Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

Ulrike Bücking ()
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Ulrike Bücking: Technische Universität Berlin, Inst. für Mathematik

A chapter in Advances in Discrete Differential Geometry, 2016, pp 133-149 from Springer

Abstract: Abstract Two triangle meshes are conformally equivalent if their edge lengths are related by scale factors associated to the vertices. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study the approximation of a given smooth conformal map f by such discrete conformal maps $$f^\varepsilon $$ f ε defined on triangular lattices. In particular, let T be an infinite triangulation of the plane with congruent strictly acute triangles. We scale this triangular lattice by $$\varepsilon >0$$ ε > 0 and approximate a compact subset of the domain of f with a portion of it. For $$\varepsilon $$ ε small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by $$\log |f'|$$ log | f ′ | on the boundary. Furthermore we show that the corresponding discrete conformal (piecewise linear) maps $$f^\varepsilon $$ f ε converge to f uniformly in $$C^1$$ C 1 with error of order $$\varepsilon $$ ε .

Keywords: Dirichlet Problem; Edge Length; Circle Pattern; Triangular Lattice; Boundary Vertex (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-50447-5_3

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DOI: 10.1007/978-3-662-50447-5_3

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