Detecting Affine Equivalences and Symmetries of Implicit Plane Algebraic Curves


Alcázar J. G. , Gözütok U., Çoban H. A. , Hermoso C.

9th International Congress on Fundamental and Applied Sciences 2022 (ICFAS2022) , İstanbul, Turkey, 28 - 30 June 2022, vol.1, pp.115

  • Publication Type: Conference Paper / Summary Text
  • Volume: 1
  • City: İstanbul
  • Country: Turkey
  • Page Numbers: pp.115

Abstract

We present a computational method to detect affine equivalences of implicit plane algebraic curves. The method proceeds by first reducing the problem to the case of affine equivalences where the orgin is fixed, using the Hessian curves of the input curves. Then, the Hessian is again used to derive certain differential invariants that allow us to compute the components of the equivalences by just using resultants and polynomial factoring. In special cases, for instance curves with infinitely many self-affine equivalences, equivalences are computed using a more direct method involving polynomial system solving, but with polynomials of smaller degree. A comparison of the algorithm with other methods, including brute force, is also given.