Abstract
Recent methods for tuning a subdivision scheme create a concentric wave pattern around the extraordinary vertex (EV). We explain it as resulting from the antagonism between the rules which would create a nice limit surface at the EV and the ordinary rules used in the surrounding regular surface.
We show that even a scheme which fulfils the most recently proposed conditions for good convergence at the EV may still produce this wave pattern.
Then, in order to smooth this antagonism, we define any new vertex as a convex combination of the ideal new vertex from the EV point of view and the one defined with ordinary rules. The weight of the extraordinary rules decreases as the new vertex is topologically farther from the EV.
The concentric wave pattern shades off whereas the expected conditions are not too much spoiled. This tuning method remains simple and useful, involving no optimisation process.
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References
Augsdörfer, U.H., Dodgson, N.A., Sabin, M.A.: Tuning subdivision by minimising Gaussian curvature variation near extraordinary vertices. Computer Graphics Forum 25(3) (2006); (Proc. Eurographics 2006)
Barthe, L., Gérot, C., Sabin, M.A., Kobbelt, L.: Simple computation of the eigencomponents of a subdivision matrix in the Fourier domain. In: Dodgson, N.A., Floater, M.S., Sabin, M.A. (eds.) Advances in Multiresolution for Geometric Modelling, pp. 245–257. Springer, Heidelberg (2005)
Barthe, L., Kobbelt, L.: Subdivision scheme tuning around extraordinary vertices. Computer Geometric Aided Design 21(6), 561–583 (2004)
Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design 10(6), 350–355 (1978)
Doo, D., Sabin, M.A.: Behaviour of recursive division surface near extraordinary points. Computer Aided Design 10(6), 356–360 (1978)
Gérot, C., Barthe, L., Dodgson, N.A., Sabin, M.A.: Subdivision as a sequence of sampled cp surfaces. In: Dodgson, N.A., Sabin, M.A. (eds.) Advances in Multiresolution for Geometric Modelling. Springer, Heidelberg (2005)
Ginkel, I., Umlauf, G.: Controlling a subdivision tuning method. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Avignon 2006, pp. 170–179. Nashboro Press (2007)
Ivrissimtzis, I., Dodgson, N.A., Sabin, M.A.: A generative classification of mesh refinement rules with lattice transformations. Computer Aided Geometric Design 21(1), 99–109 (2004)
Kobbelt, L.: \(\sqrt{3}\)-subdivision. In: SIGGRAPH 2000 Conference Proceedings, pp. 103–112 (2000)
Karčiauskas, K., Peters, J.: Concentric tessellation maps and curvature continuous guided surfaces. Computer Aided Geometric Design 24(2), 99–111 (2007)
Karčiauskas, K., Peters, J., Reif, U.: Shape characterization of subdivision surfaces–case studies. Computer Geometric Aided Design 21(6), 601–614 (2004)
Levin, A.: Modified subdivision surfaces with continuous curvature. Computer Graphics 25(3), 1035–1040 (2006); Proc. SIGGRAPH 2006
Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah (1987)
Loop, C.: Bounded curvature triangle mesh subdivision with the convex hull property. The Visual Computer 18, 316–325 (2002)
Peters, J., Reif, U.: Shape characterization of subdivision surfaces: basic principles. Computer Aided Geometric Design 21(6), 585–599 (2004)
Prautzsch, H.: Smoothness of subdivision surfaces at extraordinary points. Adv. Comput. Math. 9, 377–389 (1998)
Prautzsch, H., Umlauf, G.: A g 1 and g 2 subdivision scheme for triangular nets. International Journal on Shape Modelling 6(1), 21–35 (2000)
Reif, U.: A unified approach to subdivision algorithm near extraordinary vertices. Computer Geometric Aided Design 12, 153–174 (1995)
Reif, U.: A degree estimate for subdivision surfaces of higher regularity. Proc. Amer. Math. Soc. 124(7), 2167–2174 (1996)
Sabin, M.A.: Eigenanalysis and artifacts of subdivision curves and surfaces. In: Iske, A., Quak, E., Floater, M.S. (eds.) Tutorials on Multiresolution in Geometric Modelling, pp. 69–97. Springer, Heidelberg (2002)
Sabin, M.A., Barthe, L.: Artifacts in recursive subdivision surfaces. In: Cohen, A., Merrien, J.-L., Schumaker, L.L. (eds.) Curve and Surface Fitting: Saint-Malo 2002, pp. 353–362. Nashboro Press (2003)
Umlauf, G.: Analysis and tuning of subdivision algorithms. In: SCCG 2005: Proceedings of the 21st spring conference on Computer graphics, pp. 33–40. ACM Press, New York (2005)
Velho, L., Zorin, D.: 4-8 subdivision. Computer Geometric Aided Design 18(5), 397–428 (2001)
Warren, J., Warren, J.D., Weimer, H.: Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann Publishers Inc., San Francisco (2001)
Zorin, D.: Stationary subdivision and multiresolution surface representations. PhD thesis, California Institute of Technology (1997)
Zorin, D., Schröder, P.: Subdivision for modeling and animation. In: SIGGRAPH 2000 course notes (2000)
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Gérot, C., Destelle, F., Montanvert, A. (2010). Smoothing the Antagonism between Extraordinary Vertex and Ordinary Neighbourhood on Subdivision Surfaces. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_16
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DOI: https://doi.org/10.1007/978-3-642-11620-9_16
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