1. Animation and Games Development
, Semester 1,
12. Basic Physics
Objective
show how physics is used in games programming

2. Overview 1. Why Physics? Point-based Physics Rigid Body Physics
4. Other Types of Force
5. The World of the Car
6. JBullet Features
7. A Game Physics Textbook

3. 1. Why Physics?
Physics can make game worlds appear more natural / realistic
But...
Often games are not realistic
e.g. cars travelling at 200 mph, double jumps of Mario
Physics engines are computationally expensive
they may slow down the game too much

4. Engines Commercial Free Havok (Havok.com) Renderware (renderware.com)
NovodeX (novodex.com)
Free
Open Dynamic Engine (ODE) (ode.org)
PhysX
Bullet (Java version used in jME)
Box2D (2D)

5. Types of Physics Newtonian (this one for games) Classical Quantum
Newton’s space and time are absolute
Classical
Einstein's space-time can be “changed” by matter
Quantum
Deterministic causality questioned and multiple localities in a single instance

6. Remember this stuff….. Time A measured duration Speed
The distance travelled given some time limit
Direction
An indication of travel
Velocity
The rate of change of position over time (has speed and direction)
Acceleration
The rate of change of velocity
Force
Causes objects to accelerate (has direction as well as magnitude)
Mass
The weight of an object? (No! weight depends on gravitational pull)
The quantity of matter

7. Newton’s First Law of Motion
Every body (object) continues in its state of rest, or uniform motion in a straight line, unless there is an external force acting on it.
Often called "The Law of Inertia"

8. Newton’s Second Law of Motion
The rate of change of momentum of a body is proportional to the force acting on it, and takes place in the direction of that force.
Simplified – An object’s change in velocity is proportional to an applied force.
An object's mass m, its acceleration a, and the applied force F canbe represented by the formula:
F = ma

9. Newton’s 3rd Law of Motion
For every action there is an equal and opposite reaction.
If two objects bump into each other they will react by moving apart.

10. As Cartoons

11. Game Physics
Two main types of Newtonian physics used in games: point-based, rigid body based
Point-based physics
used in particle systems, bullet motion…
a point's mass is located inside the point
a point does not rotate: it has no rotational orientation

12. 2. Point-based Physics
Solve Newtonian equations to get the position of a point at time t:
Force applied to the point, F(t), causes acceleration
Acceleration, a(t), causes a change in the point's velocity
Velocity, V(t) causes a change in the point's position

13. Calculation Examples
Calculate a new position based on a constant velocity:
xt = x0 + (vx * t)
yt = y0 + (vy * t)
zt = z0 + (vz * t)
Velocity is a Vector, (vx, vy, vz), or (vx, vy) in 2D
Example, point at (1, 5), velocity of (4, -3)

14. Standard Equations v = u + a t s = ½ t (u + v) v2 = u2 + 2 a s
s = u t + ½ a t2
u = initial velocity; v = final velocity
a = acceleration
t = time (seconds)
s = distance (meters)

15. Velocity Vtotal = √(Vx2 + Vy2 + Vz2) for 3D games
Velocities are vectors:
Vtotal = √(Vx2 + Vy2 + Vz2) for 3D games
Vtotal = √(Vx2 + Vy2) for 2D games

16. Acceleration
Calculate a new velocity (vt) based on a constant acceleration:
vt = dx/dt = v0 + (a * dt)
Each of these calculations are in one dimension
you must perform similar calculations on y & z axes

17. Forces Forces are vectors Ftotal = √(Fx2 + Fy2 + Fz2) for 3D games
Ftotal = √(Fx2 + Fy2) for 2D games

18. Example: 3D Projectile Motion
Weight of the projectile (a point), W = mg
g: constant acceleration due to gravity (9.81m/s2)
Projectile equations of motion:
Here, the terms on the right such as g(t-tinit) indicate multiplication by (t-tinit), rather than g being a function of (t-tinit). The perils of inconsistent and contradictory notation exist throughout math and science. See Chapter Zero in Jim Blinn’s book, Notation, Notation, Notation, for further enlightenment related to this fact!
*IMPORTANT NOTE: The closed form equations are exact; HOWEVER, they are only exact when the gravitational field is truly uniform, acting in the same direction throughout the gravitational field. In real life, on real planets, this is not the case, since when you move across the surface the direction to the center of mass of the planet changes. (In real life acceleration due to gravity doesn’t even act exactly towards the center of the planet, since the mass distribution of the planet isn’t constant, and fluctuates ever so subtly). But, for objects moving relatively small distances over the surface of large planets, the assumption of constant g is quite accurate. For space craft simulation, where the distance from the object to planet center is much larger than the planet radius, the assumption of constant g breaks down and isn’t at all accurate. The next step up is to use a central force approximation and use conservation of angular momentum to model the orbital motion. Could also just calculate the force due to gravity based on the relative location of two masses, using Newton’s Law of Universal Gravitation (or something more exotic).
p() is the position function

19. Target Practice Projectile Launch Position, pinit Target
Under constant gravitional acceleration, projectile path is a parabola
Target

20. 3. Rigid Body Physics
The shape (body) has a mass that occupies volume.
We assume that the body never changes shape
The orientation of the body can change over time.

21. Rigid Body Translation
Good news: the maths of rigid body translation is the same as for points, since we can treat the center of mass of the body as a point.
This means we can reuse the maths:
F = m a
v = ∫a dt
x = ∫v dt
momentum is conserved

22. center of mass
= point

23. Rotation A rigid body has an orientation (rotated position in space)
Rotation calculations are complicated
one reason is that the order of rotation operations is important:
x-axis rotation then y-axis rotation ≠ y-axis rotation then x-axis rotation
Rotation is not commutative

24. When a body can rotate the Newtonian motion equations must be extended.
new equations are needed for rotational (angular) mass, velocity, acceleration, force, momemtum, etc.

25. Angular Velocity, ω
The angular velocity ω describes the speed of rotation and the orientation of the axis about which the rotation occurs.

27. Angular Velocity Again
Δθ = change in angular displacement
Δt = time

28. Angular Acceleration, α
Δω = change in angular velocity
Δt = time

29. Applying Force What happens when you push on a rotating body?
There are two things to consider: translation and rotation.
For translation, we can use F=ma, because the body's center of mass can be treated like a point.
But how does the force affect the body's orientation?

30. Torque, T
Torque is a measure of how much a force acting on an object causes that object to rotate.
T = r x F = r F sin(θ)
Torque is the cross product (x) between the distance vector from the pivot point (O) to the point where force vector F is applied
θ is the angle between r and F

31. Uses for Torque
The same amount of torque can be applied with less force if the distance from the pivot (fulcrum) is increased.

32. Moment of Inertia I
Moment of inertia is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation.
Moment of inertia depends on the shape of the body and may be different around different axes of rotation.
shape effect

33. easy
(torque
is small)
hard
(torque
is large)
axes effect

34. Another definition: moment of Inertia is an object’s resistance to rotating around an axis

35. Linear Momentum, p
Also called translational momentum: the product of the mass and velocity of an object
p = m v
Linear momentum is a conserved quantity:
if a closed system is not affected by external forces, then its total linear momentum cannot change
useful for calculating velocity change after collisions

36. Angular momentum, L
Angular momentum is the rotational version of linear momentum.
e.g. a tire rolling down a hill has angular momentum
Defined as the cross product of the moment of inertia I and the angular velocity ω.
L = l x ω

37. Angular momentum is a conserved quantity:
if a closed system is not affected by external torque, then its total angular momentum cannot change
useful for calculating rotational change when moments of inertia change
L = I1 x ω1
L = I2 x ω2

38. Linear vs. Angular Linear Concept Angular Version velocity v
angular velocity ω
acceleration a
angular acceleration α
F = m a
torque T = r x F'
= I α
mass m
moment of inertia I
momentum p = m v
angular momentum L = I x ω

39. Changing Coordinate Systems
The maths for rigid bodies is simpler if we use a local coordinate system for each body
e.g. place the origin at the centre of mass of the body
But, we also need to transform any global forces into each body's local coordinate system
We also have to transform any local motion back into global coordinates.

40. 4. Other Types of Force (Virtual) Springs Damping Friction
Aerodynamic Drag …

41. (Virtual) Springs
Even if you don't see very many actual springs in a game, there are likely to be many invisible virtual springs at work.
Virtual springs are useful for implementing constraints between objects:
preventing objects overlapping
cloth rendering
character animation

42. Linear Springs

43. Damping
Damping describes physical conditions such as viscosity, roughness, etc.

44. Static Friction
Not Sliding
On the Brink
Sliding

45. Kinetic Friction
Once static friction is overcome and the object is moving, friction continues to push against the relative motion of the two surfaces.
called kinetic friction

46. 5. The World of the Car
Cars exist in an environment that exerts forces.

47. Resistance
A car driving down a road experiences two (main) types of resistance.
Aerodynamic drag, rolling resistance
Once resistance has been calculated, it is possible to calculate the amount of power a car needs to move.

48. Aerodynamic Drag
Projected frontal area of car normal to direction of V
Mass density of air
Speed of car
Drag coefficient: 0.29 – 0.4: sports cars; 0.6 – 0.9: trucks

49. Rolling resistance
Tires rolling on a road experience rolling resistance. This is not friction and has a lot to do with wheel deformation. Simplifying, we can say:
Coefficient of rolling resistance: Cars ≈ 0.015; trucks ≈ – 0.01
Weight of car (assuming four identical wheels)

50. Power
Power is the measure of the amount of work done by a force, or torque, over time.
Mechanical work done by a force is equal to the force * distance an object moves under the action of that force.
Power is usually expressed in units of horsepower
1 horsepower = 550 ft-lbs/s

51. Horsepower
Horsepower needed to overcome total resistance at a given speed (we are working in feet and lbs):
Total resistance corresponding to a car’s speed (V)
horsepower

52. Engine output
The previous equation relates the power delivered to the wheels to reach speed V.
The actual power required will be higher due to mechanical loss.
Power is delivered to a wheel in the form of torque.
Force delivered by a wheel to the road to push the car along
Torque on the wheel
Radius of the wheel

53. Stopping
Stopping distance depends on the braking system and how hard the driver breaks.
the harder the brakes are applied the shorter the stopping distance
If a car skids then the stopping distance depends on the frictional force between the tyres and the road.
If travelling uphill the stopping distance will be shorter.
gravity plays a role
If travelling downhill the stopping distance will be greater.

54. Calculating Skidding Distance
Initial speed of the car
Acceleration due to gravity
Angle of the road
Coefficient of friction between wheels and the road.
Usually around 0.4

55. More Calculations Required?
In a real driving game a lot more parameters play a role.
e.g. suspension, variable engine power, road surface
Usually it is easier to use a physics engine, but this still requires an understanding of the forces that you want to model.

56. 6. JBullet Features Rigid Bodies Joints and Spring Constraints
Simple shapes, complex geometries
Joints and Spring Constraints
Dynamics Modeling
Integrating forces and torques
Collision Detection
Physical interactions between objects
Ray Tracing
Range finders and optical flow
Cloth, Soft (deformable) Bodies
what we've been
talking about in
this part
the next part

57. 7. A Game Physics Textbook
Physics for Game Developers David M Bourg, Bryan Bywalec O'Reilly, April 2013, 2nd ed.