CS 691g: Computational Geometry Voronoi Diagram and Delaunay Triangulation

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CS 691g Computational GeometryVoronoi Diagram
and Delaunay Triangulation

• Ileana Streinu Oliver Brock
• Fall 2005

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The Post Office Problem

• Which is the closest post office to every house?
  (Don Knuth)
• Given n sites in the plane
• Subdivision of planebased on proximity

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Voronoi Diagram
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Descartes in 1644 Gravitational Influence of
stars
René Descartes 1596-1650
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Distribution of McDonalds in SF
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Soap Bubble in a Frame
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Honeycomb
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Dragonflys Wing
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Graphic by D'Arcy Thompson
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Installation by Scott Snibbe, 1998
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Uses for Voronoi Diagram

• Anthropology and Archeology -- Identify the parts
  of a region under the influence of different
  Neolithic clans, chiefdoms, ceremonial centers,
  or hill forts.
• Astronomy -- Identify clusters of stars and
  clusters of galaxies (Here we saw what may be the
  earliest picture of a Voronoi diagram, drawn by
  Descartes in 1644, where the regions described
  the regions of gravitational influence of the sun
  and other stars.)
• Biology, Ecology, Forestry -- Model and analyze
  plant competition ("Area potentially available to
  a tree", "Plant polygons")
• Cartography -- Piece together satellite
  photographs into large "mosaic" maps
• Crystallography and Chemistry -- Study chemical
  properties of metallic sodium ("Wigner-Seitz
  regions") Modelling alloy structures as sphere
  packings ("Domain of an atom")
• Finite Element Analysis -- Generating finite
  element meshes which avoid small angles
• Geography -- Analyzing patterns of urban
  settlements
• Geology -- Estimation of ore reserves in a
  deposit using information obtained from bore
  holes modelling crack patterns in basalt due to
  contraction on cooling

• Geometric Modeling -- Finding "good"
  triangulations of 3D surfaces
• Marketing -- Model market of US metropolitan
  areas market area extending down to individual
  retail stores
• Mathematics -- Study of positive definite
  quadratic forms ("Dirichlet tessellation",
  "Voronoi diagram")
• Metallurgy -- Modelling "grain growth" in metal
  films
• Meteorology -- Estimate regional rainfall
  averages, given data at discrete rain gauges
  ("Thiessen polygons")
• Pattern Recognition -- Find simple descriptors
  for shapes that extract 1D characterizations from
  2D shapes ("Medial axis" or "skeleton" of a
  contour)
• Physiology -- Analysis of capillary distribution
  in cross-sections of muscle tissue to compute
  oxygen transport ("Capillary domains")
• Robotics -- Path planning in the presence of
  obstacles
• Statistics and Data Analysis -- Analyze
  statistical clustering ("Natural neighbors"
  interpolation)
• Zoology -- Model and analyze the territories of
  animals

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Delaunay Triangulation (1934)
Boris Nikolaevich Delone (1890 - 1980)
Dual of Voronoi (graph theoretic, topological,
combinatorial)
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Delaunay Triangulation Properties

• maximizes minimum angle in each triangle
• minimizes maximum radius of circumcircle and
  enclosing circle
• minimizes sum of inscribed radii
• many more

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Finite Element Analysis
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Function Interpolation
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D06 Delaunays Proof
Given a triangulation of n sites such that
for every pair of adjacent tirangles abc and
bcd a is not in the circumcircle of bcd, then
that triangulation is the Delaunay triangulation.
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d
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Sidebar The Power of a Point
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C
the power of x with respect to C is ax bx
defined to be positive if x is outside of C
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D06 Delaunays Proof
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D08 Minimum Spanning Tree
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Fortunes Algorithm
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Sweeping the Cones
? pvw ? vwu
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Parabolic Front
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Evolution of the Parabolic Front
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Site Event
a)
b)
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Circle Event
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Event Scheduling
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(No Transcript)
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