Delaunay Triangulation

Delaunay TriangulationJyun-Ming ChenReference: deBerg et al.

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Important ConceptsAngle vector of T: sorted internal anglesAngle-optimal triangulation T : A(T) A(T’) for all triangulations T’ of P

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Important ConceptsIllegal edge, edge-flip and angle-vector

Legality test:

A(T) < A(T’)

[The correctness of this criterion follows from Thales’ theorem]

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ProofWe need to show that angle vector gets bigger if illegal edge is flippedThere are six angles in current configuration:

pqr + psr > 180 are the larger ones;the 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

p

q

s

r

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qps > qpr

qsr > qpr

qsp > qrpq

p

r

sThales’ theorem

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pqs > prs

qrs > prs

sqr > rps

q

p

r

s

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Algorithm

Only flip if ijkl are convex (non-convex cannot flip)It will terminateToo slow to be interestingHow to come up with initial triangulation T ?!

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Incremental AlgorithmThis 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)

0

1111

det

22

22

22

22

yxyx

yxyx

yxyx

yxyx

ssssrrrrqqqqpppp p

q

s

rNote the triangle pqr is CCW!

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Swap Test (TRIPACK, Renka96)

21

21

sin2sin1cos2cos1sin12sin

sin2 2cos

sin1 1cos

180 if swap

qpqrsrsp

qpqrqpqr

srspsrsp

p

q

s

r

1

2

For numerical robustness (next page):Swap if sin12 - swtol Just check

the sign!!

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Numerical Robustnesscocircular Almost cocircular

[do not flip – avoid cycling]

1

2

34

[1]

[2] [3]

[4]

Don’t flipflip

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ObservationEvery 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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1

2

4

35

Create DT(P)•Mark center of circum-circle carefully•Do the in-circle test by comparing distances•Perform all necessary LegalizeEdge steps

(i)

(i)

(i)

-1

-2 -3

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1

2

4

35 (i)

-1

-2 -3

(iv)

(0)

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1

2

4

35

-1

-2 -3

(iv)

(0)

(iv)

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1

2

4

35

-1

-2 -3

(iii)(0)

(i)

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1

2

4

35

-1

-2 -3

(0)

(ii)

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1

2

4

35

-1

-2 -3

(0)

(iii)

(i)

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1

2

4

35

-1

-2 -3

(iii)

(ii)

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1

2

4

35

-1

-2 -3

(0)

(0)

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1

2

4

35

-1

-2 -3

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Edge swap complexity!?

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Choosing Initial Trianglefar away so that they don’t influence the DT of P

Don’t want to introduce huge coordinate (numerical errors)

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The Choice Suggested by Text

Choose them to contain PModify 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 i&j 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[iv]Exactly 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]

i<0 j<0 NOT( i<0 j<0)

[i,j,k,l] all positive

[i,j,k,l] one negative

[i,j,k,l] two negative: ij must be one negative

[i,j,k,l] three negative:impossible

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Point-Location SubproblemTo 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)(n2) edges, standard MST algorithm O(n2)EMST DGO(nlogn) algorithm to compute EMST for PTSP …