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By Desbrun M., et al.

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L. 2003. Discrete Surface Modeling using Geometric Flows. Tech. , University of Texas. , AND B ELYAEV, A. G. 2002. Fair Triangle Mesh Generation with Discrete Elastica. In Geometric Modeling and Processing, IEEE Computer Society, 119–123. Chapter 6: Discrete Conformal Mappings via Circle Patterns Liliya Kharevych Caltech Boris Springborn TU Berlin Peter Schr¨oder Caltech Figure 1: Examples of discrete conformal maps produced with our method. Next to each 3D texture image is a visualization of the planar region over which the surface is parameterized.

Since both the bounds constraints and the objective are convex and the additional constraints linear we have experienced no difficulty finding solutions efficiently even for large meshes (see Section 3). Figure 8: Mapping a mesh to the disk. (1) Remove a boundary vertex together with all adjacent faces (dark pink). (2) Map the mesh to a convex polygon where all the original boundary edges lie on a straight line (top). The removed vertex (which should be imagined at infinity) and its incident triangles are shown schematically (dark pink).

5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve,surface, solid and object representations; Geometric algorithms, languages, and systems. Keywords: Conformal parameterizations; discrete differential geometry; circle patterns; discrete analytic functions; meshing; texture mapping 1 Introduction Surfaces are often represented as collections of samples with connectivity, typically in the form of a simplicial mesh. It is natural and convenient to use the implied piecewise linear mesh as the basis for formulating a variety of computational algorithms, such as parameterization problems or the solution of partial differential equations for purposes of simulation.

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