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 E2 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.

Recognition of Paths and Curves in the 2-Dimensional Euclidean Geometry

“…Then every parametrization ξ ∈ Φ is a completely degenerate regular path or a non-degenerate path.Proof. It is similar to the proof of[15, Proposition 5.3]. Let Φ and Ψ are completely degenerate regular curves such that L Φ = (−∞, +∞), L Ψ = (−∞, +∞) and ξ ∈ P Φ , η ∈ P Ψ are invariant parametrizations.…”

“…(see [13,14]). In the present paper, by developing used method in the previous paper [15], for the group of Euclidean transformations in E 2 , equivalence problems, existence and rigidity theorems for regular, completely regular and non-degenerate paths and curves are given. For given two paths and curves with the common differential G-invariants, we obtain, for the first time,evident forms of all Euclidean transformations that maps one of the paths and curves to the other.…”

“…8) for all t ∈ I,resp.Proof. A proof of this theorem is obtained from Proposition 4.1 and[15, Theorem 3.6]. Let ξ(t) and η(t) be two non-degenerate I-paths in E 2 .…”

“…4.7) and(4.8),resp.Proof. A proof of this theorem is obtained from Proposition 4.1 and[15, Theorem 3.7]. (see[1, p.31],[20, p.21]) The function κ…”

Recognition of complex polynomial Bezier curves under similarity transformations

“…For only similarity group Sim(2) and linear similarity group LSim(2), the problems of equivalence two Bézier curves of degree m are investigated in [18]. For orthogonal group O(2), special orthogonal group O + (2), linear similarity group LSim(2) and orientation linear similarity group LSim + (2), the conditions of the global G-equivalence of two regular paths are given in [10,21].…”