Global invariants of paths and curves for the group of orthogonal transformations in the two-dimensional Euclidean space

Abstract 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 ∈ G , carrying x ( t ) to y ( t ), are obtained. Similar results have obtained for plane curves.

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