Computation of symmetries of rational surfaces

TL;DR

This paper introduces symbolic algorithms for computing symmetries of rational surfaces, including specialized methods for ruled surfaces, with implementations demonstrating practical efficiency in computer algebra systems.

Contribution

The paper presents novel symbolic algorithms for symmetry detection in rational and ruled surfaces, leveraging differential invariants and properties of lines of striction.

Findings

Algorithms work well for sparse parametrizations and PN surfaces.

The ruled surface algorithm reduces to symmetry analysis of rational space curves.

Implementations in Maple are publicly available and demonstrate good performance.

Abstract

In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the algorithm works well for sparse parametrizations (e.g. toric surfaces) and PN surfaces. Additionally, we provide a specific, also symbolic algorithm for computing the symmetries of ruled surfaces; this algorithm works extremely well in practice, since the problem is reduced to that of rational space curves, which can be efficiently solved by using existing methods. The algorithm for ruled surfaces is based on the fact, proven in the paper, that every symmetry of a rational surface must also be a symmetry of its line of striction, which is a rational space curve. The algorithms have been implemented in the computer algebra system Maple, and the implementations have been made public; evidence of their performance is given in the paper.