On Generalized Jacobi–Bernstein Basis Transformation: Application of Multidegree Reduction of Bأ©zier Curves and Surfaces

Abstract

This paper formulates a new explicit expression for the generalized Jacobi polynomials (GJPs) in terms of Bernstein basis. We also establish and prove the basis transformation between the GJPs basis and Bernstein basis and vice versa. This transformation embeds the perfect leastsquare performance of the GJPs with the geometrical insight of the Bernstein form. Moreover, the GJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for leastsquare approximation of Bأ©zier curves and surfaces. Application to multidegree reduction (MDR) of Bأ©zier curves and surfaces in computer aided geometric design (CAGD) is given.