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.
Be a projective space. A bijective mapping is called projectivity if and only if
In short, is a double ratio loyal collineation.
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 ".