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Mathematical Problems in Engineering
Volume 2015, Article ID 735629, 12 pages
Research Article

Shape Modification for -Bézier Curves Based on Constrained Optimization of Position and Tangent Vector

1Department of Applied Mathematics, Xi’an University of Technology, Xi’an 710054, China
2College of the Arts, Xi’an University of Technology, Xi’an 710048, China

Received 7 October 2014; Revised 15 January 2015; Accepted 23 January 2015

Academic Editor: Chih-Cheng Hung

Copyright © 2015 Gang Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Besides inheriting the properties of classical Bézier curves of degree , the corresponding -Bézier curves have a good performance on adjusting their shapes by changing shape control parameter. Specially, in the case where the shape control parameter equals zero, the -Bézier curves degenerate to the classical Bézier curves. In this paper, the shape modification of -Bézier curves by constrained optimization of position and tangent vector is investigated. The definition and properties of -Bézier curves are given in detail, and the shape modification is implemented by optimizing perturbations of control points. At the same time, the explicit formulas of modifying control points and shape parameter are obtained by Lagrange multiplier method. Using this algorithm, -Bézier curves are modified to satisfy the specified constraints of position and tangent vector, meanwhile the shape-preserving property is still retained. In order to illustrate its ability on adjusting the shape of -Bézier curves, some curve design applications are discussed, which show that the proposed method is effective and easy to implement.