Abstract
A stationary subdivision scheme such as Catmull and Clark's is described by a matrix iteration around an extraordinary point. We show how higher order smoothness of a limiting surface obtained by a stationary subdivision algorithm for tri- or quadrilateral nets depends on the spectral properties of the matrix and give necessary and sufficient conditions. The results are also useful to construct subdivision algorithms for surfaces of any smoothness order at extraordinary points.
Similar content being viewed by others
References
A.A. Ball and D.J.T. Storry, Conditions for tangent plane continuity over recursively generated B-spline surfaces, ACM Trans. Graphics 7(2) (1988) 83–102.
E. Catmull and J. Clark, Recursively generated B-spline surfaces on arbitrary topological meshes, Comput. Aided Design 10(6) (1978) 350–355.
D.W.H. Doo, A subdivision algorithm for smoothing down irregular shaped polyhedrons, in: Proc. of Conf. on Interactive Techniques in Computer Aided Design (IEEE, Bologna, 1978) pp. 157–165.
D.W.H. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Comput. Aided Design 10(6) (1978) 356–360.
N. Dyn, J. Gregory and D. Levin, A butterfly subdivision scheme for surface interpolation with tension control, ACM Trans. Graphics 9 (1990) 160–169.
J.M. Lane and R.F. Riesenfeld, A theoretical development for the computer generation and display of piecewise polynomial surfaces, IEEE Trans. Pattern Anal. Machine Intell. 2(1) (1980) 35–46.
C.T. Loop, Smooth subdivision surfaces based on triangles, Master's thesis, University of Utah (1987).
C.A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114 (1989) 841–870.
J. Peters and U. Reif, Analysis of algorithms generalizing B-spline subdivision, SIAM J. Numer. Anal. (1997, to appear).
H. Prautzsch, C k-analysis of subdivision surfaces at extraordinary points, Technical report 04/98, Fakultät für Informatik, Universität Karlsruhe (1998), available at http://i33www.ira.uka. de.
H. Prautzsch, Freeform splines, Comput. Aided Geom. Design 14 (1997) 201–206.
H. Prautzsch and U. Reif, Degree estimates for C k piecewise polynomial subdivision surfaces, to appear in Adv. Comput. Math.
H. Prautzsch and G. Umlauf, A G 2-subdivision algorithm, in: Proceedings of the Dagstuhl Conference on Geometric Modelling 1996, eds. Bieri, Brunnet and Farin, Computing Suppl. (1998), available at http://i33www.ira.uka.de/publications.html.
H. Prautzsch and G. Umlauf, Improved triangular subdivision schemes, in: Proceedings Computer Graphics International 1998, eds. Wolter and Patrikalakis (IEEE Computer Society, Los Alamitos, CA, 1998) pp. 626–632.
R. Qu, Recursive subdivision algorithms for curve and surface design, Ph.D. thesis, Brunel University (1990).
U. Reif, Neue Aspekte in der Theorie der Freiformflächen beliebiger Topologie, Ph.D. thesis, Mathematisches Institut A der Universität Stuttgart (1993).
U. Reif, Some new results on subdivision algorithms for meshes of arbitrary topology, in: Wavelets and Multilevel Approximation, eds. C.K. Chui and L.L. Schumaker, Series in Approximations and Decompositions, Vol. 2 (World Scientific, 1995) pp. 367–374.
U. Reif, A unified approach to subdivision algorithms near extraordinary vertices, Comput. Aided Geom. Design 12 (1995) 153–174.
U. Reif, Letter from December 4, 1996.
U. Reif, A degree estimate for subdivision surfaces of higher regularity, Proc. of the AMS 124(7) (1996) 2167–2174.
U. Reif, TURBS – Topologically Unrestricted Rational B-splines, Constructive Approximation 14(1) (1998) 57–78.
G. Umlauf, Verbesserung der Glattheitsordnung von Unterteilungsalgorithmen für Flächen beliebiger Topologie, Master's thesis, Universität Karlsruhe, Fakultät für Informatik (1996).
G. Umlauf, Analyzing the characteristic map of triangular subdivision schemes, Preprint (1998), available at http://i33www.ira.uka.de/publications.html (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Prautzsch, H. Smoothness of subdivision surfaces at extraordinary points. Advances in Computational Mathematics 9, 377–389 (1998). https://doi.org/10.1023/A:1018945708536
Issue Date:
DOI: https://doi.org/10.1023/A:1018945708536