Smooth convex partition of unity on uniform triangulations with Hermite interpolation using radial ERBS

摘要

In [2] a new general construction of smooth convex partition of unity was proposed for a very general class of covers and partitions of multidimensional domains providing the option of Hermite interpolation on a scattered point set consistent with domain/partition. The tensor-product based and radial-based versions of this construction were studied in further detail in [5] and [3], respectively. In all versions the underlying concept of the construction is the univariate expo-rational B-spline [7] and its generalizations [4]. One of the interesting features of the construction is that, in general, the basis functions generated via it depend on the ordering of the elements of the domain cover/partition and the respective scattered-point set, while for a narrower range of the construction parameters the basis is unique and independent of this ordering. In the present paper we consider the radial-based version of the construction from [3] in the special context of uniform triangulation in the bivariate case, and conduct exhaustive study of all possible cases of different bases obtained in the general, order-dependent, case. We provide graphical comparative visualization of the different cases of basis functions, using 2-dimensional level maps and ray-traced images in 3 dimension.