Power diagrams: properties, algorithms and applications
SIAM Journal on Computing
A relationship between Gale transforms and Voronoi diagrams
Discrete Applied Mathematics
Surfaces over Dirichlet Tessellations
Computer Aided Geometric Design
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Properties of local coordinates based on Dirichlet tessellations
Geometric modelling
Systems of coordinates associated with points scattered in the plane
Computer Aided Geometric Design
Smooth surface reconstruction via natural neighbour interpolation of distance functions
Proceedings of the sixteenth annual symposium on Computational geometry
GMP '00 Proceedings of the Geometric Modeling and Processing 2000
A general geometric construction of coordinates in a convex simplicial polytope
Computer Aided Geometric Design
Smooth natural neighbour interpolants over the whole domain
International Journal of Computational Science and Engineering
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There are two types of the discontinuity of the original Voronoi-based interpolants: one appears on the data sites and the other on the Delaunay spheres. Some techniques are known for reducing the first type of the discontinuity, but not for the second type. This is mainly because the second type of the discontinuity comes from the coordinate systems used for the interpolants. This paper proposes a sequence of new coordinate systems, called the kth-order standard coordinates, for all nonnegative integers k, and shows that the interpolant generated by the kth-order standard coordinates have C^k continuity on the Delaunay spheres. The previously known Voronoi-based interpolants coincide with the cases k=0 and k=1. Hence, the standard coordinate systems constructed in this paper can reduce the second type of the discontinuity as much as we want. In addition, this paper derives a formula for the gradient of the standard coordinates.