﻿ Homography

# Homography

As projectivity a structure-preserving, bijective mapping of a projective space is referred to themselves, ie an automorphism of a projective space in geometry and linear algebra. It is characteristic of a projectivity that she is faithful and true double ratio.

In linear algebra specially -dimensional projective space on bodies to be examined. Here projectivities can be represented ( relative to a fixed selected projective coordinate system) by a mapping matrix. This representation is also sometimes used in linear algebra on the definition of " projectivity ". The amount of these projectivities forms with respect to the sequential execution of a group, the projective linear group is a factor group of the general linear group. These groups are " General linear group " described in the main article.

In the synthetic geometry weaker conditions are applied to the projective space provided, so that projective spaces are considered, which do not belong to a vector space (on these conceptions the main article " Projective Geometry" ). Particularly projective planes are investigated. Because here it is clear from the outset what is meant by a double ratio, usually collineations are examined instead of projectivity, of which only straight fidelity is required. A collineation of any projective plane is called a projectivity if it can be represented as a composition of Perspektivitäten. In all cases the projectivities form a subgroup, and even a normal subgroup of the group of collineations.

## Definitions

### Generally

Be a projective space. A bijective mapping is called projectivity if and only if

In short, is a double ratio loyal collineation.

### Linear Algebra

Here the projective space (possibly up to isomorphism ), it can therefore be assigned as a vector space coordinate space. The general definition is still valid, but can be replaced by more manageable descriptions, a bijective mapping is called projectivity if one of the following equivalent conditions is true: The Figure

• Is a projective mapping,
• Is with respect to a projective coordinate system, a view in which the coordinate vectors of the points are mapped ( in ) linearly to the pixel coordinate vectors.

In the first condition, the projective transformation is of course a bijective self-map, in the second condition is the required linear self-map of the coordinate vector space - also by the bijectivity of required - even a Vektorraumautomorphismus. This, however, is determined only up to a " stretching ".

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