Syzygies for translational surfaces

Abstract

Implicit equationA translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces are invariant under rigid motions: translating and rotating the two generating curves translates and rotates the translational surface by the same amount. We construct three special syzygies for a translational surface from a µ-basis of one of the generating space curves, and we show how to compute the implicit equation of a translational surface from these three special syzygies. Examples are provided to illustrate our theorems and flesh out our algorithms.

Affine equivalences of surfaces of translation and minimal surfaces, and applications to symmetry detection and design

“…[19]) are surfaces generated by sliding one space curve along another space curve. Due to their simplicity, these surfaces are used in Computer-Aided Geometric Design, and efficient algorithms for computing µ-bases and implicitization are known [20], [21].…”