New Trigonometric Basis Possessing Exponential Shape Parameters

Abstract

Four new trigonometric Bernstein-like basis functions with two exponential shape pa- rameters are constructed, based on which a class of trigonometric Bezier-like curves, anal- ogous to the cubic Bezier curves, is proposed. The corner cutting algorithm for computing the trigonometric Bezier-like curves is given. Any arc of an ellipse or a parabola can be represented exactly by using the trigonometric Bezier-like curves. The corresponding trigonometric Bernstein-like operator is presented and the spectral analysis shows that the trigonometric Bezier-like curves are closer to the given control polygon than the cu- bic Bezier 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 pa- rameters 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 C 2 FC 3 continuous for a non-uniform knot vector, and C 3 or C 5 continuous for a uniform knot vector. A new class of trigonometric Bezier-like basis functions over triangular domain is also constructed. A de Casteljau-type algorithm for computing the associated trigonometric Bezier-like patch is developed. The conditions for G 1 continuous joining two trigonometric Bezier-like patches over triangular domain are deduced.