Translation Surfaces and Isotropic Transport Nets on Rational Minimal Surfaces

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

“…Let K be a field and K n the corresponding n-dimensional affine space over K. In this paper we consider K = R, C. Following [19], we equip K n with the binary operation ⊕ defined by taking the average, i.e.,…”

“…Surfaces of translation, also called translational surfaces (c.f. [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].…”

“…Minimal surfaces (c.f. [19,16] and [11,Chapters 16 and 22]) are surfaces whose mean curvature is identically zero. It was already known by Sophus Lie that such surfaces are also surfaces of translation generated by complex conjugated curves.…”

Syzygies for translational surfaces

“…The collection of translational surfaces increases if we allow the generating curves to be complex curves. For example, Vršek and Lávička (2016) generate the Enneper surface from two complex curves.…”

“…Translational surfaces defined by h * (s; t) = f * (s) + g * (t) are not translation invariant: translating both curves f * and g * by the vector v translates the surface h * by the vector 2v. One would like to define translational surfaces in such a way that translating the two generating curves by the same vector v, also translates every point on the surface by the vector v. Recently, Vršek and Lávička (2016) offer an alternative definition of translational surfaces given by the rational parametric representation h * (s; t) = f * (s)+g * (t)…”