LAUSR

Abstract In this paper, global differential G -invariants of paths in the two-dimensional Euclidean space E 2 for the similarity group G = S i m ( E 2 ) and the orientation-preserving similarity group G = S i m + ( E 2 ) are investigated. A general form of a path in terms of its global G -invariants is obtained. For given two paths ξ ( t ) and η ( t ) with the common differential G -invariants, general forms of all transformations g ∈ G , carrying ξ ( t ) to η ( t ) , are found. Similar results are given for curves. Moreover, analogous of the similarity groups in the three-dimensional space–time and in the four-dimensional space–time-mass are defined. Finally, applications to Newtonian mechanics of the above results are given.