A solution to Sheil-Small's harmonic mapping problem for Jordan polygons
Daoud Bshouty, Erik Lundberg, Allen Weitsman
TL;DR
This paper presents a straightforward computational geometry approach to solve Sheil-Small's harmonic mapping problem for Jordan polygons, enabling univalent mappings via Poisson integrals of step functions.
Contribution
It introduces an ear clipping method to construct univalent harmonic mappings of Jordan polygons, providing a practical solution to a longstanding problem.
Findings
Successfully maps Jordan polygons univalently using the proposed method
Simplifies the solution process with computational geometry techniques
Offers potential applications in geometric function theory
Abstract
The problem of mapping the interior of a Jordan polygon univalently by the Poisson integral of a step function was posed by T. Sheil-Small (1989). We describe a simple solution using "ear clipping" from computational geometry.
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Taxonomy
TopicsAnalytic and geometric function theory · Mathematics and Applications · Advanced Algebra and Geometry