1. Computer Game Design and Development
Mathematics, Collision, and Physics
Havok Destruction

2. Basic Problem Without physics: We must determine: There’s no gravity
There’s no friction
There are no collisions
Objects pass through one another
We must determine:
If objects collide (collision detection)
Where/when they collided (collision resolution)

3. Collision Detection Bullets (very fast) Characters (very complex)
Naïve algorithms run in O(n2)
Two primary methods:
Overlap testing (most common)
Intersection testing (sweeping)

4. Overlap Testing Determines if two objects overlap (duh!)
Check each vertex of an object inside another?
Exhibits the most error
Is discrete (i.e. checks an exact point in time)
Results:
Give a time of collision
Give a normal of collision
Uses bisection…

5. Overlap Bisection

6. Limitations of Overlap Testing
Design constraint? Speed of objects * time step < size of all objects? Reduce time step?

7. Intersection Testing
“Sweep” an object from one point to another (i.e. “extrude” – sphere
becomes capsule).
Can calculate t with an equation->
See book for equation
Doesn’t work well for network games

8. Collision Approximations Minkowski Sum – sweep origin of X across Y
Intersection technique.
These are the same!

9. Bounding Volumes and Boxes
AABB
OBB

10. About Bounding Volumes
Simplest is sphere
Next is capsule (2 spheres and a cylinder),
Next are AABBs
What can we do about character collisions?

11. Performance (avoiding O(n2))
Possible collision between R and B
since overlap in
all axis (2 in this case)
Subdivide such that on average one object in each cell.
Test against neighboring cells
What if all are in one cell?
Variable-sized objects?
“Plane sweep” needs to be sorted!
Which algorithm? Quicksort? Bubblesort?

12. Terrain Collision Determine which quad is easy
Heightmap (x and z are fixed, y is variable)
Which triangle in the quad?
Compare x with z

13. Resolving Collision Gross collision & subdivide if needed Three phases
Prologue – ignore or trigger other events
Collision – objects moved to point of impact, new velocities, calculate normal and penetration
Epilogue – destroy object, effects, damage, etc

14. Particle Physics Basics
Use consistent units (SI) – or bad stuff happens
Particle kinematics
Position (p) in 3-space, vector3 (x,y,z) = p(t)
Velocity (v) = change in position over time, V(t)
Acceleration (a) = change in velocity over time, a(t)
Force (f) = results in change in acceleration over time, F(t) = ma(t)
Calculus
Derivatives and integrals go from one to another

15. Center of Mass

16. Collision Response (frictionless)

17. Other Definitions Kinematic objects: those that move
Static objects: those that don’t, but affect those that do (terrain)
Rigid bodies: objects that do not deform
Soft bodies:
those that do
Center of gravity changes
Cloth, deflated ball…

18. Rigid Body Collision Center of Mass Consideration

19. More Definitions Constraints: Ragdoll physics:
Limitations on what objects can do
Joint types: ball joint, hinge joint, sliding joint
Ragdoll physics:
Animated bones replaced by rigid bodies
Good for violent death animations

20. Physics Engines Commercial Free/Shareware Game Dynamics SDK (Havok)
Renderware Physics
PhysX (NVIDIA)
NovodeX SDK
Free/Shareware
Open Dynamics Engine (ODE)
Tokamak Game Physics SDK
Newton Game Dynamics SDK
Bullet SDK

21. Demos