Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of Smooth Alignable Voss Surfaces

Arvin Rasoulzadeh

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.

Classification of Smooth Alignable Voss Surfaces

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.
Paper Structure (26 sections, 36 theorems, 153 equations, 7 figures)

This paper contains 26 sections, 36 theorems, 153 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Review on alignable surfaces
  3. Review on Voss surfaces
  4. Related works to this article:
  5. Other classical strands
  6. Contemporary works:
  7. Contribution and outline of the paper
  8. Preliminary
  9. Isometric Deformations
  10. Infinitesimal Isometric Deformation and its Rotation field
  11. V-surfaces as Rotation Fields of K-surfaces
  12. V-surfaces & V-nets
  13. Isometric Deformation of V-surfaces vs. Spectral Deformation of K-surfaces
  14. Identification of the Reciprocal-parallel K-surfaces
  15. Bour Frames, Bour Family & Bour Isometries
  16. ...and 11 more sections

Key Result

Theorem 2.6

For every infinitesimal isometric deformation $\xi$ of an immersion $\psi$ there is a unique pair of vector fields $\eta, \zeta \colon I \times J \to \mathbb R^3$ such that the following equations are satisfied:

Figures (7)

  • Figure 1: Physical prototype of an alignable net shown in its deployment sequence. Image courtesy of Davide Pellis (see AlignablePellis).
  • Figure 2: Illustration of Chebyshev, positive-alignable and negative-alignable nets in the parameter domain $\Omega$.
  • Figure 3: Illustration of the alignable deformation of alignable V-nets of the first kind with $\lambda = 1$, of negative-alignable type (a,e,i) and its corresponding catenoid (b,f,j), the positive-alignable type (c,g,k) along with the corresponding K-nets of revolution (d,h,l).
  • Figure 4: Illustration of the isometric deformation of alignable V-nets of the first kind with $k = 1$, of negative-alignable (a,e,i), positive-alignable (b,f,j) and linear combination (c,g,k) along with the corresponding spectral deformation of the K-net of the parabolic pseudosphere (d,h,l).
  • Figure 5: Bour family diagram for extracting the formulas of immersion of negative-alignable V-nets of the first kind.
  • ...and 2 more figures

Theorems & Definitions (90)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 2.8
  • Proposition 2.9
  • Definition 3.1
  • Remark 3.2
  • ...and 80 more