Kinetic Collision Detection for Convex Fat Objects

1. Mohammad Abam Mark de Berg Sheung-Hung Poon Bettina Speckmann Kinetic Collision Detectionfor Convex Fat Objects

2. Kinetic Data Structures • Simulate system of continuously moving objects. • Efficiently maintain discrete attributes of objects: • closest pair of objectst • convex hull • minimum spanning tree • binary space partition • ...

3. Example: convex hull

4. Example: convex hull

5. Dynamic data structures • Allow insertions and deletions of objects at discrete times. • Not suitable for handling moving objects.

6. Time sampling approach • Choose fixed time step. • Update the positions of moving objects at each time step. • Update the data structure with the new positions of objects.

7. t t Time sampling approach • How to choose the proper time step? • Oversampling • Undersampling

8. t Time sampling approach • Combinatorial changes occur in irregular patterns.

9. Kinetic Data Structures (Basch, Guibas, and Hershberger, 1998) • Use any static algorithm to compute the configuration function (i.e. the discrete attribute). • Turn the algorithm into a proof that this attribute is correct. • Animate this proof through time.

10. Example: convex hull Proof: a is to the left of bc d is to the left of bc b is to the right of ad c is to the left of ad c d a b c b d a a c d d b a b c

11. d Example: convex hull c a t1 CertificateFailure time a is to the left of bc d is to the left of bc b is to the right of ad c is to the left of ad t2 b never d is to the right of bc t1 t2 b is to the left of ad never

12. Kinetic Data Structures Structures • Proof of correctness of attribute (certificates) • Priority queue (event queue) Assumptions • A simple model for motion: each object follows a known flight plan with rational parameters. • Certificates are algebraic; failure is next largest root.

13. Kinetic Data Structures Efficient KDS exist for • 2d convex hull • Closest pair • Delaunay triangulation • Diameter, width of point sets • Collision detection between simple polygons in the plane, between two convex polyhedra in 3D, between multiple convex fat objects of similar sizes in 3D • …

14. Kinetic Data Structures Collision detection Set of geometric tests (elementary certificates) that certify that the input objects are disjoint.

15. Balls rolling on a plane

16. Convex fat constant-complexity objects free-flying in 3D

17. Kinetic data structure properties • A KDS is called • compactif it uses little space in addition to the input. • responsive if the data structure invariants can be restored quickly after the failure of a certificate. • local if it can be updated easily if the flight plan for an object changes. • efficient if the worst-case number of events handled by the data structure is small compared to some worst case number of external events.

18. Locality

19. Related work • Same-size convex fat objects [Basch, Guibas, and Zhang, 1997] Compactness O(n log2 n) certificates Responsiveness each event can be handled in O(log3 n) time Efficiency O(n2) events in the worst case • Two convex polyhedra[Erickson, Guibas, Stolfi, and Zhang, 1999][Guibas, Xie, and Zhang, 2001]

20. Balls rolling on a plane Balls Bi with center ci and radius ri. Certificate of non-collision: dist(ci,cj) – ri – rj > 0 Bi Bj

21. Collision tree

22. Threshold disks Smallest blue ball with radius = rmin

23. Threshold disks

24. Threshold disks

25. Threshold disks

26. Threshold disk of radius Threshold disks Bi

27. Threshold disks

28. Threshold disks

29. Threshold disks

30. Threshold disks

31. Non-collision certificates Lemma: Each point in the plane is contained in at most a constant number of threshold disks. The number of non-collision certificates is O(n).

32. Kinetic maintenance

33. Kinetic maintenance • Replace threshold disks by threshold boxes.

34. Kinetic maintenance • Replace threshold disks by threshold boxes. • Maintain two sorted lists on x- and y-coordinates of tangency points and threshold boxes. O(n) certificates.

35. Kinetic maintenance Lemma: Each point in the plane is contained in at most a constant number of threshold boxes. The number of non-collision certificates is O(n).

36. Kinetic data structure properties Balls rolling on a plane CompactnessO(n log n) certificates Responsivenesseach event can be handled in logarithmic time. EfficiencyO(n2) events in the worst case, assuming that the balls follow constant-degree algebraic trajectories.

37. Convex fat constant-complexity objects free-flying in 3D

38. Fat objects

39. Fat objects

40. Fat objects

41. Fat objects Definition: An object is called ρ-fat if ro/ri < ρ. ri ro

42. Convex fat constant-complexity objects free-flying in 3D Balls rolling on a plane: threshold disks of radius Balls free flying: threshold disks of radius ri + rmin

43. Guarding fat objects

44. Guarding fat objects LemmaFor any ρ-fat object K, there is a set G(K) of O(ρ6)guarding points such that any ρ-fat object K’ that collides with K and is at least as large as K contains a point from G(K).

45. Guarding fat objects LemmaFor any ρ-fat object K, there is a set G(K) of O(ρ6)guarding points such that any ρ-fat object K’ that collides with K and is at least as large as K contains a point from G(K).

46. Guarding fat objects LemmaFor any ρ-fat object K, there is a set G(K) of O(ρ6)guarding points such that any ρ-fat object K’ that collides with K and is at least as large as K contains a point from G(K). Approach For each g ∈ G(K) maintain the object K(g) containing g SolutionReplace objects by their bounding boxes Problem: Hard to do efficiently

47. Bounding boxes A guard g can be contained in many bounding boxesbut only the smaller objects can actually contain g maintain the smallest k objects whose bounding box contains g

48. Making it work … • A guard g.

49. Making it work … • The unit cube centered at g.

50. Making it work … • A grid whose cells have edge length .O(ρ2) grid cells.