Classification of Smooth Alignable Voss Surfaces

TL;DR

This paper classifies a special class of surfaces called alignable Voss surfaces, revealing their structure, explicit formulas, and surprising properties, thus advancing understanding of these geometrically significant surfaces.

Contribution

It provides a coordinate-free classification of alignable Voss surfaces, introduces explicit immersion formulas, and uncovers new properties including counterexamples to previous classifications.

Findings

Surfaces split into two classes with two-parameter families each.

One class admits an isothermal-conjugate geodesic net, countering earlier claims.

Explicit immersion formulas depend on deformation parameters.

Abstract

Alignable nets are grid structures that can collapse to a planar strip, which is in fact the real-world counterpart of a curve. This property simplifies on-site assembly and enables compact transport and storage. These grid structures can then be deployed by a scissor motion at each vertex in a desired location. In this article, we classify all surfaces supporting an alignable net that additionally have the geodesic and conjugate net property, namely, the alignable Voss surfaces. In doing so, we use Cartan's theory of moving-frames and we obtain a coordinate-free classification of these surfaces. In the next step we express our findings in local coordinates and at the level of the fundamental forms. We show that the alignable Voss surfaces consist of two classes where each in turn consists of two two-parameter families of surfaces. A surprising feature of one of these classes is that they admit an isothermal-conjugate geodesic net, thereby providing a counterexample to Eisenhart's earlier classification claim for Voss surfaces of this type. Finally, we derive explicit immersion formulas for one of the classes as functions of the deformation and alignability parameters. Additionally, we show that, upon disregarding certain singularities, the above immersions of alignable Voss surfaces give rise to infinitely many explicit immersions of other Voss surfaces still depending on the deformation parameter. Since explicit immersion formulas for Voss surfaces that include the deformation parameter are seldom obtainable, this provides a rare result in the literature. Finally, we examine several notable subclasses in detail, including the well known example of infinitely many geodesic-conjugate nets on a helicoid, and we give a kinematical explanation for why this phenomenon appears in computations.