@MISC{Rylov08differentrepresentations, author = {Yuri A. Rylov}, title = {Different representations of Euclidean geometry}, year = {2008} }

Share

OpenURL

Abstract

Three different representation of the proper Euclidean geometry are considered. They differ in the number of basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. V-representation contains two basic elements (point, vector) and additional structure: linear vector space. In σ-representation there is only one basic element and additional structure: world function σ = ρ 2 /2, where ρ is the distance. The concept of distance appears in all representations. However, as a structure, determining the geometry, the distance appears only in the σ-representation. The σ-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors Q0Q1,Q0Q ′ 1,..., which are equal to the vector P0P1, but they are not equal between themselves, in general.