# Affine equivalences, isometries and symmetries of ruled rational surfaces

## TL;DR

This paper introduces a direct parametric method for computing affine equivalences, isometries, and symmetries of rational ruled surfaces, avoiding implicit equations and leveraging polynomial system solving in parameter space.

## Contribution

It extends existing techniques to compute all affine and isometric symmetries of rational ruled surfaces directly in parametric form, improving efficiency and applicability.

## Key findings

- Method successfully computes affine equivalences between surfaces.
- Algorithm effectively finds isometries and symmetries of rational ruled surfaces.
- Demonstrated efficiency through multiple examples.

## Abstract

A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method translates the problem into parameter space and relies on polynomial system solving. Geometrically, the problem is related to finding the projective equivalences between two projective curves (corresponding to the directions of the rulings of the surfaces). This problem was recently addressed in a paper by Hauer and J\"uttler, and we exploit the ideas by these authors in the algorithm presented in this paper. The general idea is adapted to computing the isometries between two rational ruled surfaces, and the symmetries of a given rational ruled surface. The efficiency of the method is shown through several examples.