Mollified zone diagrams and their computation
Transactions on Computational Science XIV
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Voronoi diagram of points in the Euclidean plane and its computation is foundational to computational geometry. Polynomial root-finding is the origin of fundamental discoveries in all of mathematics and sciences.There is an intrinsic connection between polynomial root-finding in the complex plane and the approximation of Voronoi cells of its roots via a fundamental family of iteration functions, theBasic Family. For instance, the immediate basin of attraction of a root of a complex polynomial under Newton's method is arough approximation to its Voronoi cell. We formally introduce these connections through the Basic Family of iteration functions, its properties with respect to Voronoi diagrams, and a corresponding visualization called polynomiography. Polynomiography is a medium for art, math, education and science. By making use of the Basic Family we introduce a layering of the points within each Voronoi cell of a polynomial root and study its properties and potential applications. In particular, we prove some novel results about the Basic Family in connection with Voronoi diagrams.