In the first part of this work , we uncover a connection between polynomial-reproducing bases of simplex splines and certain single-element liftings of oriented matroids associated to point configurations. We use this correspondence to construct a broad family of spline bases on generic point multisets, generalizing a known result on Delaunay configurations . Our spline bases are naturally defined on finite knot sets with affine dependencies and higher multiplicities, without need for special treatment of degenerate cases. We reformulate our bases in the language of zonotopal tilings, via the Bohne-Dress theorem, obtaining a link to a known construction algorithm for bivariate spline bases based on centroid triangulations. In the second part of this work, we restrict again our attention to weighted Delaunay configurations and the associated spline bases, and we capitalize on our combinatorial viewpoint to extend, in this restricted case, the well-known bivariate construction algorithm to higher dimensions and generic point multisets. Finally, we employ this machinery to propose algorithms for the determination and evaluation of all multivariate basis spline functionssupported on a given point .
Given a point configuration A , we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical algorithms for the construction and evaluation of the associated spline functions .
R
In the first part of this work
Given a point configuration A
R
bases of simplex splines and certain single-element liftings of oriented matroids associated to point configurations. We use this correspondence to construct a broad family of spline bases on generic point multisets, generalizing a
spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a
R
. Our spline bases are naturally defined on finite knot sets with affine dependencies and higher multiplicities, without need for special treatment of degenerate cases. We reformulate our bases in the language of zonotopal tilings, via the Bohne-Dress theorem, obtaining a link to a known construction algorithm for bivariate spline bases based on centroid triangulations. In the second part of this work, we restrict again
and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn
R
weighted Delaunay configurations and the associated spline bases, and we capitalize on our combinatorial viewpoint to extend, in this restricted case, the well-known bivariate construction algorithm to higher dimensions and generic point multisets. Finally, we employ this machinery to propose
regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical
R
determination
construction
R
all multivariate basis spline functionssupported on a given point
the associated spline functions