Quasi-Bernstein–Bézier polynomials over triangular domain with multiple shape parameters

Based on a new developed recursive relation, a class of Quasi-Bernstein–Bézier polynomials over triangular domain with multiple shape parameters, which includes the classical Bernstein–Bézier polynomials and the cubic and quartic Said–Ball polynomials over triangular domain as special cases, is constructed. The given polynomials have some important and good properties for surface modeling, such as partition of unity, non-negativity, linear independence and so on. The shapes of the corresponding triangular Quasi-Bernstein–Bézier patch can be modified intuitively and foreseeable by altering the values of the shape parameters without changing the control points. In order to compute the patch stably and efficiently, a new de Casteljau-type algorithm is developed. Moreover, the conditions for G1G1 continuous smooth joining two triangular Quasi-Bernstein–Bézier patches are derived.