Fast multidimensional Bernstein-Lagrange algorithms

Abstract

In this paper we present two fast algorithms for the Bézier curves and surfaces of an arbitrary dimension. The first algorithm evaluates the Bernstein-Bézier curves and surfaces at a set of specific points by using the fast Bernstein-Lagrange transformation. The second algorithm is an inversion of the first one. Both algorithms reduce the initial problem to computation of some discrete Fourier transformations in the case of geometrical subdivisions of the d-dimensional cube. Their orders of computational complexity are proportional to those of corresponding d-dimensional FFT-algorithm, i.e. to O (N logN) + O (dN), where N denotes the order of the Bernstein-Bézier curves.