Geometric Accuracy Analysis for Discrete Surface Approximation

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 co...

• Type: Conference
• Year: 2006
• Where: GMP
• Authors: Junfei Dai, Wei Luo, Shing-Tung Yau, Xianfeng Gu
• Keywords: Delaunay Triangulation, GMP 2006, Riemannian Metric, Smooth Surfaces