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Discrete Isothermic Nets Based on Checkerboard Patterns.

Felix Dellinger1,2

  • 1Institute of Geometry, TU Graz, Kopernikusgasse 24, 8010 Graz, Austria.

Discrete & Computational Geometry
|June 4, 2024
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Summary
This summary is machine-generated.

This study explores discrete differential geometry using checkerboard patterns. It introduces principal, Koenigs, and isothermic nets, revealing their connections and properties for constructing discrete minimal surfaces.

Keywords:
Differential geometryDiscrete differential geometryIsothermic surfacesKoenigs nets

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Area of Science:

  • Discrete differential geometry
  • Möbius geometry
  • Computational geometry

Background:

  • Quadrilateral nets and checkerboard patterns are fundamental in discrete geometry.
  • Understanding principal, Koenigs, and isothermic nets is crucial for geometric analysis.

Purpose of the Study:

  • To investigate the discrete differential geometry of checkerboard patterns in quadrilateral nets.
  • To establish a framework for defining and analyzing principal, Koenigs, and isothermic nets.
  • To explore the properties and transformations of these discrete net types.

Main Methods:

  • Utilizing the checkerboard pattern formed by connecting edge midpoints of a quadrilateral net.
  • Applying concepts of orthogonality and conjugacy to define principal nets.
  • Identifying connections with sphere congruences and Möbius geometry.
  • Defining discrete Koenigs nets via the conic of Koenigs.
  • Investigating dualizability and Laplace invariants for Koenigs nets.

Main Results:

  • The checkerboard pattern serves as a versatile tool for defining various discrete nets.
  • Principal nets are identified with sphere congruences in Möbius geometry.
  • Discrete Koenigs nets exhibit dualizability and equal Laplace invariants.
  • Isothermic nets are shown to be both Koenigs and principal nets.
  • The class of isothermic nets is invariant under dualization and Möbius transformations.

Conclusions:

  • The study provides a unified approach to discrete principal, Koenigs, and isothermic nets.
  • These findings enable the natural construction of discrete minimal surfaces and their Goursat transformations.
  • The invariance properties of isothermic nets offer new avenues for geometric modeling and analysis.