Fast Algorithm for Composition of the Bernstein Polynomials

Composition of the Bernstein polynomials is an important research topic in computer-aided geometric design. This function is useful in implementing evaluation, subdivision, free-form deformation, trimming, conversion between tensor product and Bézier simplex forms, degree raising etc. To accomplish the composition, some numerically stable algorithms were introduced, such as blossoming algorithm, optimal algorithm. Nevertheless they are computational expensive. Observing that the composition results remain as Bernstein polynomials, we present a fast algorithm to evaluate the coefficients of the resultant polynomials based on polynomials interpolation. The reconstruction matrix used in interpolation is constant if the sampling points are chosen evenly in the parametric domain. Thus it can be computed in advance. To avoid numerical error, we employ a symbolic computation algorithm to evaluate the inverse matrix. The runtime analysis shows that the proposed algorithm is the fastest one among the current algorithms and it does not involve numerical stability, additional storage and code complexity problems occurred during implementation.