期刊
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
卷 20, 期 6, 页码 919-934出版社
IEEE COMPUTER SOC
DOI: 10.1109/TVCG.2013.253
关键词
Meshing; Delaunay refinement; conformal parameterization; normal cycle; curvature measures; convergence
资金
- [ANR 2010 INTB 0301 01]
- [NSF DMS-1221339]
- [NSF Nets-1016829]
- [NSF CCF-1081424]
- [NSF CCF-0830550]
Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.
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