In this paper, for the orthogonal group G = O(2) and special orthogonal group G = O+ (2) global G-invariants of plane paths and plane curves in two-dimensional Euclidean space E-2 are studied. Using complex numbers, a method to detect G-equivalences of plane paths in terms of the global G-invariants of a plane path is presented. General evident form of a plane path with the given G-invariants are obtained. For given two plane paths x(t) and y(t) with the common G-invariants, evident forms of all transformations g is an element of G, carrying x(t) to y(t), are obtained. Similar results have obtained for plane curves.