Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Abstract

L-Bases and B-bases are two important classes of polynomial bases used for representing surfaces in approximation theory and computer aided geometric design. It is well known that the Bernstein and multinomial (or Taylor) bases are special cases of both L-bases and B-bases. We establish that certain proper subclasses of bivariate Lagrange and Newton bases are L-bases. Furthermore, we present a rich collection of lattices (or point-line configurations) that admit unique Lagrange or Hermite interpolation problems which can be solved quite naturally in terms of Lagrange and Newton L-bases. A new geometric point-line duality between L-bases and B-bases is described: lines in L-bases correspond to points or vectors in B-bases and concurrent lines map to collinear points and vice versa. This duality between L-bases and B-bases is then used to establish that certain proper subclasses of power bases are B-bases and are dual to Lagrange L-bases. This geometric duality is further used to describe the lattices that admit power B-bases. B-bases dual to Newton L-bases are also investigated. Duality can also be used to develop change of basis algorithms with computational complexity O(n 3 ) between any two L-bases andÂor B-bases. We describe, in particular, a new change of basis algorithm from a bivariate Lagrange L-basis to a bivariate Bernstein basis with computational complexity O(n 3 ).