Lecture 12: Triangulation

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Lecture 12 Triangulation

• Voronoi diagrams Definitions and Examples
• Properties of Voronoi Diagrams
• Delaunay Triangulation as Dual of Voronoi Diagram
• Properties of Delaunay Triangulations
• Finding a Delaunay Triangulation

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Post Office What is the area of service?
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Definition of Voronoi Diagram

• Let S be a set of n distinct points pi (sites) in
  the plane and let q be a point not in S.
• A point q lies in the Voronoi region
  corresponding to a site pi
• V(pi) if and only if q is closest to pi than
  to any other point of S
• Euclidean_Distance( q, pi ) lt Euclidean_distance(
  q, pj ),
• for each pi ? P, j ? i.
• The Voronoi diagram of S, VD (S), is the
  collection of Voronoi regions for each point of S

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Voronoi Diagram Example1 site
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Two sites form a perpendicular bisector
Voronoi edge is locus of pointsequidistant from
both p and q, and closer to p and q than to
remaining points of S. Voronoi diagram consists
of the two half planes on eitherside H(p,q) and
H(q,p) In general, a Voronoi region is
p
q
H(p, q)
H(q, p)
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Collinear sites form a series of parallel lines
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Non-collinear sites form Voronoi half lines that
meet at a vertex
A vertex hasdegree ? 3
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Voronoi Regions and Segments
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Voronoi Regions and Segments
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Who wants to be a Millionaire?

• Which of the following is true for2-D Voronoi
  diagrams?
• Four or more non-collinear sites are
• sufficient to create a bounded region
• necessary to create a bounded region
• 1 and 2
• none of above

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Who wants to be a Millionaire?

• Which of the following is true for2-D Voronoi
  diagrams?
• Four or more non-collinear sites are
• sufficient to create a bounded region
• necessary to create a bounded region
• 1 and 2
• none of above

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Degenerate Case no bounded region!
v
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Summary of Voronoi Properties

• A point q lies on a Voronoi edge between sites
  pi and pj iff the largest empty circle centered
  at q touches only pi and pj
• A Voronoi edge is a subset of locus of points
  equidistant from pi and pj

pi site points
e Voronoi edge
v Voronoi vertex
v
pi
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Summary of Voronoi Properties

• A point q is a vertex iff the largest empty
  circle centered at q touches at least 3 sites
• A Voronoi vertex is an intersection of 3 or more
  segments, each equidistant from a pair of sites

pi site points
e Voronoi edge
v Voronoi vertex
v
pi
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Delaunay Triangulation

• Let Spi be the site points such that 4 or
  more are not co
• circular. Then the Delaunay triangulation and the
  Voronoi
• diagram are dual as plane graphs
• Every Delaunay node pi corresponds to a Voronoi
  region V(pi )
• Every Delaunay edge pi pj (i?j) corresponds to
  an edge shared by two Voronoi regions V(pi ) and
  V(pj ).
• Every Voronoi vertex v corresponds to a Delaunay
  face

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Delaunay Triangulation

• A Voronoi region

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Delaunay Triangulation

• A Voronoi diagram

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Delaunay Triangulation

• A Voronoi diagram and its dual, a
• Delaunay Triangulation. To build
• the Delaunay triangulation, draw a
• line segment between any two sites
• whose Voronoi regions share an
• edge. This procedure splits the
• convex hull of the sites into
• triangles.

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Delaunay Triangulation - Properties

• 1. Every edge belonging to the convex hull of
  Spi is an edge of any Delaunay
• triangulation
• Let e be an internal edge of the triangulation.
• 2. Max-min angle property Either the
  quadrilateral Q formed by the two triangles
• sharing the edge e is not convex or e is the
  diagonal of Q which maximizes the
• minimum of 6 internal angles associated with each
  of the two possible
• triangulations of Q.

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Delaunay Triangulation- Properties

• 3. Local empty-circle property
• The circumcircle of the 2 triangles
• sharing the edge e does not contain the
• vertex of the other triangle in its
• interior.
• Note Properties 2 and 3 are locally
• equivalent. Basically they say that
• a Delaunay triangulation is well
• balanced the triangles tend
• toward equiangularity

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Delaunay Triangulation- Properties

• It is dual to the Voronoi diagram computing one
  automatically gives you the other
• It is a planar graph by Eulers formula, it has
  at most 3n-6 edges and at most 2n-5 triangles.
  This property can be used to reduce many problems
  with quadratic size (like closest pair) down to
  linear size in the time it takes to construct the
  triangulation

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Delaunay Triangulation -- Special Case

• When more than three Voronoi sites are
  co-circular, the Delaunay triangulation is not a
  triangulation

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Delaunay Triangulation vs. Triangulation

• A triangulation of a finite point set S is a
  triangulation of the convex hull of the set S
  that uses all the points of S. The line segments
  of the triangulation may not cross, they may meet
  only at shared endpoints, points of S.
• A triangulation of a point set S is a Delaunay
  triangulation if the circumcircle of every
  triangle is point free

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Finding a Delaunay triangulationIncremental
construction

• This algorithm works by growing a current
  triangulation, triangle by triangle.
• Initially the current triangulation consists of a
  single hull edge. At completion the current
  triangulation equals the Delaunay triangulation
• At each iteration the algorithm seeks a new
  triangle which attaches to the frontier of the
  current triangulation
• Classification of edges dormant not yet
  discovered by the algorithm live it has been
  discovered but only one face is known dead
  edge has been discovered and both faces are known
• Initially one live edge, the remaining edges
  are dormant
• As the algorithm proceeds, edges transition from
  dormant to live, then from live to dead. The
  frontier at each stage consists of the set of
  live edges

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Finding a Delaunay triangulationIncremental
construction

• In each iteration, select any edge e of the
  frontier and process it, i.e. seek edges e
  unknown face. Each edge is directed such that its
  unknown face (to be sought) lies to the right of
  the edge
• This face can be some triangle t determined by e
  and some third vertex v. In this case edge e dies
  since both of its faces are known now and each of
  the other two edges of triangle t transition to
  the next state
• Vertex v the mate of edge e.
• If face turns out to be the unbounded plane, edge
  e simply dies
• Attach picture Growing a Delaunay triangulation
• Maintain a dictionary frontier of live edges.
  Look-up in the dictionary using a method edgeComp

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Finding a Delaunay triangulationIncremental
construction

• How does the frontier change from one iteration
  to the next? Method updateFrontier that updates
  the dictionary frontier of live edges
• When a new triangle t attaches to the frontier
  the state of the triangles three edges changes
• The edge of t that attaches to the frontier
  changes from live to dead remove it from the
  dictionary
• Each of the two remaining edges of t changes
  state
• from dormant to live if the edge is not already
  in the dictionary or
• From live to dead if the edge is already in the
  dictionary
• (See next slide)

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Finding a Delaunay triangulationIncremental
construction

• Here we process the live edge af and, upon
  discovering that point b is its mate, add
  triangle afb to the current triangulation
• Then look up the edge fb in the dictionary. Since
  it is not present, its state changes from dormant
  to live
• To update the dictionary, flip fb so its unknown
  face lies to its right and then insert the edge
  into the dictionary
• Next, look up the edge ba in the dictionary it
  is present (live, with known face triangle abc).
  Its unknown face, triangle afb, has just been
  discovered, so remove the edge from the
  dictionary

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Finding a Delaunay triangulationIncremental
construction

• Finding the mate of an edge
• Any edge ab determines an infinite family of
  circles whose boundaries contain both points a
  and b. The centers of the circles lie along edge
  abs perpendicular bisector. Parameterize the
  perpendicular bisector and let Cr denote the
  circle corresponding to parametric value r.
• If abs known face is unbounded, then r minus
  infinity and Cr is the half plane to the left of
  ab
• Suppose that Cr is the circumcircle of abs
  known face. Seek for the smallest value t gt r
  such that some point of S lies in the boundary of
  Ct
• If no such t exists, then edge ab has no mate
• Imagine blowing a two-dimensional bubble through
  the edge ab. If the bubble eventually reaches
  some point of S, this point is the mate of edge ab

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Finding a Delaunay triangulationIncremental
construction

• Finding the mate of an edge

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Finding a Delaunay triangulationIncremental
construction

• Efficiency
• One edge leaves the frontier at each iteration.
  Since every edge leaves the frontier exactly
  once, the number of iterations equals the number
  of edges in the Delaunay triangulation
• No more than O(n) edges in the triangulation
  O(n) iterations
• Because it spends O(n) time per iteration O(n2)

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Resources
Interactive demonstration of construction of
Voronoi diagram and Delaunay triangulation

• http//www.cs.cornell.edu/Info/People/chew/Delauna
  y.html

Animation of Fortunes plane sweep algorithm for
Voronoi diagrams
http//www.diku.dk/hjemmesider/studerende/duff/For
tune/