Curve construction based on four αβαβ-Bernstein-like basis functions

Four new αβαβ-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 αβαβ-Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding αβαβ-Bézier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar αβαβ-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 αβαβ-Bernstein-like basis, a class of αβαβ-B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated αβαβ-B-spline-like curves have C2C2 continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be C2∩FCk+3C2∩FCk+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.