Translational surfaces and iterated resultants

TL;DR

This paper introduces an alternative method using iterated homogeneous resultants to compute the implicit equations of translational surfaces, improving efficiency and applicability over previous syzygy-based approaches.

Contribution

The authors propose a new algorithm employing iterated homogeneous resultants for implicitization of translational surfaces, addressing limitations of prior methods with syzygies.

Findings

The new method often produces smaller Sylvester matrices, leading to faster computation.

It succeeds in cases where previous syzygy-based methods fail due to ill-behaved basepoints.

The approach broadens the applicability of implicitization techniques for translational surfaces.

Abstract

A translational surface is a tensor product surface constructed from two space curves by translating one along the other. These surfaces are common within geometric modeling and, since their description is parametric, it is desirable to obtain the implicit equation of such a surface. These surfaces have been studied thoroughly by Goldman and Wang, where a particular set of syzygies was identified and shown to yield the implicit equation through an inhomogeneous resultant. As this method may fail in the presence of ill-behaved basepoints of the parameterization, we offer an alternative method in this article using iterated homogeneous resultants. The algorithm presented here involves smaller Sylvester matrices overall, potentially resulting in faster computation, and succeeds in many instances where the previous method cannot be applied.