Exploring Voronoi and Delaunay Diagrams in Computational Geometry

1. An Introduction to Computational Geometry:Voronoi and Delaunay Diagrams Joseph S. B. Mitchell Stony Brook University Chapters 3,4: Devadoss-O’Rourke

2. Georgy Voronoi 1868-1908 Voronoi Diagrams

3. Historical Origins and Diagrams in Nature René Descartes 1596-1650 1644: Gravitational Influence of stars Dragonfly wing Giraffe pigmentation Honeycomb Constrained soap bubbles Ack: Streinu&Brock

4. http://uxblog.idvsolutions.com/2011/07/chalkboard-maps-united-states-of.htmlhttp://uxblog.idvsolutions.com/2011/07/chalkboard-maps-united-states-of.html Craigslist Map “A geometric pass at delineating areas within the United States potentially covered by each craigslist site -the United States of Craigslist.” Source: idvsolutions.com

5. VoronoiUSA (source: http://www.reddit.com/r/MapPorn/comments/1hi07c/voronoi_map_thiessen_polygons_of_the_usa_based_on/)

6. Voronoi Applications [Lecture 12, David Mount notes] Voronoi, together with point location search: nearest neighbor queries Facility location: Largest empty disk (centered at a Voronoi vertex) Shape description/approximation: medial axis

7. Starbucks Post Office Problem Query point Post offices

8. Voronoi Diagram Q cocircular Voronoi cell of pi is open, convex Partition the plane according to the equivalence classes: V(Q) = { (x,y) : the closest sites of S to (x,y) is exactly the set Q  S } • |Q| = 1 Voronoi cells (2-faces) • |Q| = 2 Voronoi edges (1-faces) • |Q|  3 Voronoi vertices (0-faces) “cell complex”

9. Example p p Voronoi cell of p

10. Delaunay Diagram A witness to the Delaunayhood of (pi , pj) If no 4 points cocircular (degenerate), then Delaunay diagram is a (very special) triangulation. pi pj Join pito pjiff there is an “empty circle” through piand pj • Equivalent definition: “Dual” of the Voronoi diagram • Applet [Chew]

11. Voronoi/Delaunay

12. Delaunay: An Important, Practical Triangulation Application: Piecewise-linear terrain surface interpolation

13. Terrain Interpolation from Point Sample Data

14. Delaunay: An Important, Practical Triangulation Application: Piecewise-linear terrain surface interpolation

15. Delaunay: An Important, Practical Triangulation Application: Piecewise-linear terrain surface interpolation

16. Delaunay for Terrain Interpolation

17. Voronoi and Delaunay

18. Voronoi and Delaunay

19. Delaunay’s Theorem

20. Delaunay’s Theorem

21. Voronoi and Delaunay Properties • The planar dual of Voronoi, drawn with nodes at the sites, edges straight segments, has no crossing edges [Delaunay]. It is the Delaunay diagram, D(S) (defined by empty circle property). • Combinatorial size:  3n-6 Voronoi/Delaunay edges;  2n-5 Voronoi vertices (Delaunay faces) O(n) • A Voronoi cell is unbounded iff its site is on the boundary of CH(S) • CH(S) = boundary of unbounded face of D(S) • Delaunay triangulation lexico-maximizes the angle vector among all triangulations • In particular, maximizes the min angle

23. Angle Sequence of a Triangulation

24. Angle Sequence of a Triangulation

25. Legal Edges, Triangulations

26. Edge Flips (Swaps)

27. Geometry of Circles/Triangles Thale’s Theorem

28. Legal/Illegal Edges Equivalent: • alpha2 + alpha6 > pi • InCircle(pi ,pk ,pj ,pl )

29. Legal/Illegal Edges

30. Algorithm: Legalize Triangulation

31. Example: 4 points

32. Voronoi and Delaunay Properties • In any partition of S, S=B  R, into blue/red points, any blue-red pair that is shortest B-R bridge is a Delaunay edge. • Corollaries: • D(S) is connected • MST  D(S) (MST=“min spanning tree”) • NNG  D(S) (NNG=“nearest neighbor graph”)

33. Voronoi and Delaunay Properties • In any partition of S, S=B  R, into blue/red points, any blue-red pair that is shortest B-R bridge is a Delaunay edge. • D(S) is connected • MST  D(S) (MST=“min spanning tree”) • NNG  D(S) (NNG=“nearest neighbor graph”) • Voronoi/Delaunay can be built in time O(n log n) • Divide and conquer • Sweep • Randomized incremental • VoroGlideapplet

34. Nearest Neighbor Graph: NNG Directed graph NNG(S):

35. Minimum Spanning Tree; MST • Greedy algorithms: • Prim: grows a tree • Kruskal: grows a forest: O(e log v)

36. Euclidean MST • One way to compute: Apply Prim or Kruskal to the complete graph, with edges weighted by their Euclidean lengths: O(n2 log n) • Better: Exploit the fact that MST is a subgraph of the Delaunay (which has size O(n)): • Apply Prim/Kruskal to the Delaunay: O(n log n), since e = O(n)

37. Euclidean MST

38. Traveling Salesperson Problem: TSP

40. Approximation Algorithms

41. PTAS for Geometric TSP

42. Voronoi Applet VoroGlideapplet

43. Voronoi and Delaunay: 1D Easy: Just sort the sites along the line Delaunay is a path Voronoi is a partition of the line into intervals Time: O(n log n) Lower bound: (n log n), since the Delaunay gives the sorted order of the input points, {x1 , x2 , x3 ,…}

44. Voronoi and Delaunay: 2D Algorithms Naïve O(n2 log n) algorithm: Build Voronoi cells one by one, solving n halfplane intersection problems (each in O(n log n), using divide-and-conquer) Incremental: worst-case O(n2) Divide and conquer (first O(n log n)) Lawson Edge Swap (“Legalize”): Delaunay Randomized Incremental (expected O(n log n)) Sweep [Fortune] Any algorithm that computes CH in R3 , e.g., QHull Qhull website

46. Delaunay: Edge Flip Algorithm Lawson Edge Swap “Legalize” [BKOS] d b Locally non-Delauany a c • Assume: No 4 co-circular points, for simplicity. • Start with any triangulation • Keep a list (stack) of “illegal” edges: • ab is illegal if InCircle(a,c,b,d) • iff the smallest of the 6 angles goes up if flip • Flip any illegal edge; update legality status of neighboring edges • Continue until no illegal edges • Theorem: A triangulation is Delaunay iff there are no illegal edges (i.e., it is “locally Delaunay”) • Only O(n2) flips needed.

47. Connection to Convex Hulls in 3D Delaunay diagram  lower convex hull of the lifted sites, (xi , yi , xi2 +yi2), on the paraboloid of revolution, z=x2 + y2 Upper hull  “furthest site Delaunay” 3D CH applet

50. Delaunay in 2D is Convex Hull in 3D