New Trigonometric Basis Possessing Denominator Shape Parameters

摘要

Four new trigonometric Bernstein-like bases with two denominator shape parameters (DTB-like basis) are constructed, based on which a kind of trigonometric Bezier-like curve with two denominator shape parameters (DTB-like curves) that are analogous to the cubic Bezier curves is proposed. The corner cutting algorithm for computing the DTB-like curves is given. Any arc of an ellipse or a parabola can be exactly represented by using the DTB-like curves. A new class of trigonometric B-spline-like basis function with two local denominator shape parameters (DT B-spline-like basis) is constructed according to the proposed DTB-like basis. The totally positive property of the DT B-spline-like basis is supported. For different shape parameter values, the associated trigonometric B-spline-like curves with two denominator shape parameters (DT B-spline-like curves) can be C-2 continuous for a non-uniform knot vector. For a special value, the generated curves can be C(2n-1) (n = 1, 2, 3,...) continuous for a uniform knot vector. A kind of trigonometric B-spline-like surfaces with four denominator shape parameters (DT B-spline-like surface) is shown by using the tensor product method, and the associated DT B-spline-like surfaces can be C-2 continuous for a non-uniform knot vector. When given a special value, the related surfaces can be C(2n-1) (n = 1, 2, 3,...) continuous for a uniform knot vector. A new class of trigonometric Bernstein-Bezier-like basis function with three denominator shape parameters (DT BB-like basis) over a triangular domain is also constructed. A de Casteljau-type algorithm is developed for computing the associated trigonometric Bernstein-Bezier-like patch with three denominator shape parameters(DT BB-like patch). The condition for G(1) continuous jointing two DT BB-like patches over the triangular domain is deduced.