Parts of these slides are based on

1. Introduction To Collision Detection Parts of these slides are based on www2.informatik.uni-wuerzburg.de/ mitarbeiter/holger/lehre/osss02/schmidt/vortrag.pdf by Jakob Schmidt

2. What ? The problem: The search for intersecting planes of different 3D models in a scene. Collision Detection is an important problem in fields like computer animation, virtual reality and game programming.

3. Intro The problem can be defined as if, where and when two objects intersect.

4. Intro This introduction will deal with the basic problem: IF two (stationary) objects intersect.

5. Intro The simple solution: Pairwise collision check of all polygons the objects are made of.

6. Intro Problem: • complexity O(n²) • not acceptable for reasonable number n of polygons • not applicable for realtime application

7. Bounding Volumes Solution: Bounding Volumes Reduce complexity of collision computation by substitution of the (complex) original object with a simpler object containing the original one.

8. Bounding Volumes The original objects can only intersect if the simpler ones do. Or better: if the simpler objects do NOT intersect, the original objects won’t either.

9. Bounding Volumes How to choose BVs ? • Object approximation behavior (‘Fill efficiency’) • Computational simplicity • Behavior on (non linear !) transformation (incl. deformation) • Memory efficiency

10. AABB Sphere OBB k-DOP Bounding Volumes Different BVs used in game programming: • Axes Aligned Bounding Boxes (AABB) • Oriented Bounding Boxes (OBB) • Spheres • k-Discrete Oriented Polytopes (k DOP)

11. Bounding Volumes Axes Aligned Bounding Box (AABB) • Align axes to the coordinate system • Simple to create • Computationally efficient • Unsatisfying fill efficiency • Not invariant to basic transformations, e.g. rotation

12. Bounding Volumes Oriented Bounding Box (OBB) Align box to object such that it fits optimally in terms of fill efficiency Computationally expensive Invariant to rotation Complex intersection check

13. Bounding Volumes The overlap test is based on the Separating Axes Theorem (S. Gottschalk. Separating axis theorem. Technical Report TR96-024,Department of Computer Science, UNC Chapel Hill, 1996) Two convex polytopes are disjoint iff there exists a separating axis orthogonal to a face of either polytope or orthogonal to an edge from each polytope.

14. Bounding Volumes Separating Axes Theorem Axes

15. Bounding Volumes Check for overlap on every single Axis (project polygon to axis). Two polygons intersect, if their projections overlap on EVERY axis.

16. Bounding Volumes Overlap check on single axis: Sort and Sweep (Example for xy-axis aligned boxes) sort 1 2 3 On the y-axis, all boxes overlap

17. Bounding Volumes Overlap check on single axis: Sort and Sweep 1 2 3 sort On the x-axis, boxes 2-3 and 3-1 overlap

18. Bounding Volumes Combine overlap check of single axes:

19. Bounding Volumes Sphere • Relatively complex to compute • Bad fill efficiency • Simple overlap test • invariant to rotation

20. Bounding Volumes K-DOP • Easy to compute • Good fill efficiency • Simple overlap test • Not invariant to rotation

21. Bounding Volumes k-DOP is considered to be a trade off between AABBs and OBBs. Its collision check is a general version of the AABB collision check, having k directions

22. Bounding Volumes k-DOPs are used e.g. in the game ‘Cell Damage’ (XBOX, Pseudo Interactive, 2002)

23. Bounding Volumes How to Compute and Store k-DOPs: k-directions k-directions Bi define planes (Bi is the normal to plane i) , the intersection of these planes defines the k-DOP bounding volume. Plane Pi = {x | Bi x – di <= 0} (-> Hesse Normal Form)

24. Bounding Volumes 3D Example: UNREAL-Engine

25. Bounding Volumes 2D Example for planes defining a k-DOP Normal vector Bi

26. Bounding Volumes Collision on different scales: Hierarchies

27. Hierarchies Idea: To achieve higher exactness in collision detection, build a multiscale BV representation of the object

28. Hierarchies

29. Hierarchies Use the hierarchy from coarse to fine resolution to exclude non intersecting objects

30. Hierarchies The hierarchy is stored in a tree, named by the underlying BV scheme: AABB – tree OBB – tree Sphere – tree kDOP – tree

31. Hierarchies Sphere Trees are used for example in “Gran Tourismo”

32. Hierarchies Simple example: • Binary tree • Each node contains all primitives of its subtree • Leaves contain single primitive

33. Hierarchies

34. Hierarchies

35. Hierarchies Comparison AABB / OBB