On the computation of the straight lines contained in a rational surface

TL;DR

This paper introduces an algorithm to compute all straight lines on a rational surface using differential geometry principles, implemented in Maple, and compares its efficiency with brute-force methods.

Contribution

The paper presents a novel algorithm for finding straight lines on rational surfaces based on differential geometry, with an implementation and performance comparison.

Findings

The algorithm successfully computes straight lines on rational surfaces.

It outperforms brute-force approaches in efficiency.

Implementation in Maple demonstrates practical applicability.

Abstract

In this paper we present an algorithm to compute the (real and complex) straight lines contained in a rational surface, defined by a rational parameterization. The algorithm relies on the well-known theorem of Differential Geometry that characterizes real straight lines contained in a surface as curves that are simultaneously asymptotic lines, and geodesics. We also report on an implementation carried out in Maple 18, and we compare the behavior of our algorithm with two brute-force approaches.