Syzygies for translational surfaces
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.
scientific article; zbMATH DE number 6881229
scientific article