Volume 33, Issue 6
New Trigonometric Basis Possessing Exponential Shape Parameters

Yuanpeng Zhu & Xuli Han


J. Comp. Math., 33 (2015), pp. 642-684.

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  • Abstract

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 C² ∩ FC³ 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.

  • History

Published online: 2015-12

  • AMS Subject Headings

65D07, 65D18.

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