Bézier Curves and Surfaces with three Parameters and Extensions in the Triangular Domain

To define a new basis function to obtain a basis that can inherit the excellent properties of the traditional B-spline method and Bézier method, global and locality of shape adjustment, and can accurately represent the elliptical arc and circle. Firstly, an optimal standard full positive base, the cut angle algorithm, the 1 C and 2 C continuous proof of the base under the quasi-extended Chebyshev space in this paper. Secondly, the base on the rectangular field to the triangular field to obtain the quasi-cubic triangular Bernstein-Bézier base on the triangular field. Thirdly, this base can accurately represent the elliptic arc and circle, and then gives the base cutting algorithm on the triangular domain, and reverse introduce two conditions under which the quasi-cubic triangular Bernstein-Bézier surfaces are 1 G continuous in surface splicing. After a lot of analysis and examples, the new basis function has excellent properties of traditional methods, and can also flexibly adjust the shape parameters to obtain the required curve surface, which meets the actual industrial design requirements.