Advertisement

Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3951–3966 | Cite as

The use of Jacobi wavelets for constrained approximation of rational Bézier curves

  • M. R. Eslahchi
  • Marzieh Kavoosi
Article

Abstract

This paper presents an efficient method to solve the approximation problem of the rational Bézier curve by continuous piecewise polynomial curve in \( L_{2} \)-norm. For this purpose, extended Jacobi wavelets together with the Gauss–Jacobi quadrature rules are employed. The proposed technique has some advantages such as: simplicity, high accuracy and fast computation. Also, the method in the paper performs multi-degree reduction at one time and does not need the stepwise computing. Several examples are given to illustrate the effectiveness of the algorithm.

Keywords

Rational Bézier curves Piecewise polynomial approximation Jacobi wavelets Guass–Jacobi quadrature 

Mathematics Subject Classification

65T60 65D17 41A29 

Notes

Acknowledgements

The authors are very grateful to all reviewers for carefully reading the paper and for their useful comments and suggestions.

References

  1. Cai HJ, Wang GJ (2010) Constrained approximation of rational Bézier curves based on a matrix expression of its end points continuity condition. Comput Aided D 42:495–504CrossRefMATHGoogle Scholar
  2. Chen J, Wang GJ (2011) Hybrid polynomial approximation to higher derivatives of rational curves. J Comput Appl Math 235(17):4925–4936MathSciNetCrossRefMATHGoogle Scholar
  3. Christensen O, Christensen KL (2005) Approximation theory: from taylor polynomials to wavelets. Birkhäuser, BaselCrossRefMATHGoogle Scholar
  4. Dannenberg L, Nowacki H (1985) Approximate conversion of surface representations with polynomial bases. Comput Aided Geom D 2(1–3):123–132MathSciNetCrossRefMATHGoogle Scholar
  5. Dehghan M (2002) Fully explicit finite-difference methods for two-dimensional diffusion with an integral condition. Nonlinear Anal Theory 48:637–650MathSciNetCrossRefMATHGoogle Scholar
  6. Dehghan M (2003) Identifying a control function in two-dimensional parabolic inverse problems. Appl Math Comput 143:375–391MathSciNetMATHGoogle Scholar
  7. Dehghan M, Hamedi EA, Khosravian-Arab H (2014) A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J Vib Control 22(6):1547–1559MathSciNetCrossRefMATHGoogle Scholar
  8. Farin G (1983) Algorithms for rational Bézier curves. Comput Aided D 15(2):73–77CrossRefGoogle Scholar
  9. Floater MS (2006) High order approximation of rational curves by polynomial curves. Comput Aided Geom D 23:621–628MathSciNetCrossRefMATHGoogle Scholar
  10. Forrest AR (1972) Interactive interpolation and approximation by Bézier polynomials. Comput J 15(1):71–79MathSciNetCrossRefMATHGoogle Scholar
  11. Hoschek J (1987) Approximation of spline curves. Comput Aided Geom D 4(1–2):59–66CrossRefMATHGoogle Scholar
  12. Hu Q, Xu H (2013) Constrained polynomial approximation of rational Bézier curves using reparameterization. J Comput Appl Math 249:133–143MathSciNetCrossRefMATHGoogle Scholar
  13. Huang Y, Huaming S, Hongwei L (2008) A simple method for approximating rational Bézier curve using Bézier curves. Comput Aided Geom D 25(8):697–699CrossRefMATHGoogle Scholar
  14. Krylov VI (1962) Approximate calculation of integral. Macmillan, New YorkMATHGoogle Scholar
  15. Lee BG, Park Y (1998) Approximate conversion of rational Bézier curves. J KSIAM 2:88–93Google Scholar
  16. Lewanowicz S, Woźny P, Keller P (2011) Polynomial approximation of rational Bézier curves with constraints. Numer Algorithms 59(4):607–622CrossRefMATHGoogle Scholar
  17. Liu LG, Wang GJ (2000) Recursive formulae for Hermite polynomial approximation to rational Bézier curves. In: Martin R, Wang WP (eds) Proceedings of geometry modeling and processing 2000: theory and applications. IEEE Computer Society, Los Alamitos, pp 190–197Google Scholar
  18. Lu LZ (2010) Weighted progressive iteration approximation and convergence analysis. Comput Aided Geom D 27:129–137MathSciNetCrossRefMATHGoogle Scholar
  19. Lu L (2011) Sample-based polynomial approximation of rational Bézier curves. J Comput Appl Math 235:1557–1563MathSciNetCrossRefMATHGoogle Scholar
  20. Razzaghi M, Yousefi S (2002) Legendre wavelets method for constrained optimal control problems. Math Methods Appl Sci 25:529–539MathSciNetCrossRefMATHGoogle Scholar
  21. Rong LJ, Chang P (2016) Jacobi wavelet operational matrix of fractional integration for solving fractional integro-differential equation. J Phys Conf Ser 693(1):012002.  https://doi.org/10.1088/1742-6596/693/1/012002
  22. Sederberg TW, Kakimoto M (1991) Approximating rational curves using polynomial curves. In: Farin G (ed) NURBS for curve and surface design. SIAM, Philadelphia, pp 149–158Google Scholar
  23. Shen J, Tang T, Wang L-L (2011) Spectral methods: algorithms, analysis and applications. Springer, BerlinCrossRefMATHGoogle Scholar
  24. Wang GJ, Tai CL (2008) On the convergence of hybrid polynomial approximation to higher derivatives of rational curves. J Comput Appl Math 214:163–174MathSciNetCrossRefMATHGoogle Scholar
  25. Wang GZ, Zheng JM (1997) Bounds on the moving control points of hybrid curves. CVGIP 59(1):19–25MATHGoogle Scholar
  26. Wang GJ, Sederberg TW, Chen FL (1997) On the convergence of polynomial approximation of rational functions. J Approx Theory 89:267–288MathSciNetCrossRefMATHGoogle Scholar
  27. Woźny P (2013) Construction of dual bases. J Comput Appl Math 245:75–85MathSciNetCrossRefMATHGoogle Scholar
  28. Woźny P (2014) Construction of dual B-spline functions. J Comput Appl Math 260:301–311MathSciNetCrossRefMATHGoogle Scholar
  29. Woźny P, Lewanowicz S (2009) Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials. Comput Aided Geom D 26:566–579CrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran

Personalised recommendations