Symmetries and Similarities of Rational Plane Algebraic Curves in Complex Representation

9th International Congress on Fundamental and Applied Sciences 2022 (ICFAS2022), İstanbul, Türkiye, 28 - 30 Haziran 2022, cilt.1, ss.116

This study is devoted to investigate the problem of detecting symmetries and similarities of

rational plane algebraic curves given in complex representation. Unlike other studies

addressing the same problem, we provide a different approach which uses complex

differential invariants of the input curves. Our method takes advantage of the complex

representation 𝑧(t)=x(t)+iy(t) of a rational plane curve 𝑪 and the fact that the curves 𝑪𝟏

and 𝑪𝟐 properly parametrized by 𝑧_1(t) and 𝑧_2(t) are similar if and only if there exist complex

numbers 𝑎,𝑏 and a Möbius transformation 𝜑 such that 𝑎𝑧_1(t)+b=z_2(𝜑(t)). In order to

determine 𝑎 and 𝑏, we first determine the Möbius transformation 𝜑. It can be easily done by

using an rational complex differential invariant function 𝐼(z) defined on a curve 𝑪 properly

parametrized by 𝑧. Finally the symmetries of a curve 𝑪 can be determined by the same setup

by taking |𝑎| = 1 and 𝑪𝟏 =𝑪2.