Delaunay Triangulation

1. Delaunay Triangulation Jyun-Ming Chen Reference: deBerg et al.

2. Angle vector of T: sorted internal angles Angle-optimal triangulation T : A(T)  A(T’) for all triangulations T’ of P Important Concepts

3. A(T) < A(T’) Important Concepts • Illegal edge, edge-flip and angle-vector • Legality test: [The correctness of this criterion follows from Thales’ theorem]

4. p s q r Proof • We need to show that angle vector gets bigger if illegal edge is flipped • There 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

5. qsp > qrp qsr > qpr qps > qpr Thales’ theorem q r p s

6. pqs > prs sqr > rps qrs > prs q r p s

7. 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 ?!

8. This was first suggested by Lawson(77) Incremental Algorithm

9. Recursive Call Complication on presence of p-1, p-2,p-3

10. p s q r Swap Test (Text, exercise 9.5) Note the triangle pqr is CCW!

11. q1 q2 p s q r Swap Test (TRIPACK, Renka96) For numerical robustness (next page): Swap if sin12  - swtol Just check the sign!!

12. [2] [3] Numerical Robustness 2 cocircular Almost cocircular 3 4 1 [do not flip – avoid cycling] [4] Don’t flip flip [1]

13. Observation • Every new edge created due to insertion of pr is incident to pr This is also called the Bowyer-Watsonalgorithm

14. Boywer-Watson Algorithm Bowyer-Watson triangulation: circum-circles that contain the new point, and the resulting triangulation

16. 2 5 3 1 4 • 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

17. 2 5 3 1 4 -1 (0) (i) (iv) -2 -3

18. 2 5 3 1 4 -1 (0) (iv) (iv) -2 -3

19. 2 5 3 1 4 -1 (iii) (0) (i) -2 -3

20. 2 5 3 1 4 -1 (ii) (0) -2 -3

21. 2 5 3 1 4 -1 (0) (i) (iii) -2 -3

22. 2 5 3 1 4 -1 (ii) (iii) -2 -3

23. 2 5 3 1 4 -1 (0) (0) -2 -3

24. 2 5 3 1 4 -1 -2 -3

32. Edge swap complexity!?

35. Choosing Initial Triangle • far away so that they don’t influence the DT of P • Don’t want to introduce huge coordinate (numerical errors)

36. 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}

37. 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

38. Logical Analysis [i,j,k,l] i<0  j<0 NOT( i<0  j<0) [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

39. To locate the triangle containing pr Data structure D as described in text Point-Location Subproblem

40. Alternate Point-Location Test Abstract of [Mucke et al. 96]

41. Explanation from [Kallmann et al.03]

42. Example p

43. AP: EMST (exercise 9.11) • Q(n2) edges, standard MST algorithm O(n2) • EMST  DG • O(nlogn) algorithm to compute EMST for P • TSP …