Effective algorithm to approximate rational triangular B-B surfaces using polynomial forms

This paper generalizes American computer graphics expert T. Sederberg's idea and algorithm of the Hybrid method, approximating rational Bézier curves with polynomial forms, to the case of rational surfaces defined in the triangular domain, which is adopted widely in engineering. The main work in this paper includes the following aspects: Giving a rational triangular B-B surface; representing each control point of a polynomial triangular B-B surface by a rational triangular B-B surface of the same degree, i.e., regarding each control point of the polynomial surface as a moving point on a rational triangular B-B surface with some constraints, the equivalent Hybrid surface defined in the triangular domain is constructed. With the help of the appropriate construction of Hybrid surface of degree 1 for the given rational triangular B-B surface, a recursive formula is derived for the control points between two Hybrid surfaces that are equivalent with each other but have adjacent degree. Additionally, by fixing a point in the convex hull of each moving control points of the Hybrid surface as the corresponding control points of the Hybrid polynomial B-B surface, an effective algorithm is obtained to approximate rational triangular B-B surfaces by using the corresponding polynomial forms. As an example, a numerical application is presented in this paper. All the results above can improve the feasibility of data exchange and efficiency of computation evidently in computer aided geometric design system.