In this paper, global differential G-invariants of paths in the two-dimensional Euclidean space E-2 for the similarity group G = Sim(E-2) and the orientation-preserving similarity group G = Sim(+)(E-2) are investigated. A general form of a path in terms of its global G-invariants is obtained. For given two paths xi(t) and eta(t) with the common differential G-invariants, general forms of all transformations g is an element of G, carrying xi(t) to eta(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. (C) 2020 Elsevier B.V. All rights reserved.