1. Using BSP for CD Ref: BSP in deBerg et al ’ s book (url)url

2. Fall 20102 Introduction BSP: binary space partition CD: collision detection Especially useful for indoor scenes

3. Fall 20103 Kd-Tree and BSP tree kd-tree is a special kind of bsp tree + –

4. Fall 20104 BSP Tree Construction

5. Fall 20105 Equation at a Node Defines the hyperplane,  : ax+by+cz+d=0 Subtree polygons are classified according to the coefficients stored at (internal) nodes Coefficients are often “ normalized ” for distance computation between point and plane

6. Fall 20106 Elements of BSP Construction Choosing the partition plane Auto-partition: choose the partition plane from the input set of polygons It is desirable to have a balanced tree, where each leaf contains roughly the same number of polygons. However, there is some cost in achieving this. Partitioning polygons If a polygon happens to span the partition plane, it will be split into two or more pieces. A poor choice of the partition plane can result in many such splits, and a marked increase in the number of polygons.

7. Fall 20107 Elements of BSP Construction (cont) The decision to terminate tree construction can be: when the number of polygons in a leaf node is below a maximum value. until every polygon is placed in an internal node. maximum tree depth.

8. Fall 20108 S3 S2 S3 S1 SIZE=3 Different ordering generates different trees

9. Fall 20109 S3 S2 S2b S2a S2 S3 S3a S3b S1 S2b S3bS2a S3a S1 S2b S3b S2a S3a S2bS2a S3b S3a S3b S3a S3bS3a SIZE=5

10. Fall 201010 Painter ’ s Algorithm What do we mean “sort in z”? That is for a triangle, what is its representative z value? Minimum z Maximum z Polygon ’ s centroid Work cost = sort + draw An object space visibility algorithm

11. Fall 201011 Painter ’ s Algorithm (cont) Pros: No extra memory Relatively fast Easy to understand and implement Cons: Precision issues (and additional work to handle them) Sort stage Intersecting objects

12. Fall 201012 Painter ’ s Algorithm (BSP Version) S(v) need not be drawn!

13. Fall 201013 Example Scene

14. Fall 201014 A B E C F D G

15. Fall 201015 A E C1C1 F D G C2C2 B2B2 B1B1 D A B 1 C 1 B 2 C 2 EFG + –

16. Fall 201016 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F A B 1 C 2 B 2 EG

17. Fall 201017 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A Tree completed!

18. Fall 201018 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye Rendering BSP [b-to-f]

19. Fall 201019 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V hv-hv- hv+hv+

20. Fall 201020 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V C1 hv-hv- hv+hv+

21. Fall 201021 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 hv-hv- hv+hv+

22. Fall 201022 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A

23. Fall 201023 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A D hv-hv- hv+hv+

24. Fall 201024 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A D hv-hv- hv+hv+

25. Fall 201025 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A D hv-hv- hv+hv+

26. Fall 201026 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A D B2

27. Fall 201027 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A B2 E D hv-hv- hv+hv+

28. Fall 201028 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A B2 E D G

29. Fall 201029 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A B2 E D F G hv-hv- hv+hv+

30. Fall 201030 A E C1C1 F D G C2C2 B2B2 B1B1 D C1C1 F B1B1 C 2 E G B 2 A eye V B1 C1 A B2 E D F G hv-hv- hv+hv+ C2

31. Line Equation P 1 (x 1,y 1 ) P 2 (x 2,y 2 ) L = 0 L > 0 (left side) P(x,y) (1,0) (0,1) Check: origin on + (left) side ! Left side correspond to positive side n

32. Fall 201032 BSP in Games (ref)ref To account for character size, use offset planes in BSP Problems when angles between planes are large Character-environment collision detection

33. Fall 201033 Collision Resolution Better to solve a collision by deflecting the motion rather than stopping at the impact point Stepping and climbing

34. Fall 201034 Example L1 L2 L3 L1 L2 L3 (0,4) (6,0) (6,4) L1: (0,4)  (6,0) L2: (6,0)  (6,4) L3: (6,4)  (0,4) + – solid free

35. Fall 201035 Point-Scene Test

36. Fall 201036 Example L1 L2 L3 L1 L2 L3 00 <0 d0d0 d  -r r

37. Fall 201037 Example L1 L2 L3 L1 L2 L3 00 <0 d  -r r

38. Fall 201038 Example L1 L2 L3 L1 L2 L3 00 <0 d  -r solid free

39. Fall 201039 Example L1 L2 L3 L4 L5 L6 L1 L2 L5 L4 L3 L6 + – solid free

40. Fall 201040 Ex: Tree Building 1/4 L1 + – L2 L3 L7 L5 L6 L4 L2,L3,L4,L5,L6L5,L6,L7

41. Fall 201041 Tree Building 2/4 L1 + – L2 L3 L7 L5 L6 L4 L2,L3,L4,L5,L6L5,L6,L7 L4 L2,L3 L5,L6 L7 L5,L6

42. Fall 201042 Tree Building 3/4 L1 + – L2 L3 L7 L5 L6 L4 L2,L3 L5,L6 L7 L6 L5 L2 L3 L5 L6

43. Fall 201043 Tree Building 4/4 L1 + – L2 L3 L7 L5 L6 L4 L7 L6 L5 L2 L3 L5 L6 solid

44. Fall 201044 Dynamic Scenes Moving camera: no problem Moving objects: Fuchs et. al. was to specify a bounding polygon around the area that of the scene that would be changing. Doom: use z-buffer for for dynamic objects (missiles and bullets) Adding objects: no problem Deleting objects: BIG problem (ref)ref

45. Fall 201045 [BSP & Ray Tracing] Speed up ray tracing by: Reaching leaf nodes Determine face through which ray exits Find region-binding faces: collect all parental halfplanes Compute exit points Extending ray into next region … until the ray shoots into infinity reference