A new method to detect projective equivalences and symmetries of rational [formula omitted] curves

Abstract

We present a new approach to detect projective equivalences and symmetries between two rational parametric 3D curves properly parametrized. In order to do this, we introduce two rational functions that behave nicely for Möbius transformations, which are the transformations in the parameter space associated with the projective equivalences between the curves. The Möbius transformations are found by first computing the gcd of two polynomials built from these two functions, and then searching for a special type of factors, “Möbius-like”, of this gcd. The projective equivalences themselves are easily computed from the Möbius transformations. In particular, and unlike previous approaches, we avoid solving big polynomial systems. The algorithm has been implemented in Maple™ (2021), and evidences of its efficiency as well as a comparison with previous approaches are given.