Abstract
We consider the weighted-multi-degree reduction of Bézier curves. Based on the fact that exact degree reduction is not possible, therefore approximative process to reduce a given Bézier curve of high degree n to a Bézier curve of lower degree m, \(m<n\) is needed. The weight function is used to better representing the approximative curve at some parts that need more details, and the error is greater than other parts. The \(L_2\) norm is used in the degree reduction process. Numerical results and comparisons are supported by examples. The numerical results obtained from the new method yield minimum approximation error, improve the approximation in some parts of the curve, and show up possible applications in science and engineering.
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References
Lutterkort, D., Peters, J., Reif, U.: Polynomial degree reduction in the \(L_2\)-norm equals best Euclidean approximation of Bézier coefficients. Comput. Aided Geom. Des. 16, 607–612 (1999)
Ahn, Y., Lee, B.G., Park, Y., Yoo, J.: Constrained polynomial degree reduction in the \(L_2\)-norm equals best weighted Euclidean approximation of Bézier coefficients. Comput. Aided Geom. Des. 21, 181–191 (2004)
Ait-Haddou, R.: Polynomial degree reduction in the discrete \(L_2\)-norm equals best Euclidean approximation of h-Bézier coefficients. BIT Numer, Math (2016)
Rababah, A., Lee, B.G., Yoo, J.: Multiple degree reduction and elevation of Bézier curves using Jacobi-Bernstein basis transformations. Num. Funct. Anal. Optim. 28(9–10), 1179–1196 (2007)
Rababah, A., Mann, S.: Linear methods for \(G^1\), \(G^2\), and \(G^3-\)multi-degree reduction of Bézier curves. Comput.-Aided Des. 45(2), 405–14 (2013)
Rababah, A., Ibrahim, S.: Weighted \(G^1\)-multi-degree reduction of Bézier curves. Int. J. Adv. Comput. Sci. Appl. 7(2), 540–545 (2016)
Rababah, A., Ibrahim, S.: Weighted degree reduction of Bézier curves with \(G^2\)-continuity. Int. J. Adv. Appl. Sci. 3(3), 13–18 (2016)
Rababah, A., Ibrahim, S.: Weighted \(G^0\)- and \(G^1\) multi-degree reduction of Bézier curves. In: AIP Conference Proceedings 1738, 05, vol. 7, issue 2. p. 0005 (2016). https://doi.org/10.1063/1.4951820
Woźny, P., Lewanowicz, S.: Multi-degree reduction of Bézier curves with constraints, using dual Bernstein basis polynomials. Comput. Aided Geom. Des. 26, 566–579 (2009)
Vijayaragavan, A., Visumathi, J., Shunmuganathan, K.L.: Cubic Bézier curve approach for automated offline signature verification with intrusion identification. Math. Prob. Eng. 2014(Article ID 928039), 8 pp. (2014)
Gan, H.-S., Swee, T.T., Abdul Karim, A.H., Amir Sayuti, K., Abdul Kadir, M.R., Tham, W.-T., Wong, L.-X., Chaudhary, K.T., Ali, J., Yupapin, P.P.: Medical image visual appearance improvement using bihistogram Bezier curve contrast enhancement: data from the osteoarthritis initiative. Sci. World J. 2014(Article ID 294104), 13 pp. (2014)
Lu, L., Wang, G.: Optimal multi-degree reduction of Bézier curves with \(G^2\)-continuity. Comput. Aided Geom. Des. 23, 673–683 (2006)
Lu, L.: Sample-based polynomial approximation of rational Bézier curves. J. Comput. Appl. Math. 235, 1557–1563 (2011)
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The authors would like to thank the reviewers for helpful comments.
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Rababah, A., Ibrahim, S. (2018). Geometric Degree Reduction of Bézier Curves. In: Ghosh, D., Giri, D., Mohapatra, R., Sakurai, K., Savas, E., Som, T. (eds) Mathematics and Computing. ICMC 2018. Springer Proceedings in Mathematics & Statistics, vol 253. Springer, Singapore. https://doi.org/10.1007/978-981-13-2095-8_8
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DOI: https://doi.org/10.1007/978-981-13-2095-8_8
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