CMPUT 498 Delaunay Triangulations

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CMPUT 498Delaunay Triangulations

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Terrain modeling

• Determine the height at an arbitrary point given
  a set of samples
• Improve over nearest-neighbour height

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Triangulations

• A triangulation of a set of points P is a set of
  triangles with apexes among P that partitions the
  convex hull of P
• To build a triangulation, insert edges until no
  more edges can be inserted without intersections

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Triangulations

• Intuition for a plausible triangulation
• do not use triangulations with small angles
• among all triangulations, choose the one that
  maximizes the smallest angle

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Triangulations

• The number of triangles and the number of edges
  are linear in the number of points
• If multiple triangulations have a smallest (set
  of smallest) angles
• choose the one with the largest next-smallest
  angle
• call such a triangulation angle-optimal

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Angle property(Thales theorem)

• angle(arb) gt angle(apb)
• angle(aqb) gt angle(asb)

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Edge flipping

• Execute each edge flip that results in an
  increase in the minimum angle

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Local edge flipping criterion

• To detect whether pipj should be replaced with
  pkpl
• determine whether pl is inside the circle(pi, pj,
  pk)
• or also whether pk is inside the circle(pi, pj,
  pl)

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First attempt

• for each pi, pj, pk
• if a point pl is in the circle(pi, pj, pk)
• do not use the triangle (pi, pj, pk)

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Second attempt
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Delaunay graph

• The Delaunay graph is the dual of the Voronoi
  diagram
• every Voronoi site defines a vertex
• every Voronoi edge defines a Delaunay edge
• The Delaunay triangulation is the
  straight-segment embedding

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Delaunay local characterisations

• Three points pi, pj, pk define a Delaunay
  triangle iff no other point is inside the
  circle(pi, pj, pk)
• Two points pi, pj define a Delaunay edge iff a
  circle exists that is passes by pi, pj and no
  other point

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Delaunay local characterisations

• A triangulation is Delaunay (circle(pi, pj, pk)
  is empty)
• ?
• the triangulation is legal (no edge flips
  possible)
• A triangulation is legal
• ?
• the triangulation is Delaunay

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Uniform treatment of hull edges
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Observations

• New edges are always legal
• Edges only become illegal if a site is inserted
  in their neighborhood
• Infinite loop is not possible
• every flip improves the sorted angle list
• Finding the triangle quickly
• point location

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Reference
Chap. 9