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

2. Voronoi Diagrams
Georgy Voronoi

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

4. http://uxblog. idvsolutions
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. Voronoi USA
(source:

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

7. Starbucks
Post Office Problem
Query point
Post offices

8. Voronoi Diagram
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| = Voronoi cells (2-faces)
|Q| = Voronoi edges (1-faces)
|Q|  Voronoi vertices (0-faces)
Q cocircular
Voronoi cell of pi is open, convex
“cell complex”

9. Example p p
Voronoi cell of p

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

11. Voronoi/Delaunay

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

13. Terrain Interpolation from Point Sample Data
Got to here, 10/8/13… just started to discuss the terrain interpolation problem.

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
Got to this slide, 10/25/12

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”)
Got to here 3/26/13

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
VoroGlide applet

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
VoroGlide applet
Got to here 3/26/13

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]
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. d b
Locally non-Delauany a c

47. Connection to Convex Hulls in 3D
Delaunay diagram  lower convex hull of the lifted sites, (xi , yi , xi 2 +yi 2 ), 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

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

52. Voronoi and Delaunay Algorithms: Straightforward incremental: O(n2 )
Divide and conquer (first O(n log n))
Sweep
Randomized incremental
Any algorithm that computes CH in R3 , e.g., QHull
Qhull website

53. Straightforward Incremental Algorithm: Voronoi

54. Straightforward Incremental Algorithm: Voronoi

55. Fortune’s Sweep Algorithm
AMS 545 / CSE 555
Applet
Ack: Guibas&Stolfi

56. Parabolic Front
AMS 545 / CSE 555
Applet
Ack: Guibas&Stolfi

57. Site Events a) b) c)
AMS 545 / CSE 555
Applet
Ack: Guibas&Stolfi

58. Circle Events
AMS 545 / CSE 555
Applet
Ack: Guibas&Stolfi

59. Scheduling Circle Events
AMS 545 / CSE 555
Applet
Ack: Guibas&Stolfi

62. Incremental Construction
Add sites one by one, modifying the Delaunay (Voronoi) as we go:
Join vi to 3 corners of triangle containing it
Do edge flips to restore local Delaunayhood
If added in random order, simple “Backwards analysis” shows expected time O(n log n) [Guibas, Knuth, Sharir]
old link to applet, showing Vor, Del, empty circles
AMS 545 / CSE 555

63. CS691G Computational Geometry - UMass Amherst
Example
CS691G Computational Geometry - UMass Amherst

64. Voronoi Extensions Numerous ! [see Okabe, Boots, Sugihara, Chiu]
Different metrics
Higher dimensions:
Delaunay in Rd  lower CH in Rd+1
O( n log n + n (d+1)/2  )
Order-k, furthest-site
Other sites: Voronoi of polygons, “medial axis”
Additive/multiplicative weights; power diagram
-shapes: a powerful shape representation
GeoMagic, biogeometry at Duke

65. Furthest Site Voronoi

66. Medial Axis: Convex Polygon

67. Medial Axis: Convex Polygon

68. Medial Axis: Nonconvex Polygon

69. Medial Axis: Curved Domain

70. Straight Skeleton

71. VRONI: Fast, robust Voronoi of polygonal domains
Incremental algorithm

73. Computing offsets with a Voronoi diagram

74. Alpha Shapes, Hulls
“Erase” with a ball of radius  to get -shape. Straighten edges to get -hull

76. -Shapes
Theorem: For each Delaunay edge, e=(pi,pj), there exists min(e)>0 and max(e)>0 such that e  -shape of S iff min(e)    max(e).
Thus, every alpha-hull edge is in the Delaunay, and every Delaunay edge is in some alpha-shape.
Applet