1. Robust and Fast Algorithm for a Circle Set Voronoi Diagram in a Plane
The 2001 International Conference
on Computational Science
San Francisco, CA, USA
May 28, 2001
D.-S. Kim*, D. Kim*, K. Sugihara** and J. Ryu*
* Hanyang University, Seoul, Korea
** University of Tokyo, Tokyo, Japan

2. Problem Definition Basic Idea Given : VD(P) Find : VD(C)
Get the topology of VD(C) from VD(P)
Update geometric values
Basic Idea

3. Introduction
Point Set Voronoi Diagram : VD(P)

4. Point set Voronoi diagram VD(P)
Well understood
Efficient/robust algorithm exists
Excellent code is available

5. Circle Set Voronoi Diagram
Hyperbolic arc
Star shaped polygon

6. Previous Works Kirkpatrick (79) Lee & Drysdale (81) Sharir (85)
line seg./polygon, D&C
Lee & Drysdale (81)
line seg./polygons/circles, D&C, O(nlog2n)
Sharir (85)
circles, D&C, O(nlog2n)
Yap (87)
line seg./circles, D&C, O(nlogn)
Fortune (87)
points/line seg./circles, line sweeping, O(nlogn)
Sugihara (93)
approximation
Gavrilova & Rokne (99)
swap condition of dynamic VD(C)

9. Flipping Condition
Two CC exist (No Flip)

10. Flipping Condition
Two CC exist (Flip)

11. Flipping Condition
One CC exists (Flip)

12. Flipping Condition
One CC exists (No Flip)

13. Flipping Condition
No CC exists (No Flip)

14. Summary of Flipping Condition
Both CC intersect with mates
One CC intersects with mate
No CC intersects with mate
Both CC exist
Flip
No flip
One CC exist
No CC exists

15. Two-edge Face

16. New Born Edges & Vertices

17. Removed Edges & Vertices

18. Treatment
Create 4 fictitious generators

19. Time Complexity : O(n2)
Delaunay (P)
Delaunay (C)

21. Procedure Composition
Generator-by-generator approach
greedy generator
Edge-by-edge approach
flippancy edge

22. Geometry Update Vertex geometry Edge geometry
Center of a circumcircle of three generators.
Apollonius’ Tenth Problem
Edge geometry
Bisector between two generators.

23. Edge Geometry Hyperbola (or line) Rational quadratic Bézier curve
Conditions to determine RQB.
Two end points : given by Voronoi vertices
Two tangents at the end points
One passing point

24. Tangent Line & Passing Point
Tangent Vector
≡ Angle bisector
Passing point

26. Experiments
3,500 random circles

28. 1,000 random circles

29. 800 random circles on a spiral + one large circle

30. 400 random circles on a circle + one large circle

31. 400 equi-radii circles on a circle + one large circle.

32. 400 random circles on a circle

33. Number of Flips (1,200 generators)

37. Conclusions New algorithm for VD(C) Simple to code
As robust as VD(P) algorithm
Fast :
no removal of data object
no trimming
Extendable to other generalized problems