A new method to detect projective equivalences and symmetries of rational $3D$ curves

TL;DR

This paper introduces a differential invariant-based method to efficiently detect projective equivalences and symmetries of rational 3D curves, avoiding complex polynomial system solving.

Contribution

The authors develop a novel approach using differential invariants and gcd computations to identify projective equivalences, improving efficiency over previous methods.

Findings

Algorithm implemented in Maple demonstrates high efficiency.

Comparison shows advantages over previous approaches.

Method effectively detects symmetries and equivalences in rational 3D curves.

Abstract

We present a new approach using differential invariants to detect projective equivalences and symmetries between two rational parametric $3D$ curves properly parametrized. In order to do this, we introduce two differential invariants that commute with M\"obius transformations, which are the transformations in the parameter space associated with the projective equivalences between the curves. The M\"obius transformations are found by first computing the gcd of two polynomials built from the differential invariants, and then searching for the M\"obius-like factors of this gcd. The projective equivalences themselves are easily computed from the M\"obius transformations. In particular, and unlike previous approaches, we avoid solving big polynomial systems. The algorithm has been implemented in Maple, and evidences of its efficiency as well as a comparison with previous approaches are given.