An Introduction to Computational Geometry

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An Introduction to Computational Geometry
Joseph S. B. Mitchell Stony Brook University
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Voronoi Diagrams
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Historical Origins and Diagrams in Nature
René Descartes 1596-1650 1644
Gravitational Influence of stars
Dragonfly wing
Giraffe pigmentation
Honeycomb
Constrained soap bubbles
Ack StreinuBrock
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Post Office Problem
Starbucks
Query point
Post offices
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Voronoi Diagram

• Partition the plane according to the equivalence
  classes V(Q) (x,y) the closest sites of S
  to (x,y) is exactly the set Q ? S
• Q 1 Voronoi cells (2-faces)
• Q 2 Voronoi edges (1-faces)
• Q ? 3 Voronoi vertices (0-faces)

Q cocircular
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Example
p
p
Voronoi cell of p
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Delaunay Diagram

• Join pi to pj iff there is an empty circle
  through pi and pj
• Equivalent definition Dual of the Voronoi
  diagram
• Applet Chew

A witness to the Delaunayhood of (pi , pj)
If no 4 points cocircular (degenerate), then
Delaunay diagram is a (very special)
triangulation.
pi
pj
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Voronoi and Delaunay Properties

• The planar dual of Voronoi, drawn with nodes at
  the sites, edges straight segments, has no
  crossing edges Delaunay. It is the Delaunay
  diagram, D(S) (defined by empty circle property).
• Combinatorial size ? 3n-6 Voronoi/Delaunay
  edges ? 2n-5 Voronoi vertices (Delaunay faces)
  O(n)
• A Voronoi cell is unbounded iff its site is on
  the boundary of CH(S)
• ?CH(S) boundary of unbounded face of D(S)
• Delaunay triangulation lexico-maximizes the angle
  vector among all triangulations
• In particular, maximizes the min angle

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Voronoi and Delaunay Properties

• In any partition of S, SB ? R, into blue/red
  points, any blue-red pair that is shortest B-R
  bridge is a Delaunay edge.
• D(S) is connected
• MST ? D(S) (MSTmin spanning tree)
• NNG ? D(S) (NNGnearest neighbor graph)
• Voronoi/Delaunay can be built in time O(n log n)
• Divide and conquer
• Sweep
• Randomized incremental
• VoroGlide applet

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Delaunay Edge Flip Algorithm
Lawson Edge Swap Legalize BKOS

• Assume No 4 co-circular points, for simplicity.
• Start with any triangulation
• Keep a list (stack) of illegal edges
• ab is illegal if InCircle(a,c,b,d)
• iff the smallest of the 6 angles goes up if flip
• Flip any illegal edge update legality status of
  neighboring edges
• Continue until no illegal edges
• Theorem A triangulation is Delaunay iff there
  are no illegal edges (i.e., it is locally
  Delaunay)
• Only O(n2 ) flips needed.

d
b
Locally non-Delauany
a
c
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Connection to Convex Hulls in 3D

• Delaunay diagram ? lower convex hull of the
  lifted sites, (xi , yi , xi 2 yi 2 ), on the
  paraboloid of revolution, zx2 y2
• Upper hull ? furthest site Delaunay
• 3D CH applet

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Voronoi and Delaunay

• Algorithms
• Divide and conquer (first O(n log n))
• Sweep
• Randomized incremental
• Any algorithm that computes CH in R3 , e.g.,
  QHull

Qhull website
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Fortunes Sweep Algorithm
Applet
Ack GuibasStolfi
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Parabolic Front
Applet
Ack GuibasStolfi
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Site Events
a)
b)
c)
Applet
Ack GuibasStolfi
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Circle Events
Applet
Ack GuibasStolfi
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Scheduling Circle Events
Applet
Ack GuibasStolfi
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Incremental Construction

• Add sites one by one, modifying the Delaunay
  (Voronoi) as we go
• Join vi to 3 corners of triangle containing it
• Do edge flips to restore local Delaunayhood
• If added in random order, simple Backwards
  analysis shows expected time O(n log n)
  Guibas, Knuth, Sharir

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Example
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Voronoi Extensions

• Numerous ! see Okabe, Boots, Sugihara, Chiu
• Different metrics
• Higher dimensions
• Delaunay in Rd ? lower CH in Rd1
• O( n log n n ?(d1)/2 ? )
• Order-k, furthest-site
• Other sites Voronoi of polygons, medial axis
• Additive/multiplicative weights power diagram
• ?-shapes a powerful shape representation

GeoMagic, biogeometry at Duke
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VRONI Fast, robust Voronoi of polygonal domains
Incremental algorithm
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Computing offsets with a Voronoi diagram
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Alpha Shapes, Hulls
Erase with a ball of radius ? to get ?-shape.
Straighten edges to get ?-hull
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?-Shapes

• Theorem For each Delaunay edge, e(pi,pj),
  there exists ?min(e)gt0 and ?max(e)gt0 such that e
  ? ?-shape of S iff ?min(e) ? ? ? ?max(e).
• Thus, every alpha-hull edge is in the Delaunay,
  and every Delaunay edge is in some alpha-shape.

Applet