<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, for the orthogonal group <italic>G</italic> = <italic>O</italic>(2) and special orthogonal group <italic>G</italic> = <italic>O</italic>+(2) global <italic>G</italic>-invariants of plane paths and plane curves in two-dimensional Euclidean space <italic>E</italic>2 are studied. Using complex numbers, a method to detect <italic>G</italic>-equivalences of plane paths in terms of the global <italic>G</italic>-invariants of a plane path is presented. General evident form of a plane path with the given <italic>G</italic>-invariants are obtained. For given two plane paths <italic>x</italic>(<italic>t</italic>) and <italic>y</italic>(<italic>t</italic>) with the common <italic>G</italic>-invariants, evident forms of all transformations <italic>g</italic> ∈ <italic>G</italic>, carrying <italic>x</italic>(<italic>t</italic>) to <italic>y</italic>(<italic>t</italic>), are obtained. Similar results have obtained for plane curves.</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.

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

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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...

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

Published Online: Sep 26, 2019

Page range: 37 - 65

Received: Apr 06, 2018

Accepted: Jun 30, 2018

DOI: https://doi.org/10.2478/auom-2019-0018

Keywords
Curve, Invariant, Euclidean geometry

© 2019 Djavvat Khadjiev et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Published Online: Sep 26, 2019

Page range: 37 - 65

Received: Apr 06, 2018

Accepted: Jun 30, 2018

DOI: https://doi.org/10.2478/auom-2019-0018

KeywordsCurve, Invariant, Euclidean geometry

© 2019 Djavvat Khadjiev et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Curve, Invariant, Euclidean geometry