Syzygies for translational surfaces
type: Item
instance of: scholarly article
title: Syzygies for translational surfaces
zbMATH Open document ID: 1404.14069
published in: Journal of Symbolic Computation
publication date: 2018-06-06T00:00:00Z
full work available at URL: http://hdl.handle.net/1911/103240
review text: The subject of this paper concerns the study of translational surfaces and their syzygies. The interest for translation surfaces started to grow, when it turned out that these surfaces are typical modeling surfaces in architecture and computer aided design. A translation surface, formally speaking, is a rational tensor product surface, which is generated from two rational space curves by translating one curve, along the other one. Note, that translational surfaces are invariant under rigid motions: translating and rotating the two generating curves, translates and rotates the translational surface by the same amount. In this paper, in Section 2, some background concerning the translational surfaces is given: their main properties are recalled and \(\mu\)-bases for rational curves and surfaces are considered. Further investigations lead, on the one hand, to the study of the syzygies of translational surfaces; on the other hand, to consider the relation of the syzygies of the generating curves to the syzygies of the corresponding translational surface. Finally, the construction of three special syzygies for a translational surface is given (this is done from a \(\mu\)-basis of one of the generating space curves). This motivates the main result of this paper (Section 3): it is shown how to compute the implicit equation of a translational surface, using those three special syzygies. As a an application of this result, it is shown that the techniques, used and developed, in this paper can be applied to translational surfaces of the following type: \[ h^{\star} (s;t)= af^{*}(s) + bg^{*}(t), \] where \(a,b \in\mathbb{R}\) and \(ab \neq 0\), up to some minor modifications. Throughout this paper, examples are provided to illustrate the theorems and to flesh out algorithms. In the last section (i.e. Section 5) two open problems for future research are proposed.
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 65D17
zbMATH DE Number: 6881229
zbMATH Keywords: translational surface
zbMATH Keywords: syzygy
zbMATH Keywords: \(\mu\)-basis
zbMATH Keywords: implicit equation
author: Haohao Wang
author: Ron Goldman
reviewed by: Noémie Combe
describes a project that uses: SINGULAR
describes a project that uses: Maple
describes a project that uses: Macaulay2
MaRDI profile type: Publication
OpenAlex ID: W2770465300
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DOI: 10.1016/J.JSC.2017.11.004
description: scientific article; zbMATH DE number 6881229