Approximating Planar Conformal Maps Using Regular Polygonal Meshes

Renjie Chen, Craig Gotsman

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Continuous conformal maps are typically approximated numerically using a triangle mesh which discretizes the plane. Computing a conformal map subject to user-provided constraints then reduces to a sparse linear system, minimizing a quadratic ‘conformal energy’. We address the more general case of non-triangular elements, and provide a complete analysis of the case where the plane is discretized using a mesh of regular polygons, e.g. equilateral triangles, squares and hexagons, whose interiors are mapped using barycentric coordinate functions. We demonstrate experimentally that faster convergence to continuous conformal maps may be obtained this way. We provide a formulation of the problem and its solution using complex number algebra, significantly simplifying the notation. We examine a number of common barycentric coordinate functions and demonstrate that superior approximation to harmonic coordinates of a polygon are achieved by the Moving Least Squares coordinates. We also provide a simple iterative algorithm to invert barycentric maps of regular polygon meshes, allowing to apply them in practical applications, e.g. for texture mapping.

Original languageEnglish (US)
Pages (from-to)629-642
Number of pages14
JournalComputer Graphics Forum
Volume36
Issue number8
DOIs
StatePublished - Dec 2017
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Computer Graphics and Computer-Aided Design

Keywords

  • conformal maps
  • regular polygonal mesh

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