Computing the symmetries of a ruled rational surface

TL;DR

This paper introduces a direct parametric method to compute all symmetries of rational ruled surfaces using polynomial system solving, avoiding implicit equation computation, and extends to certain implicit surfaces.

Contribution

It presents a novel parametric approach for symmetry detection in rational ruled surfaces, simplifying the process by working directly in the parameter space.

Findings

Method effectively computes all symmetries including rotations.

Simplifies symmetry detection by avoiding implicit surface equations.

Applicable to certain implicit algebraic surfaces under specific conditions.

Abstract

We present a method for computing all the symmetries of a rational ruled surface defined by a rational parametrization which works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method proceeds by translating the problem into the parameter space, and relies on polynomial system solving. If we want all the symmetries of the surface, including rotational symmetries, we need to deal with polynomial systems in four variables; if we are only interested in involutions (e.g. central symmetries, axial symmetries, reflections in a plane), we can come down to bivariate polynomial systems. An application to compute symmetries of an implicit algebraic surface under certain conditions is also provided.