Abstract
In this paper, under the framework of Extended Chebyshev space, four new generalized quasi cubic trigonometric Bernstein basis functions with two shape functions α(t) and β(t) are constructed in a generalized quasi cubic trigonometric space span{1,sin2t,(1−sint)2α(t),(1−cost)2β(t)}, which includes lots of previous work as special cases. Sufficient conditions concerning the two shape functions to guarantee the new construction of Bernstein basis functions are given, and three specific examples of the shape functions and the related applications are shown. The corresponding generalized quasi cubic trigonometric Bézier curves and the corner cutting algorithm are also given. Based on the new constructed generalized quasi cubic trigonometric Bernstein basis functions, a kind of new generalized quasi cubic trigonometric B-spline basis functions with two local shape functions αi(t) and βi(t) is also constructed in detail. Some important properties of the new generalized quasi cubic trigonometric B-spline basis functions are proven, including partition of unity, nonnegativity, linear independence, total positivity and C2 continuity. The shape of the parametric curves generated by the new proposed B-spline basis functions can be adjusted flexibly.
Highlights
The construction of basis functions with good properties, including partion of unity, nonnegativity and total positivity, is a basic subject within computer aided geometric design (GAGD) and Computer Graphics (CG)
Basis functions with desirable properties have an important role in curves and surfaces construction
In order to control the shape of the curves flexibly, various spline curves possessing shape parameters have been proposed
Summary
The construction of basis functions with good properties, including partion of unity, nonnegativity and total positivity, is a basic subject within computer aided geometric design (GAGD) and Computer Graphics (CG). Basis functions with desirable properties have an important role in curves and surfaces construction. In [1], a kind of G2 continuous rational cubic spline curves with tension shape parameters was developed. In [2], a kind of G2 continuous Beta-spline curves with local bias and tension parameters was proposed. In [3], in the polynomial space {1, t, (1 − t) p , tq }, a kind of variable degree spline curves was constructed. In [4], a kind of changeable degree spline curves was constructed. In [5], new cubic rational B-spline curves with two shape parameters were constructed. In the space span 1, 3t2 − 2t3 , (1 − t)α , t β , a class of αβ–Bernstein basis functions and the related
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