K Let E-2 be the 2-dimensional Euclidean space, LSim(2) be the group of all linear similarities of E-2 and LSim(+)(2) be the group of all orientation-preserving linear similarities of E-2. The present paper is devoted to solutions of problems of global G-equivalence of paths and curves in E-2 for the groups G = LSim(2), LSim(+)(2). Complete systems of global G-invariants of a path and a curve in E-2 are obtained. Existence and uniqueness theorems are given. Evident forms of a path and a curve with the given global invariants are obtained.