Homography

A homography is a mathematical mapping that preserves the structure of projective spaces. In geometry, it is a type of bijection, which means every point maps to exactly one unique point. This mapping is also a collineation, a term used for transformations that always map lines to lines. While some collineations are not homographies, the fundamental theorem of projective geometry states they are the same in real projective spaces with at least two dimensions. You might also hear these called projective transformations or projective collineations.

To understand how a homography works, consider the concept of visual perspective. Imagine looking at a flat object from different viewpoints in three-dimensional Euclidean space. A central projection uses a center point, labeled O, to map points from one plane onto another. If you have a point A, you draw a line from O through A to see where it hits a target plane. This process is called a perspectivity. A single perspectivity is only a partial function because it cannot map points that lie on a plane passing through the center. However, if you perform one perspectivity from plane P to plane Q, and then another from Q back to P using a different center, the resulting move is a homography.

There are two main ways to define these spaces and their movements. One approach is algebraic, where a projective space is viewed as the set of lines in a vector space. This method allows mathematicians to use linear algebra to study homographies. The second approach is synthetic, using a set of axioms called incidence geometry. In this context, collineations are defined first, and homographies are described as specific types of projective collineations. These two different ways of looking at the math have been shown to be equivalent.

Historically, homographies were introduced to study how projections work in Euclidean geometry. The name itself comes from a word meaning "similar drawing." At the end of the 19th century, mathematicians expanded these ideas by creating formal definitions of projective spaces. They added new elements called points at infinity to extend existing Euclidean and affine spaces. This allowed the math to handle directions that seem to go on forever. This expansion led to the modern term "projective transformation."

In algebra, a homography can be expressed using a nonsingular matrix. This matrix uses homogeneous coordinates to represent points. If a projective space has a dimension of n, the vector space used to build it has a dimension of n plus one. A point in the projective space is represented by the coordinates of any nonzero point on its corresponding line. Because of this, the matrix is defined up to multiplication by a nonzero element of the field.

On a projective line, these mappings have special names. They are often called homographic functions or linear fractional transformations. In the specific case of the complex projective line, which is identified with the Riemann sphere, they are called Möbius transformations. These transformations are unique because they are conformal, meaning they preserve angles, and they preserve orientation. This makes them very important in complex analysis.

Another way to view homographies is through central collineations. A central collineation is a bijection that has a fixed axis and a fixed center. The axis is a hyperplane where every point stays in its original place. The center is a point where every line passing through it is mapped to itself. There are two distinct types: homologies and elations. A homology occurs when the center is not part of the axis. An elation occurs when the center is incident with the axis.

Finally, homographies are deeply connected to the concept of a projective frame. A frame is an ordered set of points where no hyperplane contains all of them. This frame acts like a coordinate system for the space. A very important rule in projective geometry is that there is exactly one homography that can map one specific frame onto another. This connection ensures that the geometric structure remains consistent and predictable across different mathematical systems.