Delaunay Triangulation

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Delaunay Triangulation

• Jyun-Ming Chen
• Reference deBerg et al.

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Important Concepts

• Angle vector of T sorted internal angles
• Angle-optimal triangulation T A(T) ? A(T) for
  all triangulations T of P

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Important Concepts

• Illegal edge, edge-flip and angle-vector
• Legality test

The correctness of this criterion follows from
Thales theorem
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Proof

• We need to show that angle vector gets bigger if
  illegal edge is flipped
• There are six angles in current configuration
• ?pqr ?psr gt 180 are the larger onesthe rest
  four are the smaller ones
• Need to show that once flipped, all six new
  angles are larger than the four old smaller ones
• The ordering of angle vector is determined by
  smallest angle

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q
r
p
s
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q
r
p
s
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Algorithm

• Only flip if ijkl are convex (non-convex cannot
  flip)
• It will terminate
• Too slow to be interesting
• How to come up with initial triangulation T ?!

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Incremental Algorithm
This was first suggested by Lawson(77)
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Recursive Call
Complication on presence of p-1, p-2, p-3
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Swap Test (Text, exercise 9.5)
Note the triangle ?pqr is CCW!
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Swap Test (TRIPACK, Renka96)
For numerical robustness (next page) Swap if
sin12 ? - swtol
Just check the sign!!
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Numerical Robustness
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cocircular
Almost cocircular
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4
1
do not flip avoid cycling
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Dont flip
flip
1
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Observation

• Every new edge created due to insertion of pr is
  incident to pr

This is also called the Bowyer-Watson algorithm
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Boywer-Watson Algorithm
Bowyer-Watson triangulation circum-circles that
contain the new point, and the resulting
triangulation
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• Create DT(P)
• Mark center of circum-circle carefully
• Do the in-circle test by comparing distances
• Perform all necessary LegalizeEdge steps

-1
(i)
(i)
-2
(i)
-3
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-1
(0)
(i)
(iv)
-2
-3
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-1
(0)
(iv)
(iv)
-2
-3
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-1
(iii)
(0)
(i)
-2
-3
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-1
(ii)
(0)
-2
-3
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-1
(0)
(i)
(iii)
-2
-3
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-1
(ii)
(iii)
-2
-3
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-1
(0)
(0)
-2
-3
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-1
-2
-3
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Edge swap complexity!?
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Choosing Initial Triangle

• far away so that they dont influence the DT of P
• Dont want to introduce huge coordinate
  (numerical errors)

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The Choice Suggested by Text

• Choose them to contain P
• Modify illegal edge test whenever p-i are
  involved
• p-1 is outside any circles defined by P
• p-2 is outside any circles defined by P?p-1
• p-3 is outside any circles defined by P?p-1,p-2

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Case Analysis Is pipj legal?

• 0 Polygon concave legal
• i Both ij negative legal
• ii i,j,k,l positive normal case
• iii Exactly one of (i,j,k,l) negative choose
  the one with positive as legal
• ivExactly two of (i,j,k,l) negatives choose
  the one with smaller negative as legal
• v Exactly three of (i,j,k,l) negatives cannot
  occur

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Logical Analysis
i,j,k,l
ilt0 ? jlt0
NOT( ilt0 ? jlt0)
i,j,k,l two negative i?j must be one negative
i,j,k,l all positive
i,j,k,l one negative
i,j,k,l three negative impossible
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Point-Location Subproblem

• To locate the triangle containing pr
• Data structure D as described in text

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Alternate Point-Location Test
Abstract of Mucke et al. 96
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Explanation from Kallmann et al.03
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Example
p
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AP EMST (exercise 9.11)

• Q(n2) edges, standard MST algorithm O(n2)
• EMST ? DG
• O(nlogn) algorithm to compute EMST for P
• TSP