Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

## Transformations in Affine Spaces

Henryk Oryszczyszyn
Warsaw University, Bialystok
Krzysztof Prazmowski
Warsaw University, Bialystok

### Summary.

Two classes of bijections of its point universe are correlated with every affine structure. The first class consists of the transformations, called formal isometries, which map every segment onto congruent segment, the second class consists of the automorphisms of such a structure. Each of these two classes of bijections forms a group for a given affine structure, if it satisfies a very weak axiom system (models of these axioms are called congruence spaces); formal isometries form a normal subgroup in the group of automorphism. In particular ordered affine spaces and affine spaces are congruence spaces; therefore formal isometries of these structures can be considered. They are called positive dilatations and dilatations, resp. For convenience the class of negative dilatations, transformations which map every ``vector" onto parallel ``vector", but with opposite sense, is singled out. The class of translations is distinguished as well. Basic facts concerning all these types of transformations are established, like rigidity, decomposition principle, introductory group-theoretical properties. At the end collineations of affine spaces and their properties are investigated; for affine planes it is proved that the class of collineations coincides with the class of bijections preserving lines.

Supported by RPBP.III-24.C2.

#### MML Identifier: TRANSGEO

The terminology and notation used in this paper have been introduced in the following articles           

Contents (PDF format)

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