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A discrete Schwarzian derivative via circle packing

Kenneth Stephenson

TL;DR

This work defines a discrete Schwarzian derivative for circle-packings, introducing an edge-based invariant $\Sigma_F(e)$ and a local intrinsic Schwarzian $s$ for patches, both Möbius invariant. It establishes a concrete computational framework centered on circle-packing flowers, enabling parameterization and layout of petals via explicit functions $\mathfrak U_n$ and constraints $\mathfrak C_j$, and demonstrates rich structure for un-branched, univalent, and branched flowers. The paper also analyzes several special flower classes (uniform, extremal, soccerball, Doyle, Ring Lemma), deriving closed-form formulas and highlighting deep connections to classical circle-packing phenomena, Farey sequences, and the golden ratio. Overall, it provides foundational tools and computational strategies that open new avenues for understanding and constructing circle packings on spheres and projective surfaces through schwarzians.

Abstract

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current lack of circle packing algorithms in spherical geometry, and the discrete Schwarzian derivative may provide for new approaches. A companion localized notion called an intrinsic schwarzian is also investigated. The main concrete results of the paper are limited to circle packing flowers. A parameterization by intrinsic schwarzians is established, providing an essential packing criterion for flowers. The paper closes with the study of special classes of flowers that occur in the circle packing literature. As usual in circle packing, there are pleasant surprises at nearly every turn, so those not interested in circle packing theory may still enjoy the new and elementary geometry seen in these flowers.

A discrete Schwarzian derivative via circle packing

TL;DR

This work defines a discrete Schwarzian derivative for circle-packings, introducing an edge-based invariant and a local intrinsic Schwarzian for patches, both Möbius invariant. It establishes a concrete computational framework centered on circle-packing flowers, enabling parameterization and layout of petals via explicit functions and constraints , and demonstrates rich structure for un-branched, univalent, and branched flowers. The paper also analyzes several special flower classes (uniform, extremal, soccerball, Doyle, Ring Lemma), deriving closed-form formulas and highlighting deep connections to classical circle-packing phenomena, Farey sequences, and the golden ratio. Overall, it provides foundational tools and computational strategies that open new avenues for understanding and constructing circle packings on spheres and projective surfaces through schwarzians.

Abstract

There exists an extensive and fairly comprehensive discrete analytic function theory which is based on circle packing. This paper introduces a faithful discrete analogue of the classical Schwarzian derivative to this theory and develops its basic properties. The motivation comes from the current lack of circle packing algorithms in spherical geometry, and the discrete Schwarzian derivative may provide for new approaches. A companion localized notion called an intrinsic schwarzian is also investigated. The main concrete results of the paper are limited to circle packing flowers. A parameterization by intrinsic schwarzians is established, providing an essential packing criterion for flowers. The paper closes with the study of special classes of flowers that occur in the circle packing literature. As usual in circle packing, there are pleasant surprises at nearly every turn, so those not interested in circle packing theory may still enjoy the new and elementary geometry seen in these flowers.

Paper Structure

This paper contains 31 sections, 7 theorems, 62 equations, 24 figures.

Table of Contents

  1. Background
  2. On Circle Packing
  3. On Discrete Analytic Functions
  4. On Möbius Transformations
  5. On Software
  6. Classical, Discrete, and Intrinsic
  7. Patches
  8. Intrinsic Schwarzians
  9. Packing Layouts
  10. The Difficulty
  11. Flowers
  12. Notation and Preliminaries
  13. Flower Layouts
  14. Important Observations
  15. Un-branched Flowers
  16. ...and 16 more sections

Key Result

Theorem 4.1

Given schwarzians $\{s_1,\cdots,s_{n-3}\}$, the Layout Process results in a legitimate $n$-flower except in the following two situations: (a) when $c_{n-2}$ is tangent to $C$ at infinity or (b) when the computed $s_0$ exceeds 1.

Figures (24)

  • Figure 1: Example of a discrete Riemann mapping
  • Figure 2: Examples of a discrete meromorphic functions
  • Figure 3: The discrete Schwarzian derivative for an edge
  • Figure 4: Sampling some intrinsic schwarzians
  • Figure 5: Notations for normalized flower layouts
  • ...and 19 more figures

Theorems & Definitions (12)

  • Definition
  • Definition
  • Definition
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.5
  • Theorem 4.6
  • ...and 2 more