Efficient Detection of Symmetries of Polynomially Parametrized Curves

TL;DR

This paper introduces fast algorithms for detecting central and mirror symmetries in algebraic curves defined by polynomial parametrizations, enabling quick analysis even for high-degree curves.

Contribution

The paper presents novel, efficient algorithms based on linear relationships between polynomial parametrizations for symmetry detection in algebraic curves.

Findings

Algorithms can analyze high-degree polynomial curves in seconds

Provides theoretical results on polynomial curve symmetries

Offers closed-form formulas for symmetry centers and axes

Abstract

We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on the existence of a linear relationship between two proper polynomial parametrizations of the curve, which leads to a triangular polynomial system (with complex unknowns) that can be solved in a very fast way; in particular, curves parametrized by polynomials of serious degrees can be analyzed in a few seconds. In our analysis we provide a good number of theoretical results on symmetries of polynomial curves, algorithms for detecting rotation and mirror symmetry, and closed formulae to determine the symmetry center and the symmetry axis, when they exist. A complexity analysis of the algorithms is also given.