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The Voronoi diagram of circles and its application to the visualization of the growth of particles

Abstract:

Circles are frequently used for modelling the growth of particle aggregates through the Johnson-Mehl tessellation, that is a special instance of the Voronoi diagram of circles. Voronoi diagrams allow one to answer proximity queries after locating a query point in the Voronoi zone it belongs to. The dual graph of the Voronoi diagram is called the Delaunay graph. In this paper, we first show a necessary and sufficient condition of connectivity of the Voronoi diagram of circles. Then, we show how the Delaunay graph of circles (the dual graph of the Voronoi diagram of circles) can be computed exactly, and in a much simpler way, by computing the eigenvalues of a two by two matrix. Finally, we present how the Voronoi diagram of circles can be used to model the growth of particle aggregates. We use the Poisson point process in the Voronoi diagram of circles to generate the Johnson-Mehl tesselation. The Johnson-Mehl model is a Poisson Voronoi growth model, in which nuclei are generated asynchronously using a Poisson point process, and grow at the same radial speed. Growth models produce spatial patterns as a result of simple growth processes and their visualization is important in many technical processes. © Springer-Verlag Berlin Heidelberg 2009.