Conversion from NURBS to Bézier representation

With the help of the Cox-de Boor recursion formula and the recurrence relation of the Bernstein polynomials, two categories of recursive algorithms for calculating the conversion matrix from an arbitrary non-uniform B-spline basis to a Bernstein polynomial basis of the same degree are presented. One is to calculate the elements of the matrix one by one, and the other is to calculate the elements of the matrix in two blocks. Interestingly, the weights in the two most basic recursion formulas are directly related to the weights in the recursion definition of the B-spline basis functions. The conversion matrix is exactly the Bézier extraction operator in isogeometric analysis, and we obtain the local extraction operator directly. With the aid of the conversion matrix, it is very convenient to determine the Bézier representation of NURBS curves and surfaces on any specified domain, that is, the isogeometric Bézier elements of these curves and surfaces.