space, the geometric structure of that space can be used to impose a geometric structure on the line. Thus, in synthetic geometry, the homographies and the collineations of the projective line that are considered are those obtained by restrictions to the line of collineations and homographies of spaces of higher dimension. This means that the fundamental theorem of projective geometry remains valid in the one-dimensional setting. A homography of a projective line may also be properly defined by insisting that the mapping preserves cross-ratios.

Projective frame and coordinates

A projective frame or projective basis of a projective space of dimension is an ordered set of points such that no hyperplane contains of them. A projective frame is sometimes called a simplex, although a simplex in a space of dimension has at most vertices.
Projective spaces over a commutative field are considered in this section, although most results may be generalized to projective spaces over a division ring.
Let be a projective space of dimension, where is a K-vector space of dimension, and be the canonical projection that maps a nonzero vector to the vector line that contains it.
For every frame of, there exists a basis of V such that the frame is, and this basis is unique up to the multiplication of all its elements by the same nonzero element of K. Conversely, if is a basis of V, then is a frame of
It follows that, given two frames, there is exactly one homography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry. It is sometimes called the first fundamental theorem of projective geometry.
Every frame allows to define projective coordinates, also known as homogeneous coordinates: every point may be written as ; the projective coordinates of on this frame are the coordinates of on the base. It is not difficult to verify that changing the e and, without changing the frame nor p, results in multiplying the projective coordinates by the same nonzero element of K.
The projective space has a canonical frame consisting of the image by of the canonical basis of , and. On this basis, the homogeneous coordinates of are simply the entries of the tuple. Given another projective space of the same dimension, and a frame of it, there is one and only one homography mapping onto the canonical frame of. The projective coordinates of a point on the frame are the homogeneous coordinates of on the canonical frame of.

Central collineations

In above sections, homographies have been defined through linear algebra. In synthetic geometry, they are traditionally defined as the composition of one or several special homographies called central collineations. It is a part of the fundamental theorem of projective geometry that the two definitions are equivalent.
In a projective space, P, of dimension, a collineation of P is a bijection from P onto P that maps lines onto lines. A central collineation is a bijection α from P to P, such that there exists a hyperplane H, which is fixed pointwise by α and a point O, which is fixed linewise by α. There are two types of central collineations. Elations are the central collineations in which the center is incident with the axis and homologies are those in which the center is not incident with the axis. A central collineation is uniquely defined by its center, its axis, and the image α of any given point P that differs from the center O and does not belong to the axis. of any other point Q is the intersection of the line defined by O and Q and the line passing through α
A central collineation is a homography defined by a × matrix that has an eigenspace of dimension n. It is a homology, if the matrix has another eigenvalue and is therefore diagonalizable. It is an elation, if all the eigenvalues are equal and the matrix is not diagonalizable.
The geometric view of a central collineation is easiest to see in a projective plane. Given a central collineation α, consider a line ℓ that does not pass through the center O, and its image under α,. Setting, the axis of α is some line M through R. The image of any point A of ℓ under α is the intersection of OA with ℓ. The image of a point B that does not belong to ℓ may be constructed in the following way: let, then.
The composition of two central collineations, while still a homography in general, is not a central collineation. In fact, every homography is the composition of a finite number of central collineations. In synthetic geometry, this property, which is a part of the fundamental theory of projective geometry is taken as the definition of homographies.