1. Convex Hulls Computational Geometry, WS 2007/08 Lecture 2 – Supplementary Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

2. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann2 Area of Triangle Underlying idea: –For every edge, compute between it and a given line (or point). –Sum the area in a predetermined order. Generalizable for any polygon.

3. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann3 Area of Triangle Use x-axis line as reference. Compute in clockwise order. (x0,y0)(x0,y0) (x1,y1)(x1,y1) (x2,y2)(x2,y2)

4. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann4 Area of Triangle ½ ( x 1 – x 0 )( y 0 + y 1 ) (x0,y0)(x0,y0) (x1,y1)(x1,y1) (x2,y2)(x2,y2)

5. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann5 Area of Triangle ½ ( x 1 – x 0 )( y 0 + y 1 ) + ½ ( x 2 – x 1 )( y 1 + y 2 ) (x0,y0)(x0,y0) (x1,y1)(x1,y1) (x2,y2)(x2,y2)

6. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann6 Area of Triangle ½ ( x 1 – x 0 )( y 0 + y 1 ) + ½ ( x 2 – x 1 )( y 1 + y 2 ) + ½ ( x 0 – x 2 )( y 2 + y 0 ) (x0,y0)(x0,y0) (x1,y1)(x1,y1) (x2,y2)(x2,y2)

7. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann7 Area of Triangle ½ ( x 1 – x 0 )( y 0 + y 1 ) + ½ ( x 2 – x 1 )( y 1 + y 2 ) + ½ ( x 0 – x 2 )( y 2 + y 0 ) (x0,y0)(x0,y0) (x1,y1)(x1,y1) (x2,y2)(x2,y2) = ½ [( x 1 – x 0 )( y 0 + y 1 ) + ( x 2 – x 1 )( y 1 + y 2 ) + ( x 0 – x 2 )( y 2 + y 0 )] = ½ [ x 1 y 0 + x 1 y 1 – x 0 y 0 – x 0 y 1 + x 2 y 1 + x 2 y 2 – x 1 y 1 – x 1 y 2 + x 0 y 2 + x 0 y 0 – x 2 y 2 – x 2 y 0 ] = ½ [ x 1 y 0 + x 2 y 1 + x 0 y 2 – x 0 y 1 – x 1 y 2 – x 2 y 0 ]

8. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann8 Area of Triangle (x0,y0)(x0,y0) (x1,y1)(x1,y1) (x2,y2)(x2,y2) | x 0 y 0 1 | | x 1 y 1 1 | | x 2 y 2 1 | Area of triangle = ½

9. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann9 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

10. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann10 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

11. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann11 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

12. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann12 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

13. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann13 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

14. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann14 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

15. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann15 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

16. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann16 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

17. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann17 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

18. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann18 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

19. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann19 Convex Hull – Divide & Conquer Split set into two, compute convex hull of both, combine.

20. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann20 Convex Hull – Divide & Conquer Merging two convex hulls.

21. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann21 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

22. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann22 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

23. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann23 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

24. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann24 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

25. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann25 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

26. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann26 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

27. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann27 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

28. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann28 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

29. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann29 Convex Hull – Divide & Conquer Merging two convex hulls: (i) Find the lower tangent.

30. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann30 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

31. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann31 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

32. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann32 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

33. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann33 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

34. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann34 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

35. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann35 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

36. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann36 Convex Hull – Divide & Conquer Merging two convex hulls: (ii) Find the upper tangent.

37. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann37 Convex Hull – Divide & Conquer Merging two convex hulls: (iii) Eliminate non-hull edges.