Volume 55, pp. 671-686, 2022.

Constructing diffeomorphisms between simply connected plane domains

Kendall Atkinson, David Chien, and Olaf Hansen

Abstract

Consider a simply connected domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with boundary $\partial \mathrm{\Omega }$ that is given by a smooth function $\phi :[a,b]\mapsto {\mathbb{R}}^2$. Our goal is to calculate a diffeomorphism $\mathrm{\Phi }:{\mathbb{B}}_1(0)\mapsto \mathrm{\Omega }$, ${\mathbb{B}}_1(0)$ the open unit disk in ${\mathbb{R}}^2$. We present two different methods where both methods are able to handle boundaries $\partial \mathrm{\Omega }$ that are not star-shaped. The first method is based on an optimization algorithm that optimizes the curvature of the boundary, and the second method is based on the physical principle of minimizing a potential energy. Both methods construct first a homotopy between the boundary $\partial {\mathbb{B}}_1(0)$ and $\partial \mathrm{\Omega }$ and then extend the boundary homotopy to the inside of the domains. Numerical examples show that the method is applicable to a wide variety of domains $\mathrm{\Omega }$.

Full Text (PDF) [954 KB], BibTeX , DOI: 10.1553/etna_vol55s671

Key words

domain transformations, constructing diffeomophisms, shape blending

AMS subject classifications

65D05, 49Q10

Links to the cited ETNA articles

[2]	Vol. 39 (2012), pp. 202-230 Kendall Atkinson and Olaf Hansen: Creating domain mappings

ETNA articles which cite this article

Vol. 60 (2024), pp. 351-363 Kendall Atkinson, David Chien, and Olaf Hansen: Constructing diffeomorphisms between simply connected plane domains-part 2