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Isothermic nets with spherical parameter lines from discrete holomorphic maps

Tim Hoffmann, Gudrun Szewieczek

Abstract

We prove that all discrete isothermic nets with a family of planar or spherical lines of curvature can be obtained from special discrete holomorphic maps via lifted-folding. This novel approach is a generalization and discretization of a classical method to create planar curvature lines on smooth surfaces. In particular, this technique provides an efficient way to construct discrete isothermic topological tori composed of fundamental pieces from discrete periodic holomorphic maps.

Isothermic nets with spherical parameter lines from discrete holomorphic maps

Abstract

We prove that all discrete isothermic nets with a family of planar or spherical lines of curvature can be obtained from special discrete holomorphic maps via lifted-folding. This novel approach is a generalization and discretization of a classical method to create planar curvature lines on smooth surfaces. In particular, this technique provides an efficient way to construct discrete isothermic topological tori composed of fundamental pieces from discrete periodic holomorphic maps.
Paper Structure (14 sections, 20 theorems, 62 equations, 14 figures)

This paper contains 14 sections, 20 theorems, 62 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Discrete planar and spherical curvature lines
  3. Ribaucour transformations of discrete curves
  4. Lifted-folding
  5. Spherical curvature lines and 2-dimensional circular nets
  6. Discrete isothermic surfaces with a family of spherical curvature lines
  7. Lifted-foldings of isothermic nets
  8. Darboux transforms of $(\mathfrak{m})$-type
  9. Discrete holomorphic maps of $(M)$-type and constrained elastic curves.
  10. Isothermic nets with a family of planar lines of curvature
  11. Special solutions: quasi-periodicity, symmetries and closedness
  12. (Quasi-)periodic discrete holomorphic maps related to isothermic nets with spherical curvature lines
  13. Discrete isothermic topological cylinders and tori with symmetries
  14. Light cone model

Key Result

Proposition 3

Let $f$ and $g$ be two discrete spherical curves with curve points lying on two distinct spheres $s^1, s^2 \in \mathbb{P}(\mathcal{L})$, that is, If $(f, g)$ is a discrete Ribaucour pair, then all elements in the 3-dimensional subspace provide fixed points for all M-inversions in the R-evolution map.

Figures (14)

  • Figure 1: A discrete holomorphic map with circular folding axes and two discrete isothermic nets with a family of spherical curvature lines obtained via lifted-folding.
  • Figure 2: Left. A planar sequence of $(\mathfrak{m}^{ij})$-type Ribaucour transformed discrete curves together with their folding axes (dashed lines). Middle and Right. The nets in 3-space with planar curvature lines are generated from the planar net via lifted-foldings. The planes are rotated around their corresponding folding axes.
  • Figure 3: Circular nets with a family of spherical lines of curvature obtained from a planar circular net of $(M)$-type via lifted-foldings.
  • Figure 4: Discrete Joachimsthal surfaces generated via lifted-folding from discrete orthogonal trajectories of a 2-dimensional discrete cyclic system discrete_cyclic.
  • Figure 5: Iterative construction of a discrete curve (blue) with two Darboux evolution maps of $(\mathfrak{m}_1)$-type and $(\mathfrak{m}_2)$-type (see Construction 2).
  • ...and 9 more figures

Theorems & Definitions (45)

  • Definition 1
  • proof
  • Proposition 3
  • proof
  • Definition 4
  • Corollary 5
  • proof
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 35 more