Computational Geometry
The study of algorithms for
combinatorial, topological, and metric
problems concerning sets of points,
typically in Euclidean space.
Representative areas of research
include geometric search, convexity,
proximity, intersection, and linear
programming.
Online Computing Dictionary
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CS691G Computational Geometry - UMass Amherst
Discrete Geometry
Covering
Tiling
Packing
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CS691G Computational Geometry - UMass Amherst
Video Games
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CS691G Computational Geometry - UMass Amherst
What we saw…
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Walking through large model
Collisions
Dynamic simulation
(Compare with automated movie
generation)
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CS691G Computational Geometry - UMass Amherst
What to look for…
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Algorithms
Complexity
Data structures
Geometric primitives
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CS691G Computational Geometry - UMass Amherst
Proximity Queries
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CS691G Computational Geometry - UMass Amherst
Dynamic Simulation
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CS691G Computational Geometry - UMass Amherst
Dynamic Simulation
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CS691G Computational Geometry - UMass Amherst
Multi-Player Games
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CS691G Computational Geometry - UMass Amherst
Multi-Player Games
• Some players might be computer
generated (animations)
• Distributed state representation
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CS691G Computational Geometry - UMass Amherst
Motion Planning
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CS691G Computational Geometry - UMass Amherst
Kinetic Data Structures
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CS691G Computational Geometry - UMass Amherst
The Post Office Problem
• Which is the closest post office to every
house? (Don Knuth)
• Given n sites in the plane
• Subdivision of plane
based on proximity
Georgy Voronoi
1868-1908
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CS691G Computational Geometry - UMass Amherst
Voronoi Diagram
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CS691G Computational Geometry - UMass Amherst
Descartes in 1644: Gravitational
Influence of stars
René Descartes
1596-1650
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CS691G Computational Geometry - UMass Amherst
Distribution of McDonalds in SF
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CS691G Computational Geometry - UMass Amherst
Soap Bubble in a Frame
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CS691G Computational Geometry - UMass Amherst
Honeycomb
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CS691G Computational Geometry - UMass Amherst
Dragonfly’s Wing
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CS691G Computational Geometry - UMass Amherst
Graphic by D'Arcy Thompson
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CS691G Computational Geometry - UMass Amherst
Installation by Scott Snibbe, 1998
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CS691G Computational Geometry - UMass Amherst
Uses for Voronoi Diagram
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Anthropology and Archeology -- Identify the parts
of a region under the influence of different Neolithic
clans, chiefdoms, ceremonial centers, or hill forts.
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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.)
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Biology, Ecology, Forestry -- Model and analyze
plant competition ("Area potentially available to a
tree", "Plant polygons")
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Cartography -- Piece together satellite
photographs into large "mosaic" maps
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Crystallography and Chemistry -- Study chemical
properties of metallic sodium ("Wigner-Seitz
regions"); Modelling alloy structures as sphere
packings ("Domain of an atom")
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Finite Element Analysis -- Generating finite
element meshes which avoid small angles
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Geography -- Analyzing patterns of urban
settlements
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Geology -- Estimation of ore reserves in a deposit
using information obtained from bore holes;
modelling crack patterns in basalt due to contraction
on cooling
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Geometric Modeling -- Finding "good"
triangulations of 3D surfaces
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Marketing -- Model market of US metropolitan
areas; market area extending down to individual
retail stores
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Mathematics -- Study of positive definite quadratic
forms ("Dirichlet tessellation", "Voronoi diagram")
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Metallurgy -- Modelling "grain growth" in metal
films
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Meteorology -- Estimate regional rainfall averages,
given data at discrete rain gauges ("Thiessen
polygons")
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Pattern Recognition -- Find simple descriptors for
shapes that extract 1D characterizations from 2D
shapes ("Medial axis" or "skeleton" of a contour)
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Physiology -- Analysis of capillary distribution in
cross-sections of muscle tissue to compute oxygen
transport ("Capillary domains")
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Robotics -- Path planning in the presence of
obstacles
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Statistics and Data Analysis -- Analyze statistical
clustering ("Natural neighbors" interpolation)
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Zoology -- Model and analyze the territories of
animals
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CS691G Computational Geometry - UMass Amherst
Voronoi Graph
• Voronoi region Vor(p) (p in set S)
– the set of points on the plane that are closer
to p than to any othe rpoint in S
• Voronoi Graph VOR(S)
– dual to voronoi region graph
– two points are adjacent if their voronoi
regions have common contiguous boundary
(segment)
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CS691G Computational Geometry - UMass Amherst
Voronoi Graph
• Voronoi Graph in the rectilinear plane
• Rectilinear distance: p = (x, y); p’=(x’,y’)
Voronoi region of b
ab
b
a
bc
c
ac
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CS691G Computational Geometry - UMass Amherst
THEOREM: For any set S of points in the plane, MST is
subgraph of the Voronoi Graph VG(S)
PROOF
Let an edge XY between two points X and Y does not belong to
the Voronoi graph VG(S). We will show that there is an X-Ypath in VG(S) which contains edges e1, e2,…,ek each shorter
than XY, this will imply that XY not belong to MST.
Indeed, for each edge eI (I=1,…,k) there is an MST path pI
connecting ends of eI consisting of MST edges each no longer
than eI. The path obtained by concatenating paths p1,…,pk
connects X to Y and contains MST edges each shorter than XY.
Thus XY does not belong to MST.
Now we will find such X-Y-path e1, e2,…,ek in VG(S)
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CS691G Computational Geometry - UMass Amherst
Delaunay Triangulation (1934)
Boris Nikolaevich Delone
(1890 - 1980)
Dual of Voronoi (graph theoretic, topological, combinatorial)
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CS691G Computational Geometry - UMass Amherst
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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CS691G Computational Geometry - UMass Amherst
Finite Element Analysis
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CS691G Computational Geometry - UMass Amherst