Science China Information Sciences

, Volume 56, Issue 10, pp 1–22 | Cite as

A framework for improving uniformity of parameterizations of curves

  • Hoon Hong
  • DongMing Wang
  • Jing Yang
Research Paper


We define quasi-speed as a generalization of linear speed and angular speed for parameterizations of curves and use the uniformity of quasi-speed to measure the quality of the parameterizations. With such conceptual setting, a general framework is developed for studying uniformity behaviors under reparameterization via proper parameter transformation and for computing reparameterizations with improved uniformity of quasispeed by means of optimal single-piece, C 0 piecewise, and C 1 piecewise Möbius transformations. Algorithms are described for uniformity-improved reparameterization using different Möbius transformations with different optimization techniques. Examples are presented to illustrate the concepts, the framework, and the algorithms. Experimental results are provided to validate the framework and to show the efficiency of the algorithms.


parametric curve framework quasi-speed uniform parameterization uniformity-improved reparameterization optimal Möbius transformation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cattiaux-Huillard I, Albrecht G, Hernández-Mederos V. Optimal parameterization of rational quadratic curves. Comput Aided Geom Des, 2009, 26: 725–732CrossRefzbMATHGoogle Scholar
  2. 2.
    Costantini P, Farouki R, Manni C, et al. Computation of optimal composite re-parameterizations. Comput Aided Geom Des, 2001, 18: 875–897MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Farouki R. Optimal parameterizations. Comput Aided Geom Des, 1997, 14: 153–168MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jüttler, B. A vegetarian approach to optimal parameterizations. Comput Aided Geom Des, 1997, 14: 887–890CrossRefzbMATHGoogle Scholar
  5. 5.
    Liang X, Zhang C, Zhong L, et al. C 1 continuous rational re-parameterization using monotonic parametric speed partition. In: Proceedings of the 9th International Conference on Computer-Aided Design and Computer Graphics, Hong Kong, 2005. 16–21Google Scholar
  6. 6.
    Patterson R, Bajaj C. Curvature Adjusted Parameterization of Curves. Computer Science Technical Report CSD-TR-907. Purdue University, 1997. 773Google Scholar
  7. 7.
    Yang J, Wang D M, Hong H. ImUp: a Maple package for uniformity-improved reparameterization of plane curves. In: Proceedings of the 10th Asian Symposium on Computer Mathematics, Beijing, in pressGoogle Scholar
  8. 8.
    Yang J, Wang D M, Hong H. Improving angular speed uniformity by reparameterization. Comput Aided Geom Des, 2013, 30: 636–652MathSciNetCrossRefGoogle Scholar
  9. 9.
    Yang J, Wang D M, Hong H. Improving angular speed uniformity by C 1 piecewise reparameterization. In: Proceedings of the 9th International Workshop on Automated Deduction in Geometry, Edinburgh, 2012. 17–32Google Scholar
  10. 10.
    Yang J, Wang D M, Hong H. Improving angular speed uniformity by optimal C 0 piecewise reparameterization. In: Proceedings of the 14th International Workshop on Computer Algebra in Scientific Computing (LNCS 7442), Maribor, 2012. 349–360CrossRefGoogle Scholar
  11. 11.
    Farouki R, Sakkalis T. Real rational curves are not unit speed. Comput Aided Geom Des, 1991, 8: 151–157MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Farouki R, Sakkalis T. Real space curves are not “unit speed”. Comput Aided Geom Des, 2007, 24: 238–240MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang D M. ed. Selected Lectures in Symbolic Computation (in Chinese). Beijing: Tsinghua University Press, 2003Google Scholar
  14. 14.
    Zoutendijk G. Methods of Feasible Directions: A Study in Linear and Nonlinear Programming. Amsterdam/New York: Elsevier Publishing Company, 1960Google Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  2. 2.LIP6, CNRS—Université Pierre et Marie CurieParis cedex 05France
  3. 3.LMIB-School of Mathematics and Systems ScienceBeihang UniversityBeijingChina

Personalised recommendations