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

Abstract

L-Bases andB-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 bothL-bases andB-bases. We establish that certain proper subclasses of bivariate Lagrange and Newton bases areL-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 NewtonL-bases. A new geometric point-line duality betweenL-bases andB-bases is described: lines inL-bases correspond to points or vectors inB-bases and concurrent lines map to collinear points and vice versa. This duality betweenL-bases andB-bases is then used to establish that certain proper subclasses of power bases areB-bases and are dual to LagrangeL-bases. This geometric duality is further used to describe the lattices that admit powerB-bases.B-bases dual to NewtonL-bases are also investigated. Duality can also be used to develop change of basis algorithms with computational complexityO(n3) between any twoL-bases and/orB-bases. We describe, in particular, a new change of basis algorithm from a bivariate LagrangeL-basis to a bivariate Bernstein basis with computational complexityO(n3).