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Mathematical Problems in Engineering
Volume 2019, Article ID 3057134, 9 pages
Research Article

A Generalized Cubic Exponential B-Spline Scheme with Shape Control

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Baoxing Zhang; ten.haey@gnahzgnixoab

Received 30 January 2019; Accepted 2 May 2019; Published 3 June 2019

Academic Editor: Nhon Nguyen-Thanh

Copyright © 2019 Baoxing Zhang 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.


In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters and . By changing the values of and , the sensitivity of the shape of the subdivision curve to the initial control value can be changed and different kinds of curves can then be obtained by adjusting the value of . For this new scheme, we show that, with any admissible choice of and , it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.