International Journal of CAD/CAM

Society for Computational Design and Engineering (한국CDE학회)
권호 표지

[ $C^1$ ] Continuous Piecewise Rational Re-parameterization

  • Liang, Xiuxia (College of Computer Science and Technology, Shandong University) ;
  • Zhang, Caiming (College of Computer Science and Technology, Shandong University) ;
  • Zhong, Li (College of Computer Science and Technology, Yantai normal University) ;
  • Liu, Yi (College of Computer Science and Technology, Shandong University)
  • Published : 2006.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

A new method to obtain explicit re-parameterization that preserves the curve degree and parametric domain is presented in this paper. The re-parameterization brings a curve very close to the arc length parameterization under $L_2$ norm but with less segmentation. The re-parameterization functions we used are $C^1$ continuous piecewise rational linear functions, which provide more flexibility and can be easily identified by solving a quadratic equation. Based on the outstanding performance of Mobius transformation on modifying pieces with monotonic parametric speed, we first create a partition of the original curve, in which the parametric speed of each segment is of monotonic variation. The values of new parameters corresponding to the subdivision points are specified a priori as the ratio of its cumulative arc length and its total arc length. $C^1$ continuity conditions are imposed to each segment, thus, with respect to the new parameters, the objective function is linear and admits a closed-form optimization. Illustrative examples are also given to assess the performance of our new method.

Keywords

References

  1. Farouki, R.T., and Sakkalis, T. (1991), Real rational curves are not 'unit Speed'. Computer Aided Geometric Design 8, 151-157 https://doi.org/10.1016/0167-8396(91)90040-I
  2. Ong, B.H. (1996), An extraction of almost arc-length parameterization for parametric curves. Ann. Number.Math.3, 305-316
  3. Wang, F.C., and Wright, P.K. (1998), Open architecture controllers for machine tools, Part 2: A real time quintic spline interpolator. ASME J. Manufacturing Science and Engineering 120, 415-432
  4. Wang, F.C., Wright, P.K., Barsky, B.A.., and Yang, D.C.H., (1999), Approximately arc-length parameterized $C^3$ quintic interpolating splines. ASME J. Mech. Design 121,430-439 https://doi.org/10.1115/1.2829479
  5. Faitsch, F.N., and Nielson, G.. M.(1992), On the problem of detemining the distance between parametric curves. In: Hangen, H., Curves and Surface Design. SIAM, 123-141
  6. Victoria H.M., and Jorge E. S. (2003), Sampling points on regular parametric curves with control of their distribution, Computer Aided Geometric Design 20, 363-382 https://doi.org/10.1016/S0167-8396(03)00079-7
  7. Farouki, R.T. (1997), Optimal parameterizations. Computer Aided Geometric Design 14,153-168 https://doi.org/10.1016/S0167-8396(96)00026-X
  8. Juttler B. (1997), A vegetarian approach to optimal parameterizations. Computer Aided Geometric Design 14,887-890 https://doi.org/10.1016/S0167-8396(97)00044-7
  9. Costantini P., Farouki R.T., Manni C., and Sestini A. (2001), Computation of optimal composite re-parameterization, Computer Aided Geometric Design 16, 875-897