Curve construction based on four [formula omitted]-Bernstein-like basis functions

Abstract

Four new αβ-Bernstein-like basis functions with two exponential shape parameters, are constructed in this paper, which include the cubic Said–Ball basis functions and the cubic Bernstein basis functions. Within the general framework of Quasi Extended Chebyshev space, we prove that the proposed αβ-Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding αβ-Bézier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar αβ-Bézier-like curve having single or double inflection points, a loop or a cusp, or be locally or globally convex in terms of the relative position of its control polygons’ side vectors. Based on the new proposed αβ-Bernstein-like basis, a class of αβ-B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated αβ-B-spline-like curves have C2 continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be C2∩FCk+3 (k∈Z+) continuous for particular choice of shape parameters. The exponential shape parameters serve as tension shape parameters and play a predictable adjusting role on generating curves.