Name
Abstract | ||
|---|---|---|
In geometric modeling and processing, computer graphics and computer vision, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surfaces. A fundamental problem is to design rigorous algorithms to guarantee the geometric approximation accuracy by controlling the sampling density. This paper gives explicit formulae to the bounds of Hausdorff distance, normal distance and Riemannian metric distortion between the smooth surface and the discrete mesh in terms of principle curvature and the radii of geodesic circum-circle of the triangles. These formulae can be directly applied to design sampling density for data acquisitions and surface reconstructions. Furthermore, we prove that the meshes induced from the Delaunay triangulations of the dense samples on a smooth surface are convergent to the smooth surface under both Hausdorff distance and normal fields. The Riemannian metrics and the Laplace–Beltrami operators on the meshes are also convergent to those on the smooth surfaces. These theoretical results lay down the foundation for a broad class of reconstruction and approximation algorithms in geometric modeling and processing. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1016/j.cagd.2007.04.004 | Computer Aided Geometric Design |
Keywords | Field | DocType |
Riemannian metric,Hausdorff distance,Normal distance,Delaunay triangulation,Principle curvature,Discrete mesh | Discrete geometry,Topology,Computational geometry,Geometric modeling,Principal curvature,Hausdorff distance,Voronoi diagram,Geodesic,Mathematics,Delaunay triangulation | Journal |
Volume | Issue | ISSN |
24 | 6 | 0167-8396 |
Citations | PageRank | References |
15 | 0.89 | 24 |
Authors | ||
7 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Junfei Dai | 1 | 32 | 2.13 |
| Wei Luo | 2 | 24 | 1.95 |
| Miao Jin | 3 | 650 | 35.98 |
| Wei Zeng | 4 | 84 | 4.97 |
| Ying He | 5 | 1264 | 105.35 |
| Shing-tung Yau | 6 | 925 | 92.69 |
| Xianfeng Gu | 7 | 2997 | 189.71 |