Curve construction based on four alpha beta-Bernstein-like basis functions

Authors：Zhu Yuanpeng; Han Xuli; Liu Shengjun^*

Source：Journal of Computational and Applied Mathematics, 2015, 273: 160-181.

DOI：10.1016/j.cam.2014.06.014

• Said-Ball basis
• Bernstein basis
• Quasi Extended Chebyshev space
• B-spline curve
• Shape parameter
• Totally positive property

Four new alpha beta-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 alpha beta-Bernstein-like basis is an optimal normalized totally positive basis. In order to compute the corresponding alpha beta-Bezier-like curves stably and efficiently, a new corner cutting algorithm is developed. Necessary and sufficient conditions are derived for the planar alpha beta-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 alpha beta-Bernstein-like basis, a class of alpha beta-B-spline-like basis functions with two local exponential shape parameters is constructed. Their totally positive property is also proved. The associated alpha beta-B-splinelike curves have C-2 continuity at single knots and include the cubic non-uniform B-spline curves as a special case, and can be C-2 boolean AND FCk+3 (k is an element of 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.

• Institution
  中南大学