Abstract
A well-documented problem of Catmull and Clark subdivision is that, in the neighborhood of extraordinary point, the curvature is unbounded and fluctuates. In fact, since one of the eigenvalues that determines elliptic shape is too small, the limit surface can have a saddle point when the designer’s input mesh suggests a convex shape. Here, we replace, near the extraordinary point, Catmull-Clark subdivision by another set of rules derived by refining each bi-cubic B-spline into nine. This provides many localized degrees of freedom for special rules so that we need not reach out to, possibly irregular, neighbor vertices when trying to improve, or tune the behavior. We illustrate a strategy how to sensibly set such degrees of freedom and exhibit tuned ternary quad subdivision that yields surfaces with bounded curvature, nonnegative weights and full contribution of elliptic and hyperbolic shape components.
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References
Barthe, L., Kobbelt, L.: Subdivision Scheme tuning around extraordinary vertices. Computer Aided Geometric Design 21, 561–583
Catmull, E., Clark, J.: Recursively Generated B-spline Surfaces on Arbitrary Topological Meshes. Computer Aided Design 10(6), 350–355 (1978)
Loop, C.: Smooth Ternary Subdivision of Triangle Meshes. Curve and Surface Fitting (Saint-Malo 2002)
Loop, C.: Bounded curvature triangle Mesh subdivision with the convex hull property. The Visual Computer 18(5-6), 316–325
Karciauskas, K., Peters, J., Reif, U.: Shape Characterization of Subdivision Surfaces – Case Studies. Comp. Aided Geom. Design 21(6), 601–614 (2004)
Nasri, A., Hasbini, I., Zheng, J., Sederberg, T.: Quad-based Ternary Subdivision. In: Presentation Dagstuhl Seminar on Geometric Modeling, May 29 - June 03 (2005)
Reif, U., Peters, J.: Structural Analysis of Subdivision Surfaces – A Summary. In: Jetter, K., et al. (eds.) Topics in Multivariate Approximation and Interpolation, pp. 149–190. Elsevier Science Ltd., Amsterdam (2005)
Umlauf, G.: Analysis and Tuning of Subdivision Algorithms. In: Proceedings of the 21st spring conference on Computer Graphics, pp. 33–40 (2005)
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© 2006 Springer-Verlag Berlin Heidelberg
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Ni, T., Nasri, A.H. (2006). Tuned Ternary Quad Subdivision. In: Kim, MS., Shimada, K. (eds) Geometric Modeling and Processing - GMP 2006. GMP 2006. Lecture Notes in Computer Science, vol 4077. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11802914_31
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DOI: https://doi.org/10.1007/11802914_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36711-6
Online ISBN: 978-3-540-36865-6
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