Four new   αβ     α  β       -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     α  β       -Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding   αβ     α  β       -Bezier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar   αβ     α  β       -Bezier-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     α  β       -Bernstein-like basis, a class of   αβ     α  β       -B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated   αβ     α  β       -B-spline-like curves have   C 2        C    2          continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be   C 2 ∩FC k+3        C    2    ∩  F    C    k  +  3          (  k∈Z +      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.
['Yuanpeng Zhu', 'Xuli Han', 'Shengjun Liu']
Curve construction based on four αβ-Bernstein-like basis functions