Grids Based on Circles and Spheres
Leonardo Traversoni (Universidad Autonoma Metropolitana, Mexico)
Based on Voronoi tessellation, Robin Sibson invented in 1980 for geographic
the Natural Neighbor Interpolant. Its pointwise definition, which is purely
geometric, made it
impractical for broader use although it seemed to have very good properties.
In 1998 Watson and Sukumar published a method to use Natural Neighbors on Finite
that they baptized the Natural Element Method, partially based on an idea I
published in 1994
as they cite. However, even when they cite me, they don't use the main idea of
that paper and they remain attached to triangles for the partition.
The main idea I want to show is that there is a dual of the Voronoi tessellation
that I call Covering Spheres, far less known than the Delaunay triangles, that
very useful to be the base of a partition that is, in each of its tiles, the
natural neighbor interpolant and can be used with advantage on finite elements
plane, the sphere and the space. I also show how this can be used in a multigrid
or multilevel approximation.