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

Contents (PDF format)

Bibliography

[1] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[2] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.

[3] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.

[4] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.

[5] Henryk Oryszczyszyn and Krzysztof Prazmowski. Analytical ordered affine spaces. Journal of Formalized Mathematics, 2, 1990.

[6] Henryk Oryszczyszyn and Krzysztof Prazmowski. Ordered affine spaces defined in terms of directed parallelity --- part I. Journal of Formalized Mathematics, 2, 1990.

[7] Henryk Oryszczyszyn and Krzysztof Prazmowski. Parallelity and lines in affine spaces. Journal of Formalized Mathematics, 2, 1990.

[8] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[9] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[10] Edmund Woronowicz. Relations defined on sets. Journal of Formalized Mathematics, 1, 1989.

[11] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Journal of Formalized Mathematics, 1, 1989.