Surface reconstruction from unorganized points
SIGGRAPH '92 Proceedings of the 19th annual conference on Computer graphics and interactive techniques
Automatic reconstruction of surfaces and scalar fields from 3D scans
SIGGRAPH '95 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
Automatic reconstruction of B-spline surfaces of arbitrary topological type
SIGGRAPH '96 Proceedings of the 23rd annual conference on Computer graphics and interactive techniques
A new Voronoi-based surface reconstruction algorithm
Proceedings of the 25th annual conference on Computer graphics and interactive techniques
Delaunay triangulations and Voronoi diagrams for Riemannian manifolds
Proceedings of the sixteenth annual symposium on Computational geometry
Meshless parameterization and surface reconstruction
Computer Aided Geometric Design
Dual contouring of hermite data
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Interactive geometry remeshing
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
On the approximation of a smooth surface with a triangulated mesh
Computational Geometry: Theory and Applications
Global conformal surface parameterization
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Surface Reconstruction with Triangular B-splines
GMP '04 Proceedings of the Geometric Modeling and Processing 2004
On the optimality of spectral compression of mesh data
ACM Transactions on Graphics (TOG)
Fast exact and approximate geodesics on meshes
ACM SIGGRAPH 2005 Papers
Computing surface hyperbolic structure and real projective structure
Proceedings of the 2006 ACM symposium on Solid and physical modeling
Graphical Models - Special issue on SPM 05
Convergence of discrete Laplace-Beltrami operators over surfaces
Computers & Mathematics with Applications
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Metric Methods in Surface Triangulation
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
Computing the Hausdorff distance between two B-spline curves
Computer-Aided Design
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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. Practical algorithms for approximating surface Delaunay triangulations are introduced based on global conformal surface parameterizations and planar Delaunay triangulations. Thorough experiments are conducted to support the theoretical results.