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Discrete Koenigs nets, inscribed quadrics and autoconjugate curves

Niklas Christoph Affolter, Alexander Yves Fairley

Abstract

Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and Fairley that Koenigs nets can be characterized by the existence of touching inscribed conics. We generalize the touching inscribed conics by showing the existence of higher-dimensional inscribed quadrics for Koenigs nets. Additionally, we study Koenigs d-grids, which are Koenigs nets with parameter lines that are contained in d-dimensional subspaces. We show that Koenigs d-grids have a remarkable global property: there is a special inscribed quadric that all parameter spaces are tangent to. Finally, we establish a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.

Discrete Koenigs nets, inscribed quadrics and autoconjugate curves

Abstract

Discrete Koenigs nets are a special class of discrete surfaces that play a fundamental role in discrete differential geometry, in particular in the study of discrete isothermic and minimal surfaces. Recently, it was shown by Bobenko and Fairley that Koenigs nets can be characterized by the existence of touching inscribed conics. We generalize the touching inscribed conics by showing the existence of higher-dimensional inscribed quadrics for Koenigs nets. Additionally, we study Koenigs d-grids, which are Koenigs nets with parameter lines that are contained in d-dimensional subspaces. We show that Koenigs d-grids have a remarkable global property: there is a special inscribed quadric that all parameter spaces are tangent to. Finally, we establish a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.

Paper Structure

This paper contains 29 sections, 33 theorems, 168 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Geometry of ${\mathbb{R}\mathrm{P}}^n$
  3. Main results
  4. Q-nets
  5. Kœnigs nets
  6. Kœnigs grids
  7. Autoconjugate curves
  8. Laplace transformations
  9. Touching conics
  10. A connection to Kœnigs binets
  11. The quadric gluing lemma
  12. Inscribed quadrics
  13. Existence and Uniqueness
  14. Singular points
  15. Diagonal intersection nets
  16. ...and 14 more sections

Key Result

Lemma 3.4

For every non-degenerate Q-net $P$ defined on a finite patch $\Sigma_{a,b}$ there is an extensive Q-net $\hat{P}$ (called a lift of $P$), such that there is a central projection $\pi$ with

Figures (9)

  • Figure 1: Any Kœnigs net has a $1$-parameter family of touching conics. The existence of one instance of touching conics (as shown on the left) is sufficient to ensure the existence of a $1$-parameter family of touching conics (as shown on the right).
  • Figure 2: Two inscribed quadrics (red) for the same extensive Kœnigs net but with different instances of touching conics (green). The seven edges (black) are tangent to the inscribed quadrics.
  • Figure 3: A generic Kœnigs $1$-grid with an instance of touching conics. All the parameter spaces (the gridlines) are tangent to a non-degenerate conic.
  • Figure 4: The Laplace transform $\mathcal{L}_-P = P_{-1}$ (green) of a Q-net $P$ (black).
  • Figure 5: Combinatorial picture of the Laplace invariants $H$ (left) and $K$ (right).
  • ...and 4 more figures

Theorems & Definitions (82)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 3.7
  • Remark 3.8
  • Definition 3.9
  • Theorem 3.10
  • ...and 72 more