1. Parts of these slides are based on
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. The search for intersecting planes of different 3D models in a scene.
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. The problem can be defined as
Intro
The problem can be defined as
if, where and when
two objects intersect.

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

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

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

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

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

11. Axes Aligned Bounding Box (AABB)
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. Oriented Bounding Box (OBB)
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. The overlap test is based on the Separating Axes Theorem
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. Separating Axes Theorem
Bounding Volumes
Separating Axes Theorem
Axes

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

16. On the y-axis, all boxes overlap
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. On the x-axis, boxes 2-3 and 3-1 overlap
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. Combine overlap check of single axes:
Bounding Volumes
Combine overlap check of single axes: x 1 2 3 y 1 2 3 = 1 2 3

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

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

21. k-DOP is considered to be a trade off between AABBs and OBBs.
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. k-DOPs are used e.g. in the game (XBOX, Pseudo Interactive, 2002)
Bounding Volumes
k-DOPs are used e.g. in the game
‘Cell Damage’
(XBOX, Pseudo Interactive, 2002)

23. k-directions How to Compute and Store k-DOPs:
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. 3D Example: UNREAL-Engine
Bounding Volumes
3D Example: UNREAL-Engine

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

26. Collision on different scales:
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. The hierarchy is stored in a tree, named by the underlying BV scheme:
Hierarchies
The hierarchy is stored in a tree, named by the underlying BV scheme:
AABB – tree
OBB – tree
Sphere – tree
kDOP – tree

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

32. Each node contains all primitives of its subtree
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