Approximating Planar Conformal Maps Using Regular Polygonal Meshes

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dc.contributor.author Chen, Renjie en_US
dc.contributor.author Gotsman, Craig en_US
dc.contributor.editor Chen, Min and Zhang, Hao (Richard) en_US
dc.date.accessioned 2018-01-10T07:43:26Z
dc.date.available 2018-01-10T07:43:26Z
dc.date.issued 2017
dc.identifier.issn 1467-8659
dc.identifier.uri http://dx.doi.org/10.1111/cgf.13157
dc.identifier.uri https://diglib.eg.org:443/handle/10.1111/cgf13157
dc.description.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.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 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. en_US
dc.publisher © 2017 The Eurographics Association and John Wiley & Sons Ltd. en_US
dc.subject conformal maps
dc.subject regular polygonal mesh
dc.subject I.3.8 [Computer Graphics]: Applications
dc.title Approximating Planar Conformal Maps Using Regular Polygonal Meshes en_US
dc.description.seriesinformation Computer Graphics Forum
dc.description.sectionheaders Articles
dc.description.volume 36
dc.description.number 8
dc.identifier.doi 10.1111/cgf.13157
dc.identifier.pages 629-642


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