death_map_2008_upg


death_map_2008_upg, "The Death Map", a talk given at the University of Pittsburgh at Greensburg (UPG) on 14 November 2008 to an undergraduate audience, sponsored by Professor Gary Hart.

Despite the gruesome title, the subject was primarily Voronoi diagrams, with a little information on their properties and construction, and the creation of "centered" or "centroidal" Voronoi diagrams.

Abstract:

The markings on giraffes are an example of a peculiar kind of irregular pattern that recurs in many other situations, including cracks in drying mud, the territories of competing "tribes" of fire ants, and the network of discrete churning "cells" in boiling liquid. A mathematical structure can be abstracted from these different cases, and used to classify and understand new problems as well. The Voronoi diagram is one tool of the field of computational geometry, a subject you didn't study in high school! We will suggest some areas where the Voronoi diagram can provide insight, we will suggest some of its mathematical properties, talk about its extensions to other geometries, distances and dimensions. We will show a special kind of Voronoi diagram that arises when the defining center points are allowed to adjust themselves, and we will show how random numbers can be used to approximate this shape.

Reference:

  1. The John Snow web site maintained by the UCLA School of Public Health at http://www.ph.ucla.edu/epi/snow.html.
  2. Eldridge Adams,
    Territory Size and Shape in Fire Ants: A Model Based on Neighborhood Interactions,
    Ecology,
    Volume 79, Number 4, June 1998, pages 1125-1134.
  3. Franz Aurenhammer,
    Voronoi diagrams - a study of a fundamental geometric data structure,
    ACM Computing Surveys,
    Volume 23, Number 3, September 1991, pages 345-405.
  4. George Barlow,
    Hexagonal Territories,
    Animal Behavior,
    Volume 22, 1974, pages 876-878.
  5. John Byers,
    Dirichlet Tessellation of Bark Beetle Spatial Attack Points,
    Journal of Animal Ecology,
    Volume 61, 1992, pages 759-768.
  6. John Byers,
    Correct Calculation of Dirichlet Polygon Areas,
    Journal of Animal Ecology,
    Volume 65, 1996, pages 528-529.
  7. Marc deBerg, Marc Krevald, Mark Overmars, Otfried Schwarzkopf,
    Computational Geometry,
    Springer, 2000,
    ISBN: 3-540-65620-0,
    LC: QA448.D38.C65.
  8. Qiang Du, Vance Faber, Max Gunzburger,
    Centroidal Voronoi Tessellations: Applications and Algorithms,
    SIAM Review,
    Volume 41, Number 4, December 1999, pages 637-676.
  9. Herbert Edelsbrunner,
    Geometry and Topology for Mesh Generation,
    Cambridge, 2001,
    ISBN: 0-521-79309-2,
    LC: QA377.E36.
  10. Sandra Hempel,
    The Strange Case of the Broad Street Pump,
    University of California, 2007,
    ISBN13: 978-0520250499,
    LC: RA644.C3.H46.
  11. Christian Icking, Rolf Klein, Peter Koellner, Lihong Ma,
    A Java Applet for the Dynamic Visualization of Voronoi Diagrams,
    https://www.pi6.fernuni-hagen.de/GeomLab/VoroGlide
  12. Steven Johnson,
    The Ghost Map,
    Riverhead, 2006,
    ISBN-13: 978-1594489259,
    LC: RC133.G6.J64.
  13. Lili Ju, Qiang Du, Max Gunzburger,
    Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations,
    Parallel Computing,
    Volume 28, Number 10, October 2002, pages 1477-1500.
  14. Atsuyuki Okabe, Barry Boots, Kokichi Sugihara, Sung Nok Chiu,
    Spatial Tessellations: Concepts and Applications of Voronoi Diagrams,
    Second Edition,
    Wiley, 2000,
    ISBN: 0-471-98635-6,
    LC: QA278.2.O36.
  15. Joseph ORourke,
    Computational Geometry,
    Second Edition,
    Cambridge, 1998,
    ISBN: 0521649765,
    LC: QA448.D38.
  16. Robert Renka,
    Algorithm 772: STRIPACK: Delaunay Triangulation and Voronoi Diagram on the Surface of a Sphere,
    ACM Transactions on Mathematical Software,
    Volume 23, Number 3, September 1997, pages 416-434.
  17. Peter Vinten-Johansen, Howard Brody, Nigel Paneth, Stephen Rachman, Michael Rip,
    Cholera, Chloroform, and the Science of Medicine: A Life of John Snow,
    Oxford University Press, 2003,
    ISBN: 019513544X,
    LC: RA649.5.S66.S647.

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Last revised on 02 February 2024.