Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A degree estimate for subdivision surfaces of higher regularity

Author: Ulrich Reif
Journal: Proc. Amer. Math. Soc. 124 (1996), 2167-2174
MSC (1991): Primary 65D17, 65D07, 68U07
MathSciNet review: 1327042
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Subdivision algorithms can be used to construct smooth surfaces from control meshes of arbitrary topological structure. In contrast to tangent plane continuity, which is well understood, very little is known about the generation of subdivision surfaces of higher regularity. This work presents a degree estimate for piecewise polynomial subdivision surfaces saying that curvature continuity is possible only if the bi-degree $d$ of the patches satisfies $d \ge 2k+2$, where $k$ is the order of smoothness on the regular part of the surface. This result applies to any stationary or non-stationary scheme consisting of masks of arbitrary size provided that some generic symmetry and regularity assumptions are fulfilled.

References [Enhancements On Off] (What's this?)

  • [CC78] E. Catmull and J. Clark, Recursively generated B-spline surfaces on arbitrary topological meshes, Computer Aided Design 10 (1978), 350 -- 355.
  • [DHK93] T. DeRose, M. Halstead, and M. Kass, Efficient, fair interpolation using Catmull-Clark surfaces, Proceedings of Siggraph '93, Apple Computer, Inc., 1993, pp. 35 -- 44.
  • [DLG90] N. Dyn, D. Levin, and J. A. Gregory, A butterfly subdivision scheme for surface interpolation with tension control, ACM Transactions on Graphics 9 (1990), 160 -- 169.
  • [DLW93] T. DeRose, M. Lounsbery, and J. Warren, Multiresolution analysis for surfaces of arbitrary topological type, Tech. Report 93-10-05, University of Washington, 1993.
  • [DS78] D. Doo and M. A. Sabin, Behaviour of recursive subdivision surfaces near extraordinary points, Computer Aided Design 10 (1978), 356 -- 360.
  • [Loo87] Ch. T. Loop, Smooth subdivision for surfaces based on triangles, Master's thesis, University of Utah, 1987.
  • [Rei93] Ulrich Reif, Neue Aspekte in der Theorie der Freiformflächen beliebiger Topologie, Mathematisches Institut A der Universität Stuttgart, Stuttgart, 1993 (German, with German summary). Dissertation, Universität Stuttgart, Stuttgart, 1993. MR 1280716
  • [Rei95] U. Reif, A unified approach to sudivision algorithms near extraordinary vertices, Computer Aided Geometric Design 12 (1995), 153 -- 174.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 65D17, 65D07, 68U07

Retrieve articles in all journals with MSC (1991): 65D17, 65D07, 68U07

Additional Information

Ulrich Reif
Affiliation: Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart Germany

Keywords: Subdivision, arbitrary topology, extraordinary vertex, curvature continuity, piecewise polynomial surface
Received by editor(s): December 6, 1994
Additional Notes: This work was supported by BMFT Projekt 03–HO7STU–2.
Communicated by: Peter Li
Article copyright: © Copyright 1996 American Mathematical Society