Delaunay Triangulations

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Delaunay Triangulations

• 2000 TC Lab CG Study
• Jo, Byung-Cheol

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Contents

• Terrain Triangulation
• Legal edge and Legal Triangulation
• Delaunay Triangulation and its property
• Computing Delaunay Triangulation
• Analysis of Randomized Incremental algorithm of
  Delaunay Triangulation

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Terrain Triangulation

• Terrain
• 2D Surface Approximation of 3D Surface
• Perspective View, contour lines
• How can make terrain?
• With polynomial meshes (triangle)
• Triangulate sample points.

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Flipping Triangulation Edge

• See Figure 9.3.
• For two adjacent triangles, select a cross edge
  that maximize the minimum angle
• For we make triangle terrain, we need
  Triangulation algorithm with Legal Edge.

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Triangulation of Planar Points

• (Th9.1) Any Triangulation of P has 2n - 2 - k
  triangles and 3n - 3 - k edges.
• Proof) by Eulers formula

Number of triangles m
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Legal Illegal Edge

• (Th.9.2.) In p.184 Figure,
• Proof) See Figure.
• Legal Illegal Edge
• If miminum angle is smaller than mininum angle of
  flipping edge e, then e is illegal edge
• Legalize Edge by flipping (Fig. 9.4.)

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Legal Triangulation

• (Ob.9.3.) If an illegal edge e is flipped, then
  the triangulation is more angle-optimal
• (Lemma.9.4.) for p,q,r,s, pq is illegal lt-gt r is
  in the circle pqs
• Legal Triangulation Algorithm (p.185)
• But we cant guarentee that algorithm terminate...

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Delaunay Triangulation

• Revisit Voronoi Diagram
• Make the dual graph of Voronoi Subdivision
• Region Vertex
• Two regions share a edge Get a edge between
• The Dual graph of Voronoi Diagram DG(P), we say
  that Delaunay Graph or triangulation
• See Figure 9.5, 9.6

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Delaunay Triangulation

• Delaunay Triangulation is
• A Plane Graph (Th.9.5.)
• All edges are Legal. (Th.9.6., Th.9.7., Th.9.8.)
• Most Angle-optimal Triangulation (Th.9.9.)

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Computing Delaunay Triangulation

• Randomized incremental algorithm
• Start with LARGE Virtual Triangle (p.189 Fig.)
• Add point select with random sequence, and
  legalize edges opposite of added point p.(with
  only two case - Fig. 9.7.)
• How to find the triangle containing p? -gt
  Directed Graph Structure D represents
  Triangles split. (Fig. 9.9.)

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Computing Delaunay Triangulation

• How to make LARGE initial triangle?
• See p.194 Figure
• That ensures that P is contained in this
  triangle.
• How to decide about the legality of the edge
  contain the point of this outline triangle?
• 5-case of decision method (Figures.)

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The Analysis

• (Lemma 9.11.)
• of triangles in DG(P) at most 9n 1 gt of
  triangles created in one step at most 9
• proof)
• In one step, 2(k-3) 3 2k - 3 traingle made(k
  is the of edges incident to p)
• of edges in DG(P) at most 3(r3) - 6 gt Total
  degree of vertices of P 6r
• ( of triangle create in one step) 26 - 3 9

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The Analysis

• (Th.9.12.)
• Expected Time O(n lg n)
• Expected Strogae O(n)
• Proof)
• O(n) storage of triangle is O(n)
• Time expected
• Locating points O(n)
• Destroy triangle create new Sum of card(K(?))

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The Analysis

• Sum of card(K(?))
• For a triangle ?, K(?) is the subset of P that
  lie in the triangle ?.
• (Lemma 9.13.)
• Proof)