Surface algorithms using bounds on derivatives
Computer Aided Geometric Design
A unified approach to subdivision algorithms near extraordinary vertices
Computer Aided Geometric Design
Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values
Proceedings of the 25th annual conference on Computer graphics and interactive techniques
Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon
Computer Aided Geometric Design - Special issue dedicated to Paul de Faget de Casteljau
Best bounds on the approximation of polynomials and splines by their control structure
Computer Aided Geometric Design
Normal bounds for subdivision-surface interference detection
Proceedings of the conference on Visualization '01
Tight Bounding Volumes for Subdivision Surfaces
PG '98 Proceedings of the 6th Pacific Conference on Computer Graphics and Applications
Estimating subdivision depth of Catmull-Clark surfaces
Journal of Computer Science and Technology - Special issue on computer graphics and computer-aided design
Differentiable parameterization of Catmull-Clark subdivision surfaces
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Mesh Refinement Based on Euler Encoding
SMI '05 Proceedings of the International Conference on Shape Modeling and Applications 2005
Computational formula of depth for Catmull-Clark subdivision surfaces
Journal of Computational and Applied Mathematics - Special issue: The international symposium on computing and information (ISCI2004)
Distance between a Catmull-Clark subdivision surface and its limit mesh
Proceedings of the 2007 ACM symposium on Solid and physical modeling
GMP'06 Proceedings of the 4th international conference on Geometric Modeling and Processing
Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface Methods
Computer Graphics Forum
Proceedings of Graphics Interface 2012
Graphics Interaction: 5-6-7 Meshes: Remeshing and analysis
Computers and Graphics
| Hi-index | 0.00 |
For standard subdivision algorithms and generic input data, near an extraordinary point the distance from the limit surface to the control polyhedron after m subdivision steps is shown to decay dominated by the mth power of the subsubdominant (third largest) eigenvalue. Conversely, for Loop subdivision we exhibit generic input data so that the Hausdorff distance at the mth step is greater than or equal to the mth power of the subsubdominant eigenvalue. In practice, it is important to closely predict the number of subdivision steps necessary so that the control polyhedron approximates the surface to within a fixed distance. Based on the above analysis, two such predictions are evaluated. The first is a popular heuristic that analyzes the distance only for control points and not for the facets of the control polyhedron. For a set of test polyhedra this prediction is remarkably close to the true distance. However, a concrete example shows that the prediction is not safe but can prescribe too few steps. The second approach is to first locally, per vertex neighborhood, subdivide the input net and then apply tabulated bounds on the eigenfunctions of the subdivision algorithm. This yields always safe predictions that are within one step for a set of typical test surfaces.