Schwarz-Christoffel Maps onto Periodic Comb Domains Open Access

Abstract

Schwarz–Christoffel maps are a type of conformal maps, where the target domain typically consists of a regular polygon. For such a configuration, it is known that the map can be derived from the vertices of the polygon; the primary challenge in doing this is a computational one. This thesis investigates the parameter problem for periodic comb domains, defined as the upper half-plane with an infinite series of periodic, vertical slits removed from the real axis. Computing conformal maps onto these domains is difficult due to the accumulation of integrable singularities at the bases of the slits, which compound as the domain extends to infinity. To address this, we derive a novel periodic Schwarz–Christoffel formula specifically tailored to comb domains. By exploiting the geometric properties of the turning angles, we reduce the infinite mapping product to a highly efficient and numerically stable exponential reformulation. To solve the corresponding parameter problem, we develop a robust computa tional framework that utilizes Gauss–Jacobi quadrature to extract boundary singularities. Finally, we demonstrate the success of this method by computing and visualizing highly accurate conformal grids for various examples, while formally analyzing the fundamental computational limits of the algorithm induced by the crowding phenomenon. 

Table of Contents

1 Introduction (2)

  1.1 A Brief History of Schwarz–Christoffel Maps (2)

  1.2 Numerical Approach to Schwarz–Christoffel Mapping (3)

  1.3 The Significance of Comb Domains (4)

2 Background (6)

  2.1 Conformal Mapping (6)

  2.1.1 Canonical Domains (7)

  2.1.2 The Riemann Mapping Theorem (8)

  2.1.3 The Carathéodory Theorem (9)

  2.2 Schwarz–Christoffel Maps (11)

  2.2.1 The Schwarz–Christoffel Formula (12)

  2.2.2 The Parameter Problem (13)

3 Mapping to Comb Domains (15)

  3.1 Comb Domains (15)

  3.2 Angular Derivative (19)

  3.3 Periodic Regions (20)

  3.3.1 Periodicity of Prevertices (20)

  3.3.2 Periodic Strip Map (22)

  3.4 Periodic Combs (23)

  3.4.1 Main Theorem (23)

  3.4.2 Numerical Simplification (25)

  3.4.3 Numerical Stability (26)

4 Numerical Methods (27)

  4.1 Parameter Problem for a Comb (27)

  4.1.1 Physical Boundary Lengths (27)

  4.1.2 Extracting Endpoint Singularities (28)

  4.2 Gauss-Jacobi Quadrature (29)

  4.2.1 Application to Parameter Problem (30)

  4.3 Solving the Parameter Problem (30)

  4.3.1 Geometry of the Slits (31)

  4.3.2 Residual Optimization Problem (31)

  4.3.3 Computing Scaling Constant (32)

  4.3.4 Integration Constant (33)

  4.4 Plotting the Map (34)

  4.4.1 Change of Variables (34)

  4.4.2 Branch Cut Evasion via High Contours (35)

  4.5 Mapping Parameters and Performance (36)

  4.5.1 Examples (36)

5 Conclusion (41)

A Appendix (43)

  A.1 User Guide (43)

  A.2 Python Implementation of the Periodic Comb Map (44)

B Bibliography (54)

About this Honors Thesis

Rights statement

• Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.

School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Numerical stability Periodic regions Schwarz-Christoffel Comb domain Riemann mapping Gauss-Jacobi quadrature Conformal map
Committee Chair / Thesis Advisor	Milivoje Lukic, Emory University
Committee Members	Maja Taskovic, Emory University Alessandro Veneziani, Emory University

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