Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Translational surfaces and iterated resultants

Matthew Weaver

TL;DR

This article offers an alternative method using iterated homogeneous resultants that involves smaller Sylvester matrices overall, potentially resulting in faster computation, and succeeds in many instances where the previous method cannot be applied.

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.

Translational surfaces and iterated resultants

TL;DR

This article offers an alternative method using iterated homogeneous resultants that involves smaller Sylvester matrices overall, potentially resulting in faster computation, and succeeds in many instances where the previous method cannot be applied.

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.
Paper Structure (10 sections, 17 theorems, 38 equations)

This paper contains 10 sections, 17 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Properties of translational surfaces
  3. Translational surfaces and syzygies
  4. Moving planes and surfaces
  5. Basepoints
  6. Iterated resultants and translational surfaces
  7. Iterated resultants
  8. Planar curves and ruled surfaces
  9. Ruled surfaces
  10. Planar curves

Key Result

Lemma 2.1

With ${\bf f}$ and ${\bf h}$ above, we have the following.

Theorems & Definitions (38)

  • Lemma 2.1: GW18syz
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5: GW18syz
  • Lemma 2.6: GW18syz
  • Proposition 2.7
  • proof
  • Proposition 3.3
  • proof
  • ...and 28 more