Summary
This article deals with practical conditions for tuning a subdivision scheme in order to control its artifacts in the vicinity of a mark point. To do so, we look for good behaviour of the limit vertices rather than good mathematical properties of the limit surface. The good behaviour of the limit vertices is characterised with the definition of C2-convergence of a scheme. We propose necessary explicit conditions for C2-convergence of a scheme in the vicinity of any mark point being a vertex of valency greater or equal to three.
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References
A. A. Ball and D. J. T. Storry. Conditions for tangent plane continuity over recursively generated B-spline surfaces. ACM Transactions on Graphics, 7(2):83–102, 1988.
L. Barthe, C. Gérot, M. A. Sabin, and L. Kobbelt. Simple computation of the eigencomponents of a subdivision matrix in the fourier domain. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 245–257 (this book). Springer-Verlag, 2004.
L. Barthe and L. Kobbelt. Subdivision scheme tuning around extraordinary vertices. Submitted.
E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design, 10(6):350–355, 1978.
D. Doo and M. A. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10(6):356–360, 1978.
C. Gérot, L. Barthe, N. A. Dodgson, and M. A. Sabin. Subdivision as sequence of sampled Cp-surfaces and conditions for tuning schemes. Technical Report 583, University of Cambridge Computer Laboratory, Mar 2004. http://www.cl.cam.ac.uk/TechReports/UCAM-CL-TR-583.pdf.
C. T. Loop. Smooth subdivision surfaces based on triangles. Master's thesis, University of Utah, 1987.
J. Peters and U. Reif. Analysis of algorithms generalizing B-spline subdivision. SIAM J. Numer. Anal., 35(2):728–748, 1998.
H. Prautzsch. Smoothness of subdivision surfaces at extraordinary points. Adv. Comput. Math, 9:377–389, 1998.
H. Prautzsch and G. Umlauf. Improved triangular subdivision schemes. In Proc. Computer Graphics International, pages 626–632, 1998.
U. Reif. A unified approach to subdivision algorithm near extraordinary vertices. Computer Geometric Aided Design, 12:153–174, 1995.
M. A. Sabin. Eigenanalysis and artifacts of subdivision curves and surfaces. In A. Iske, E. Quak, and M. S. Floater, editors, Tutorials on Multiresolution in Geometric Modelling, pages 69–97. Springer-Verlag, 2002.
M. A. Sabin. Recent progress in subdivision — a survey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 203–230 (this book). Springer-Verlag, 2004.
M. A. Sabin and L. Barthe. Artifacts in recursive subdivision surfaces. In A. Cohen, J.-L. Merrien, and L. L. Schumaker, editors, Curve and Surface Fitting: Saint-Malo 2002, pages 353–362. Nashboro Press, 2003.
J. Stam. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. Proc. ACM SIGGRAPH '98, pages 395–404, 1998.
D. Zorin. Stationary subdivision and multiresolution surface representations. PhD thesis, California Institute of Technology, 1997
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Gérot, C., Barthe, L., Dodgson, N.A., Sabin, M. (2005). Subdivision as a Sequence of Sampled Cp Surfaces. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds) Advances in Multiresolution for Geometric Modelling. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-26808-1_14
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DOI: https://doi.org/10.1007/3-540-26808-1_14
Publisher Name: Springer, Berlin, Heidelberg
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