### NEW TRIGONOMETRIC BASIS POSSESSING EXPONENTIAL SHAPE PARAMETERS

Yuanpeng Zhu1,2, Xuli Han3

1. 1 School of Mathematics and Statistics, Central South University, Changsha 410083, China;
2 School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China;
3 School of Mathematics and Statistics, Central South University, Changsha 410083, China
• Received:2013-07-25 Revised:2014-10-14 Online:2015-11-15 Published:2015-11-15
• Supported by:

The research is supported by the National Natural Science Foun-dation of China(No. 60970097, No. 11271376), Postdoctoral Science Foundation of China(2015M571931), and Graduate Students Scientific Research Innovation Project of Hunan Province(No. CX2012B111).

Yuanpeng Zhu, Xuli Han. NEW TRIGONOMETRIC BASIS POSSESSING EXPONENTIAL SHAPE PARAMETERS[J]. Journal of Computational Mathematics, 2015, 33(6): 642-684.

Four new trigonometric Bernstein-like basis functions with two exponential shape parameters are constructed, based on which a class of trigonometric Bézier-like curves, analogous to the cubic Bézier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bézier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bézier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bézier-like curves are closer to the given control polygon than the cubic Bézier curves. Based on the new proposed trigonometric Bernstein-like basis, a new class of trigonometric B-spline-like basis functions with two local exponential shape parameters is constructed. The totally positive property of the trigonometric B-spline-like basis is proved. For different values of the shape parameters, the associated trigonometric B-spline-like curves can be C2FC3 continuous for a non-uniform knot vector, and C3 or C5 continuous for a uniform knot vector. A new class of trigonometric Bézier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bézier-like patch is developed. The conditions for G1 continuous joining two trigonometric Bézier-like patches over triangular domain are deduced.

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