1. Voronoi Diagram (Supplemental)
The Universal Spatial Data Structure (Franz Aurenhammer)

2. Outline Voronoi and Delaunay Facility location problem
Nearest neighbor
Fortune’s algorithm revisited
Generalized Voronoi diagrams
Fall 2005

3. Voronoi Diagram
Fall 2005

4. Dual: Delaunay Triangulation
Fall 2005

5. Facility Location Problems
Determine a location to minimize the distance to its furthest customer
Minimum enclosing circle
Determine a location whose distance to nearest store is as large as possible
Largest empty circle q
Fall 2005

6. Facility Location (version 2)
Seek location for new grocery store, whose distance to nearest store is as large as possible — center of largest empty circle
One restriction: center in convex hull of the sites
Fall 2005

7. Facility Location (cont)
Center in hull: p must be coincident with a voronoi vertex
Center on hull: p must lie on a voronoi edge
Fall 2005

8. Largest Empty Circle
Fall 2005

9. Nearest Neighbor Search
A special case of point-location problem where every face in the subdivision is monotone
Use chain method to get O(log n) time complexity for query
Fall 2005

10. Fall 2005

11. 1 2 3 4 5 6 8 7
Fall 2005

12. 1 2 3 4 5 6 8 7
Fall 2005

13. 1 2 3 4 5 6 8 7
Fall 2005

14. 1 2 3 4 5 6 8 7
Fall 2005

15. 1 2 3 4 5 6 8 7
Fall 2005

16. 1 2 3 4 5 6 8 7
Fall 2005

17. 1 2 3 4 5 6 8 7
Fall 2005

18. Cluster Analysis
Fall 2005

19. Closest Pairs
In collision detection, two closest sites are in greatest danger of collision
Naïve approach: Q(n2)
Each site and its closest pair share an edge check all Voronoi edges O(n)
Furthest pair cannot be derived directly from the diagram
Fall 2005

20. Motion Planning (translational)
Collision avoidance: stay away from obstacle
Fall 2005

21. Fortune’s Algorithm Revisited
Cones
Idea
H/W implementation
The curve of intersection of two cones projects to a line.
Fall 2005

22. 45 deg Cone
distance=height
site
Fall 2005

23. Cone (cont)
intersection of cone  equal-distance point
Fall 2005

24. Cone (cont)
When viewed from –Z, we got colored V-cells
Fall 2005

25. Nearest Distance Function
Viewed from here
[less than]
Fall 2005

26. Furthest Distance Function
Viewed from here
[greater than]
Fall 2005

27. Fortune’s Algorithm (Cont)
Cone slicing
Cone cut up by sweep plane and L are sweeping toward the right.
Fall 2005

28. Fortune’s Algorithm (Cont)
Viewed from
z = -, The heavy curve is the parabolic front.
How the 2D algorithm and the 3D cones are related…
Fall 2005

29. Generalized Voronoi Diagram
V(points), Euclidean distance
V(points, lines, curves, …)
Distance function: Euclidean,
weighted, farthest
Fall 2005

30. Brute Force Method Record ID of the closest site to each sample point
Coarse point-sampling result
Finer point-sampling result
Fall 2005

31. Graphics Hardware Acceleration
Simply rasterize the cones using graphics hardware
Our 2-part discrete Voronoi diagram representation
Color Buffer
Depth Buffer
Site IDs
Distance
Haeberli90, Woo97
Fall 2005

32. Algorithm
Associate each primitive with the corresponding distance mesh
Render each distance mesh with depth test on
Voronoi edges: found by continuation methods
Fall 2005

33. Ex: Voronoi diagram between a point and a line
Fall 2005

34. Distance Meshes
line
polygon
curve
Fall 2005

35. Applications (Mosaic)
Fall 2005

36. Hausner01, siggraph
Fall 2005

37. Medial Axis Computation
Medial axes as part of Voronoi diagram
Fall 2005

38. Piano Mover: Real-time Motion Planning (static and dynamic)
Plan motion of piano through 100K triangle model
Distance buffer of floorplan used as potential field
Fall 2005

39. Variety of Voronoi Diagram
(regular) Voronoi diagram
Furthest distance Voronoi diagram
Fall 2005

40. Minimum Enclosing Circle
Center of MEC is at the vertex of furthest site Voronoi diagram
Fall 2005