skip to main content
research-article

Deducing interpolating subdivision schemes from approximating subdivision schemes

Published: 01 December 2008 Publication History

Abstract

In this paper we describe a method for directly deducing new interpolating subdivision masks for meshes from corresponding approximating subdivision masks. The purpose is to avoid complex computation for producing interpolating subdivision masks on extraordinary vertices. The method can be applied to produce new interpolating subdivision schemes, solve some limitations in existing interpolating subdivision schemes and satisfy some application needs. As cases, in this paper a new interpolating subdivision scheme for polygonal meshes is produced by deducing from the Catmull-Clark subdivision scheme. It can directly operate on polygonal meshes, which solves the limitation of Kobbelt's interpolating subdivision scheme. A new √3 interpolating subdivision scheme for triangle meshes and a new √2 interpolating subdivision scheme for quadrilateral meshes are also presented in the paper by deducing from √3 subdivision schemes and 4-8 subdivision schemes respectively. They both produce C1 continuous limit surfaces and avoid the blemish in the existing interpolating √3 and √2 subdivision masks where the weight coefficients on extraordinary vertices can not be described by formulation explicitly. In addition, by adding a parameter to control the transition from approximation to interpolation, they can produce surfaces intervening between approximating and interpolating which can be used to solve the "popping effect" problem when switching between meshes at different levels of resolution. They can also force surfaces to interpolate chosen vertices.

Supplementary Material

JPG File (a146-lin-mp4_hi.jpg)
MOV File (a146-lin-mp4_hi.mov)

References

[1]
Catmull, E., and Clark, J. 1978. Recursively generated b-spline surfaces on arbitrary topological meshes. Computer Aided Design 10, 6, 350--355.
[2]
Claes, J., Beets, K., Reeth, F., Iones, A., and Krupkin, A. 2001. Turning the approximating catmull-clark subdivision scheme into a locally interpolating surface modeling tool. Proceeding of SMI 2001, 42--48.
[3]
Doo, D., and Sabin, M. 1978. Behavior of recursive division surfaces near extraordinary points. Computer Aided Design 10, 6, 356--360.
[4]
Dyn, N., and Levin, D. 1990. A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9, 160--169.
[5]
Halstead, M., Kass, M., and DeRose, T. 1993. Efficient, fair interpolating using catmull-clark surfaces. 35--44. (Proc. SIGGRAPH'93).
[6]
J. Stam. 1998. Exact evaluation of catmull-clark subdivision surfaces at arbitrary parameter values. Proceedings of SIGGRAPH 1998, 395--404.
[7]
Kobbelt, L. 1996. Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum 15, 3, 409--410. (Proc. EUROGRAPHICS'96).
[8]
Kobbelt, L. 2000. √3 subdivision. 103--112. (Computer Graphics Proceedings, SIGGRAPH'00).
[9]
Labsik, U., and Greiner, G. 2000. Interpolatory √3 subdivision. Computer Graphics Forum 19, 3, 131--138. (Proc. Eurographics'00).
[10]
Li, G., and Ma, W. 2007. A method for constructing interpolatory subdivisions and blending subdivisions. Computer Graphics Forum 26, 2, 185--201.
[11]
Li, G., Ma, W., and H. Bao. 2004. Interpolatory √2 subdivision. In proceeding of Geometric Modeling and Processing 2004, 185--194.
[12]
Lin, S., and Luo, X. 2007. A unified √3 interpolatory and approximation subdivision scheme. Eurographics 07. Short paper, In Eurographics Digital Library.
[13]
Loop, C. 1987. Smooth subdivision surfaces based on triangles. Master's thesis, University of Utah.
[14]
Loop, C. 2002. Bounded curvature triangle mesh subdivision with the convex hull property. The Visual Computer 18, 5--6, 316C325.
[15]
Maillot, J., and Stam, J. 2001. A unified subdivision scheme for polygonal modeling. Computer Graphics Forum 20, 3, 471--479. (Proc. Eurographics'01).
[16]
Nasri, A. 1987. Polyhedral subdivision methods for free-form surfaces. ACM Trans Graph 6, 1, 29--73.
[17]
Reif, U. 1995. A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometry Design 12, 153--174.
[18]
Velho, L., and Zorin, D. 2000. 4--8 subdivision. Computer Aided Geometric Design 18, 5, 397--427.
[19]
Zorin, D., Schroder, P., and Sweldens, W. 1996. Interpolating subdivision for meshes with arbitrary topology. Computer Graphics 30, 189--192. (Proc. SIGGRAPH'96).
[20]
Zorin, D. 1997. c k continuity of subdivision surfaces. Thesis, California Institute of Technology.
[21]
Zorin, D. 2000. Smoothness of stationary subdivision on irregular meshes. Constructive Approximation 16, 3, 359--397.

Cited By

View all
  • (2025)A hybrid family of non-uniform ternary schemes with mixed symmetry for approximating shapesJournal of Applied Mathematics and Computing10.1007/s12190-024-02342-7Online publication date: 4-Jan-2025
  • (2021)Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomialsOpen Mathematics10.1515/math-2021-005819:1(909-926)Online publication date: 28-Aug-2021
  • (2020)A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision SchemeMathematical Problems in Engineering10.1155/2020/60965452020(1-17)Online publication date: 1-Oct-2020
  • Show More Cited By

Index Terms

  1. Deducing interpolating subdivision schemes from approximating subdivision schemes

      Recommendations

      Comments

      Information & Contributors

      Information

      Published In

      cover image ACM Transactions on Graphics
      ACM Transactions on Graphics  Volume 27, Issue 5
      December 2008
      552 pages
      ISSN:0730-0301
      EISSN:1557-7368
      DOI:10.1145/1409060
      Issue’s Table of Contents
      Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      Published: 01 December 2008
      Published in TOG Volume 27, Issue 5

      Permissions

      Request permissions for this article.

      Check for updates

      Author Tags

      1. approximating subdivision
      2. interpolating subdivision

      Qualifiers

      • Research-article

      Funding Sources

      Contributors

      Other Metrics

      Bibliometrics & Citations

      Bibliometrics

      Article Metrics

      • Downloads (Last 12 months)4
      • Downloads (Last 6 weeks)0
      Reflects downloads up to 17 Feb 2025

      Other Metrics

      Citations

      Cited By

      View all
      • (2025)A hybrid family of non-uniform ternary schemes with mixed symmetry for approximating shapesJournal of Applied Mathematics and Computing10.1007/s12190-024-02342-7Online publication date: 4-Jan-2025
      • (2021)Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomialsOpen Mathematics10.1515/math-2021-005819:1(909-926)Online publication date: 28-Aug-2021
      • (2020)A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision SchemeMathematical Problems in Engineering10.1155/2020/60965452020(1-17)Online publication date: 1-Oct-2020
      • (2020)A few conjectures on a four-point interpolatory subdivision schemeComputer Aided Geometric Design10.1016/j.cagd.2020.101933(101933)Online publication date: Sep-2020
      • (2019)Nonstationary interpolatory subdivision schemes reproducing high-order exponential polynomialsRocky Mountain Journal of Mathematics10.1216/RMJ-2019-49-7-242949:7Online publication date: 1-Nov-2019
      • (2019)A Family of Binary Univariate Nonstationary Quasi‐Interpolatory Subdivision Reproducing Exponential PolynomialsMathematical Problems in Engineering10.1155/2019/76315082019:1Online publication date: 4-Nov-2019
      • (2019)Biorthogonal Wavelet Transforms and Applications Based on Generalized Progressive Catmull-Clark Subdivision with Shape ControlIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2018.284588725:7(2392-2403)Online publication date: 1-Jul-2019
      • (2019)Interpolatory subdivision schemes with the optimal approximation orderApplied Mathematics and Computation10.1016/j.amc.2018.10.078347(1-14)Online publication date: Apr-2019
      • (2018)Shading‐Based Surface Recovery Using Subdivision‐Based RepresentationComputer Graphics Forum10.1111/cgf.1353938:1(417-428)Online publication date: 20-Sep-2018
      • (2017)A non-stationary combined subdivision scheme generating exponential polynomialsApplied Mathematics and Computation10.1016/j.amc.2017.05.066313:C(209-221)Online publication date: 15-Nov-2017
      • Show More Cited By

      View Options

      Login options

      Full Access

      View options

      PDF

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      Figures

      Tables

      Media

      Share

      Share

      Share this Publication link

      Share on social media