1. Phun with Physics The basic ideas Vector calculus
Mass, acceleration, position, …

2. Some defs… Kinematics Dynamics The status of an object
Position, orientation, acceleration, speed
Describes the motion of objects without considering factors that cause or affect the motion
Dynamics
The effects of forces on the motion of objects.

3. The basics Let p(t) be the position in time. Other values:
We’ll drop the (t) and just say p
Other values:
v – velocity
a - acceleration
Velocity is the derivative of position
Acceleration is the derivative of velocity

4. Vector calculus Really, p(t) is a triple, right?
Think of these equations as three equations, one for each dimension
This is vector calculus

5. Newton’s law and Momentum
We know this one…
Momentum
Note: Force is the derivative of momentum

6. What about real objects?
Think of a real object as a bunch of points, each with a momentum
We can find a “center of mass” for the object and treat the object as a point with the total mass
We have:
Mass, position vector, acceleration vector, velocity vector

7. Example: Air Resistance
The resistance of air is proportional to the velocity
F = -kv
We know F = ma, so: ma = -kv
So, how can we solve?

8. Euler’s method

9. What about orientation?
We’ll start in 2D…
Let W be the orientation (angle)
Easy in 2D, not so easy in 3D we’ll be back
w is the angular velocity around center of gravity
a is the angular acceleration
Then…

10. What’s the velocity at a point?
Radians are important, here
Perpendicular to cp
Same length!
Perpendicularized radius vector p
Chasles’ Theorem: Velocity at any point is the sum of linear and rotational components.

11. Angular momentum Angular momentum At a point? For the whole thing?
M=mv
Momentum equals mass times velocity
What does this mean?

12. We want to know how much of the momentum is “around” the center
Angular Momentum
of a point around c
rcp

13. Torque – Angular force
Torque at a point

14. Total angular momentum and moment of inertia
I is the moment of inertia for the object.

15. Moment of inertia What are the characteristics?
What does large vs. small mean?
How to we get this value?

16. Relation of torque and moment of inertia

17. Simple dynamics
Calculate/define center of mass (CM) and moment of inertia (I)
Set initial position, orientation, linear, and angular velocities
Determine all forces on the body
Linear acceleration is sum of forces divided by mass
Angular acceleration is sum of torques divided by I
Numerically integrate to update position, orientation, and velocities

18. Object collisions Imagine objects A and B colliding
Assume collision is point on plane A n

19. What happens?
??? B A n

20. Our pal Newton…
Newton’s law of restitution for instantaneous collisions with no friction
e is a coefficient of restitution
0 is total plastic condition, all energy absorbed
1 is total elastic condition, all energy reflected
What’s the consequence of “no friction”?

21. Impulse felt by each object
Newton’s third law: equal and opposite forces
Force on A is jn (n is normal, j is amount of force)
Force on B is –jn
So… B
We’ll need to know j A n
No rotation for now…

22. Solving for j…
Then plug j into…

23. What about rotation?

24. How to determine the time of collision?
What do I mean?
Ideas?
Avoid “tunneling”: objects that move through each other in a time step

25. Euler’s method
We can write…

26. The problem with Euler’s method
Taylor series:
We’re throwing this away!
Error is O(Dt2)
Half step size cuts error to ¼!

27. Improving accuracy Supposed we know the second derivative?
But, we would like to achieve this error without computing the second derivative.
We can do this using the Midpoint Method

28. Midpoint Method 1. Compute an Euler step 2. Evaluate f at the midpoint
3. Take a step using the midpoint value.

29. Example
Step 1
Step 2
Step 3

30. Moving to 3D Some things remain the same The killer: Orientation
Position, velocity, acceleration
Just make them 3D instead of 2D
The killer: Orientation
“It’s possible to prove that no three-scaler parameterization of 3D orientation exists that doesn’t ‘suck’, forsome suitably mathematically rigorous definition of ‘suck’.” Chris Hecker

31. Orientation options Quaternions Matricies We’ll use later
We’ll use this here.

32. Variables x(t) v(t) q(t) Spatial position in time (3D)
Center of mass at time t
Assume object has center of mass at (0,0,0)
v(t)
Velocity in space
q(t)
Orientation in time (quaternion)

33. Angular velocity w(t) Angular velocity at any point in time
This is a rotation rate times a rotation vector
Strange, huh?

34. Note Angular velocity is instantaneous
We’ll compute it later in equations, but we won’t keep it around.
Angular velocity is not necessarily constant for a spinning object!
Interesting! Can you visualize why?

35. Angular velocity and vectors
Rate of change of a vector is:
This is why they like that notation
Change is orthogonal to the normal and vector
Magnitude of change is vector length times magnitude of w

36. Derivative of the rotation quaternion
Woa…
Normalize this sucker and we can take a Newton step.
Be sure to scale w(t) by Dt.

37. Velocity of a point (think vertex)
Not just v(t), must include rotation!!!
Position of point at time t:
Just a reminder:
Velocity of point at time t:
And the magic:
Angular part
Linear part

38. Force and torque Fi(t) Force on particle i at time t Vector, of course
Torque on point i:
Total Force:
Total Torque:

39. Linear Momentum
Momentum:
And, the derivative of momentum is force:

40. Angular Momentum
Angular momentum is preserved if no torque is applied.
L(t) is the angular momentum

41. Inertia Tensor
All that work typing this sucker in and it’s pretty much useless. Orientation dependent!

42. Inertia Tensor for a base orientation

43. Base inertia tensor to current inertia tensor
Relation of angular momentum and inertia tensor:

44. Bringing it all together
Current state:
Position
Orientation
Linear momentum
Angular momentum
Velocity (may itself be updated)

45. 3D Collisions There are now two possible types of collisions
Not just vertex to face

46. 3D collisions
Vertex to face
Edge to edge

47. We need… 3D collision detection Determine when the collision occurred
We’ll do that later
Determine when the collision occurred
Binary search for the time
Bisection technique

48. Velocity at a point Position (center of gravity) Position of point
Angular velocity
Velocity (linear)

49. Relative velocity Normal to object b Vertex to face: normal for face
Edge to edge: cross product of edge directions
Positive vrel means moving apart, ignore it
Negative vrel means interpenetrating, process it
What does zero vrel mean?

50. New velocities for a and b
Vector from center of mass to point
Let:
Normal
Velocity (linear) update
Mass
Angular velocity update
Inertia tensor inverse
Equivalent equations for b

51. And j…

52. Contrast 2D/3D 2D 3D

53. Questions? 2D had two different j’s 3D gets by with only one.
Linear velocity and angular velocity
3D gets by with only one.
Why is this?

54. Other issues Resting contact
Bodies in contact, but –e < vrel < e
Must deal with motion transfer