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 $\Omega\subset\mathbb{R}^2$ with boundary $\partial \Omega$ that is given by a smooth function $\varphi:[a,b]\mapsto \mathbb{R}^2$. Our goal is to calculate a diffeomorphism $\Phi:\mathbb{B}_1(0)\mapsto \Omega$, $\mathbb{B}_1(0)$ the open unit disk in $\mathbb{R}^2$. We present two different methods where both methods are able to handle boundaries $\partial \Omega$ 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 $\partial \Omega$ 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 $\Omega$.

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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

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