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

Article

First Online:

- 100 Downloads
- 1 Citations

## 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

- 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 42:495–504CrossRefzbMATHGoogle Scholar
- Chen J, Wang GJ (2011) Hybrid polynomial approximation to higher derivatives of rational curves. J Comput Appl Math 235(17):4925–4936MathSciNetCrossRefzbMATHGoogle Scholar
- Christensen O, Christensen KL (2005) Approximation theory: from taylor polynomials to wavelets. Birkhäuser, BaselCrossRefzbMATHGoogle Scholar
- Dannenberg L, Nowacki H (1985) Approximate conversion of surface representations with polynomial bases. Comput Aided Geom D 2(1–3):123–132MathSciNetCrossRefzbMATHGoogle Scholar
- Dehghan M (2002) Fully explicit finite-difference methods for two-dimensional diffusion with an integral condition. Nonlinear Anal Theory 48:637–650MathSciNetCrossRefzbMATHGoogle Scholar
- Dehghan M (2003) Identifying a control function in two-dimensional parabolic inverse problems. Appl Math Comput 143:375–391MathSciNetzbMATHGoogle Scholar
- 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):1547–1559MathSciNetCrossRefzbMATHGoogle Scholar
- Farin G (1983) Algorithms for rational Bézier curves. Comput Aided D 15(2):73–77CrossRefGoogle Scholar
- Floater MS (2006) High order approximation of rational curves by polynomial curves. Comput Aided Geom D 23:621–628MathSciNetCrossRefzbMATHGoogle Scholar
- Forrest AR (1972) Interactive interpolation and approximation by Bézier polynomials. Comput J 15(1):71–79MathSciNetCrossRefzbMATHGoogle Scholar
- Hoschek J (1987) Approximation of spline curves. Comput Aided Geom D 4(1–2):59–66CrossRefzbMATHGoogle Scholar
- Hu Q, Xu H (2013) Constrained polynomial approximation of rational Bézier curves using reparameterization. J Comput Appl Math 249:133–143MathSciNetCrossRefzbMATHGoogle Scholar
- Huang Y, Huaming S, Hongwei L (2008) A simple method for approximating rational Bézier curve using Bézier curves. Comput Aided Geom D 25(8):697–699CrossRefzbMATHGoogle Scholar
- Krylov VI (1962) Approximate calculation of integral. Macmillan, New YorkzbMATHGoogle Scholar
- Lee BG, Park Y (1998) Approximate conversion of rational Bézier curves. J KSIAM 2:88–93Google Scholar
- Lewanowicz S, Woźny P, Keller P (2011) Polynomial approximation of rational Bézier curves with constraints. Numer Algorithms 59(4):607–622CrossRefzbMATHGoogle Scholar
- 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
- Lu LZ (2010) Weighted progressive iteration approximation and convergence analysis. Comput Aided Geom D 27:129–137MathSciNetCrossRefzbMATHGoogle Scholar
- Lu L (2011) Sample-based polynomial approximation of rational Bézier curves. J Comput Appl Math 235:1557–1563MathSciNetCrossRefzbMATHGoogle Scholar
- Razzaghi M, Yousefi S (2002) Legendre wavelets method for constrained optimal control problems. Math Methods Appl Sci 25:529–539MathSciNetCrossRefzbMATHGoogle Scholar
- 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
- 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
- Shen J, Tang T, Wang L-L (2011) Spectral methods: algorithms, analysis and applications. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Wang GJ, Tai CL (2008) On the convergence of hybrid polynomial approximation to higher derivatives of rational curves. J Comput Appl Math 214:163–174MathSciNetCrossRefzbMATHGoogle Scholar
- Wang GZ, Zheng JM (1997) Bounds on the moving control points of hybrid curves. CVGIP 59(1):19–25zbMATHGoogle Scholar
- Wang GJ, Sederberg TW, Chen FL (1997) On the convergence of polynomial approximation of rational functions. J Approx Theory 89:267–288MathSciNetCrossRefzbMATHGoogle Scholar
- Woźny P (2013) Construction of dual bases. J Comput Appl Math 245:75–85MathSciNetCrossRefzbMATHGoogle Scholar
- Woźny P (2014) Construction of dual B-spline functions. J Comput Appl Math 260:301–311MathSciNetCrossRefzbMATHGoogle Scholar
- Woźny P, Lewanowicz S (2009) Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials. Comput Aided Geom D 26:566–579CrossRefzbMATHGoogle Scholar

## Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017