9th International Congress on Fundamental and Applied Sciences 2022 (ICFAS2022), İstanbul, Turkey, 28 - 30 June 2022, vol.1, pp.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.