1. Planar Convex Hull 2013 / 5 / 9 Group 4 Sungheon Park Jeongho Son CS504 Presentation [CS504 Presentation]

2. Contents CS504 Presentation Definition of convex hull Bruteforce algorithm Graham’s scan Divide and conquer Quickhull Jarvis’ method

3. What is convex hull? CS504 Presentation Let S be a set of points in the plane. Intuition: Imagine the points of S as being pegs; the convex hull of S is the shape of a rubber-band stretched around the pegs.

4. What is convex hull? CS504 Presentation

5. Applications of convex hull CS504 Presentation computer visualization, ray tracing path finding Geographical Information Systems (GIS) Visual pattern matching

6. Orientation test CS504 Presentation

7. Graham’s Scan CS504 Presentation

8. Graham’s Scan CS504 Presentation

9. Graham’s Scan CS504 Presentation

10. Original Graham’s scan CS504 Presentation Initially, points are sorted in increasing angular value If the point is not convex (concave), it removes the current point from the perimeter list

11. Divide-and-Conquer CS504 Presentation

12. Divide-and-Conquer CS504 Presentation

13. Quickhull CS504 Presentation

14. Quickhull CS504 Presentation

15. Quickhull CS504 Presentation

16. Jarvis’s March CS504 Presentation Build the hull using “gift wrapping” process

17. Jarvis’s March CS504 Presentation

18. Jarvis’s March CS504 Presentation

19. Applet CS504 Presentation Java applet –http://www.cse.unsw.edu.au/~lambert/java/3d/ hull.htmlhttp://www.cse.unsw.edu.au/~lambert/java/3d/ hull.html –http://www.cs.princeton.edu/courses/archive/s pr09/cos226/demo/ah/JarvisMarch.htmlhttp://www.cs.princeton.edu/courses/archive/s pr09/cos226/demo/ah/JarvisMarch.html

20. CS504 Presentation Chan’s algorithm Jeongho Son

21. Planar Convex Hull CS504 Presentation

22. Chan’s Algorithm CS504 Presentation

23. Chan’s Algorithm CS504 Presentation

24. Chan’s Algorithm CS504 Presentation Stage 1 n = 32 Set m = 8

25. Chan’s Algorithm CS504 Presentation Stage 1 n = 32 Set m = 8 r = 4

26. Chan’s Algorithm CS504 Presentation

27. Chan’s Algorithm CS504 Presentation Stage 2 (After Stage 1) m = 8 r = 4

28. Chan’s Algorithm CS504 Presentation Stage 2 Using Graham’s Scan

29. Chan’s Algorithm CS504 Presentation Stage3 : Jarvis’s March How to merge these r hulls into a single hull? IDEA : treat each hull as a “fat point” and run Jarvis’s March! # of iteration is at most m –to guarantee the time complexity O(nlogh)

30. Chan’s Algorithm CS504 Presentation (-inf,0) -> lowest pt lowest pt

31. Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1

32. Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1 2

33. Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1 2 3

34. Chan’s Algorithm CS504 Presentation

35. Chan’s Algorithm CS504 Presentation FAIL EXAMPLE – too small value m m = 4 4 iteration

36. Chan’s Algorithm CS504 Presentation In 4(a), how to find such points?

37. Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in each hull lowest pt 1

38. Chan’s Algorithm CS504 Presentation Find the point that maximize the angle in a hull

39. Chan’s Algorithm CS504 Presentation 1 2 3 4 5

40. Chan’s Algorithm CS504 Presentation

41. Chan’s Algorithm CS504 Presentation

42. Chan’s Algorithm CS504 Presentation

43. Chan’s Algorithm CS504 Presentation

44. Chan’s Algorithm CS504 Presentation

45. Lower bound for convex hull CS504 Presentation n points in the x-axis

46. Lower bound for convex hull CS504 Presentation lifting up to 2D plane

47. Lower bound for convex hull CS504 Presentation lower convex hull

48. Quiz CS504 Presentation

49. Quiz CS504 Presentation

50. Summary CS504 Presentation Finding the convex hull of a set of points is an important problem that is often part of a larger problem Many different algorithms –Graham’s Scan –Quickhull –Divide-and-Conquer –Jarvis’s March –Chan’s algorithm

51. Q&A CS504 Presentation Any question?