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Transactions on Computational Science III pp 20–54Cite as

The Voronoi Diagram of Circles and Its Application to the Visualization of the Growth of Particles

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Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 5300))

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.

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Author information

Authors and Affiliations

  1. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Richard Petersens Plads, DK-2800, Kongens Lyngby, Denmark

    François Anton

  2. Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3

    Darka Mioc

  3. School of Computing, University of Glamorgan, Pontypridd, CF37 1DL, UK

    Christopher Gold

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  1. François Anton
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  2. Darka Mioc
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  3. Christopher Gold
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W., T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  2. OptimaNumerics Ltd., Cathedral House, 23-31 Waring Street, BT1 2DX, Belfast, UK

    C. J. Kenneth Tan

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Anton, F., Mioc, D., Gold, C. (2009). The Voronoi Diagram of Circles and Its Application to the Visualization of the Growth of Particles. In: Gavrilova, M.L., Tan, C.J.K. (eds) Transactions on Computational Science III. Lecture Notes in Computer Science, vol 5300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00212-0_2

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  • DOI: https://doi.org/10.1007/978-3-642-00212-0_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00211-3

  • Online ISBN: 978-3-642-00212-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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