1. Collision Detection
Spring 2004

2. Content Classification Types of bounding volumes
Discrete, or continuous
Types of bounding volumes
Bounding volume hierarchy

3. We talked about … Taxonomy Collision detection Continuous, discrete
Collision detection, response
Collision detection
Particle vs. plane
Response and Contact problem
Sphere vs. plane (particle vs. offset plane)

4. Point-Plane Collision Detectionx e p v
(p: any point on plane)
Collision might occur if
1. Close enough
and
2. Head toward plane
Fall 2014

5. Point-Plane Collision Response 1/2x- x+
The projection of x- on the plane
Fall 2014

6. Point-Plane Collision Response 2/2x v vT vN
frictionless
CR: Coefficient of Restitution
0  CR  1
Fall 2014

7. Sphere-Plane Collisionx n p v
and e
Fall 2014

8. Contact Problem

9. Types of Bounding Volumes (ref)
Sphere
Ellipsoid
Cylinder
OBB (oriented bounding box)
AABB (axis-aligned bounding box)
K-DOP’s (K-discrete orientation polytopes)
Sphere packing
Swept sphere volume
Spherical shells
Convex hull

10. Types of Bounding Volumes

11. Oriented Bounding Boxes
An oriented bounding box is always smaller (maybe equal to) an axis-aligned bounding box.

12. Bounding Volumes Hierarchy
We used simple geometric entities to bound our polytopes.
We keep different levels of bounding volumes in a hierarchy tree.

13. Bounding Volume Hierarchies
So far, we have had subdivisions that break the world into cell
General Bounding Volume Hierarchies (BVHs) start with a bounding volume for each object
Many possibilities: Spheres, AABBs, OBBs, k-dops, …DOP (discrete oriented polytope)
More on these later
Parents have a bound that bounds their children’s bounds
Typically, parent’s bound is of the same type as the children’s
Can use fixed or variable number of children per node
No notion of cells in this structure

14. BVH Example

15. BVH Construction Top-down: simplest to build
Bound everything
Choose a split plane (or more), divide objects into sets
Recurse on child sets
Can also be built incrementally
Insert one bound at a time, growing as required
Good for environments where things are created dynamically
Can also be built bottom-up
Bound individual objects, group them into sets, create parent, recurse
What’s the hardest part about this?

16. BVH Operations
Some of the operations we’ve looked at so far work with BVHs
Frustum culling
Collision detection
BVHs are good for moving objects
Updating the tree is easier than for other methods
Incremental construction also helps (don’t need to rebuild the whole tree if something changes)
But, BVHs lack some convenient properties
For example, not all space is filled, so algorithms that “walk” through cells won’t work

17. Hierarchical bounding volume
AABB
OBB
Convex polytopes (polyhedra)

18. Separation of Convex PolytopesH
Try to find a hyperplane that separates the two polytopes. A B

19. Separation of Bounding Boxesl
Two boxes do not intersect ↔ there exists a line l such that their projections onto this line do not overlap.

20. The Separating Axis Theorem
The separating axis of two oriented boxes is either perpendicular to one of the faces, or can be obtained as the vector multiplication of two box axes.
The interference query between two oriented boxes can be answered by checking 15 axes.

21. Separating Axis (Vertex-Vertex)u v w L A1 A2 A3 B1 B2 B3
We can show that there exists a vector L such that all three terms have the same sign. Thus:

22. OBB Intersection
Exhaustive search of separation plane

23. Temporal Coherence

24. Interference of OBBs
Using careful analysis for the various 15 possible axes, it is possible to implement the interference test for two OBBs using about 200 floating-point operations.
This method has been implemented and successfully applied to large models.

25. Problem: Fitting OBBs
The OBB fitting problem requires finding the orientation of the box that best fits the data
Ideally, you want the minimum volume for the box, but that’s hard
Instead, turn to an idea from statistics: principal components
Point sample the convex hull of the geometry to be bound
Find the mean and covariance matrix of the samples
The mean will be the center of the box
The eigenvectors of the covariance matrix are the principal directions – they are used for the axes of the box
The principal directions tend to align along the longest axis, then the next longest that is orthogonal, and then the other orthogonal axis

26. OBB (Oriented Bounding Box)

27. Collision of n Bodies Naïve pairwise comparison: time complexity O(n2)
Sweep and prune
Sort: O(nlogn)
Sweep: O(n)
Output: O(k)
Further improvement is possible by considering time coherence (later)

28. Sort/Sweep [one-dimensional]

29. Coherance-based Method
If two intervals i and j overlap at the previous time step, but not at the current time step, one or more exchanges involving either a bi or ei value and a bj or ej value must occur.

30. Three-Dimensional Cases
Can be more complicated
Considering AABB: perform 3 one-dimensional tests