Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multi-dimensional consistency of principal binets

Niklas C. Affolter, Jan Techter

Abstract

Principal binets are a discretization of curvature line parametrized surfaces defined on the vertices and faces of the square lattice $\Z^2$. They generalize the previously established discretizations given by circular nets, conical nets, and principal contact element nets. We show that principal binets constitute a discrete integrable system in the sense of multi-dimensional consistency. In particular, they generalize to higher-dimensional square lattices $\Z^N$. We also discuss relations to the notion of discrete orthogonal coordinate systems as previously established for discrete confocal quadrics.

Multi-dimensional consistency of principal binets

Abstract

Principal binets are a discretization of curvature line parametrized surfaces defined on the vertices and faces of the square lattice . They generalize the previously established discretizations given by circular nets, conical nets, and principal contact element nets. We show that principal binets constitute a discrete integrable system in the sense of multi-dimensional consistency. In particular, they generalize to higher-dimensional square lattices . We also discuss relations to the notion of discrete orthogonal coordinate systems as previously established for discrete confocal quadrics.
Paper Structure (13 sections, 18 theorems, 43 equations, 10 figures)

This paper contains 13 sections, 18 theorems, 43 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Conjugate binets
  3. Technical Preliminaries
  4. Correspondence between conjugate face-nets and conjugate vertex-nets
  5. Multi-dimensional consistency
  6. Polar conjugate binets
  7. Principal binets
  8. Circular, conical and principal contact element nets
  9. Circular nets and Ribaucour transformations
  10. Conical nets as a consistent reduction
  11. Principal contact element nets and line complexes
  12. Discrete orthogonal coordinate systems
  13. Smooth orthogonal coordinate systems

Key Result

Theorem 1.2

Let $1 \leq i < j \leq N$. For every conjugate face-net defined on $F_N$ the restriction to $F_N^{ij}$ is a conjugate vertex-net. Conversely, every conjugate vertex-net defined on $F_N^{ij}$ is the restriction of a unique conjugate face-net defined on $F_N$.

Figures (10)

  • Figure 1: A (two-dimensional) binet is a map defined on both the vertices and faces of $\mathbb{Z}^2$.
  • Figure 2: Combinatorics for the conditions of conjugate vertex-nets and conjugate face-nets in $\mathbb{Z}^3$. Left: A face $f \in F_3$ has four incident vertices. The corresponding points of a conjugate vertex-net are coplanar. Right: A vertex $v \in V_3$ has twelve incident faces. The corresponding points of a conjugate face-net are coplanar.
  • Figure 3: Identification of $F_3^{12}$ with $\mathbb{Z}^3$. In blue two layers of $12$-faces of $\mathbb{Z}^3$. In red $F_3^{12} \simeq \mathbb{Z}^3$. The dashed edges are the missing edges for the identification with $\mathbb{Z}^3$.
  • Figure 4: The two types of crosses in $\mathbb{Z}^3$. This amounts to a total of six crosses per edge of $\mathbb{Z}^3$.
  • Figure 5: Different face types of $F_N^{12} \cong V_N$. Labels in the figure correspond to the notation used in the proof of Proposition \ref{['prop:conjugatefacetoconjugate']}. Left: $12$-face of $F_N^{12}$. Middle: $13$-face of $F_N^{12}$ with an associated $23$-face of $\mathbb{Z}^N$. Right: $34$-face of $F_N^{12}$ with a surrounding 4-cube of $\mathbb{Z}^N$.
  • ...and 5 more figures

Theorems & Definitions (51)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 41 more