Change of basis algorithms for surfaces in CAGD Original Research Article

The computational complexity of general change of basis algorithms from one bivariate polynomial basis of degree n to another bivariate polynomial basis of degree n using matrix multiplication is O(n4). Applying blossoming and duality, we derive change of basis algorithms with computational complexity O(n3) between two important classes of polynomial bases used for representing surfaces in CAGD: B-bases and L-bases. Change of basis algorithms for B-bases follow from their blossoming property; change of basis algorithms for L-bases follow from the duality between L-bases and B-bases. The Bézier and multinomial bases are special cases of both B-bases and L-bases, so these algorithms can be used to convert between the Bézier and multinomial forms. We also show that the bivariate Horner evaluation algorithm for the multinomial basis is dual to the bivariate de Boor evaluation algorithm for B-patches.