Modified Jacobi–Bernstein basis transformation and its application to multi-degree reduction of Bézier curves

• We establish and prove the basis transformation between the modified Jacobi polynomials (MJPs) and Bernstein polynomials and vice versa.
• This transformation is merging the perfect Least-square performance of the MJPs with the geometrical insight of the Bernstein form.
• The MJPs with indexes corresponding to the number of endpoint constraints are the natural basis functions for Least-square approximation of Bezier curves.
• Several numerical results are considered.

This paper reports new modified Jacobi polynomials (MJPs). We derive the basis transformation between MJPs and Bernstein polynomials and vice versa. This transformation is merging the perfect Least-square performance of the new polynomials together with the geometrical insight of Bernstein polynomials. The MJPs with indexes corresponding to the number of endpoints constraints are the natural basis functions for Least-square approximation of Bézier curves. Using MJPs leads us to deal with the constrained Jacobi polynomials and the unconstrained Jacobi polynomials as orthogonal polynomials. The MJPs are automatically satisfying the homogeneous boundary conditions. Thereby, the main advantage of using MJPs, in multi-degree reduction of Bézier curves on computer aided geometric design (CAGD), is that the constraints in CAGD are also satisfied and that decreases the steps of multi-degree reduction algorithm. Several numerical results for the multi-degree reduction of Bézier curves on CAGD are given.