Foundations of Computational Mathematics

, Volume 17, Issue 2, pp 497–526 | Cite as

Computing Teichmüller Maps Between Polygons

  • Mayank Goswami
  • Xianfeng Gu
  • Vamsi P. Pingali
  • Gaurish Telang
Article
  • 251 Downloads

Abstract

By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle-preserving manner). However, when this map is extended to the boundary, it need not necessarily map the vertices of P to those of Q. For many applications, it is important to find the “best” vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichmüller maps. There are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a \((1+\varepsilon )\)-approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichmüller maps. This paper solves the problem of finding a finite-time procedure for approximating Teichmüller maps in the continuous setting. Our construction is via an iterative procedure that is proven to converge in \(O(\text {poly}(1/\varepsilon ))\) iterations to a map whose dilatation is at most \(\varepsilon \) more than that of the Teichmüller map, for any \(\varepsilon >0\). We reduce the problem of finding an approximation algorithm for computing Teichmüller maps to two basic subroutines, namely, computing discrete (1) compositions and (2) inverses of discretely represented quasiconformal maps. Assuming finite-time solvers for these subroutines, we provide an approximation algorithm with an additive error of at most \(\varepsilon \).

Keywords

Teichmüller map Polygons Approximation algorithm Infinitesimally extremal Surface registration 

Mathematics Subject Classification

65D18 68U05 30F60 

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Copyright information

© SFoCM 2015

Authors and Affiliations

  • Mayank Goswami
    • 1
  • Xianfeng Gu
    • 2
  • Vamsi P. Pingali
    • 3
  • Gaurish Telang
    • 4
  1. 1.AG1, Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Department of Computer ScienceStony Brook UniversityStony BrookUSA
  3. 3.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA
  4. 4.Department of Applied Mathematics and StatisticsStony Brook UniversityStony BrookUSA

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