1. Rigid Body Motion

2. Game Physics “Linear physics”– physics of points –particle systems, ballistic motion… –key simplification: no orientation “Rotational physics” –orientation can change

4. Rigid Bodies No longer points: distribution of mass instead. Rigid bodies: distances between mass elements never change. Orientation of body can change over time.

5. Rigid Body Translation Can treat translational motion of rigid bodies exactly the same as points Single position (position of center of mass) F=ma (external forces) v = ∫a dt x = ∫v dt momentum conservation

7. Rotation Rigid bodies also have orientation Treating rotation properly is complicated Rotation is not a vector (rotations do not commute, i.e., order of rotations matters) No analog to x, v, a in rotations?

8. Angular velocity Infinitesimally small rotations do commute Suppose we have a rigid body rotating about an axis Can use a notion of angular velocity: ω = dθ/dt

9. Angular velocity Connection between linear and angular velocity Magnitudes: v = ωr perp Want vector relation Nice to have angular velocity about axis of rotation (so it doesn't have to change all the time for an object spinning in place) Let v = ω x r

10. Angular velocity v = ω x r Or, ω = r x v / |r| 2 Note: ω, r, v vectors Angular velocity defined this way so that constant angular velocity behaves sensibly –spinning top has constant ω

11. Applying force What happens when you push on a spinning object? (exert force) F=ma, so we know the movement of the centre of mass How does the force affect orientation?

12. Torque T = r x F r is vector from origin to location where force applied –for convenience, often take origin to be center of mass of object F is force Magnitude proportional to force, proportional to distance from origin

13. Intuition for Torque Larger the larger from the centre Lever action: small force yields equivalent torque far from fulcrum

14. Direction of Torque T = r x F Perpendicular to both location and force vectors Direction is along axis about which rotation is induced Right hand rule: thumb along axis, fingers curl in direction of rotation

15. single particle T = r F sinθ T = r F t F t = ma t = mrα T = mr 2 α Let I = mr 2 T = Iα

16. Many particles Real objects are (pretty much) continuous Game objects: distribution of point masses –not always, but common Can get reasonable behaviour with (e.g.) four point masses per rigid body Single orientation for body Single centre of mass (of course)

17. Changing Coordinate Systems We dealt with changing coordinate systems all the time before Rigid bodies are much simpler if we treat them in a natural coordinate system –origin at the centre of mass of the body –or, some other sensible origin: hinge of door Need to transform forces into body coordinate system to calculate torque Transform motion back to world space

18. Angular momentum Define angular momentum similarly to torque: L = r x p Note that with this definition, T = dL/dt, just as F = dp/dt

19. Force and Torque Note: a force is a force and a torque Moves body linearly: F=ma, changes linear momentum Rotates body: produces torque, changes angular momentum

20. Linear vs. Angular linear quantityangular quantity velocity vangular velocity ω acceleration aangular acc. α mass mmoment of inertia I p = mvL = Iω F = maT = Iα

21. Conservation of Angular Momentum Consequence of T = dL/dt: –If net torque is zero, angular momentum is unchanged Responsible for gyroscopes' unintuitive behaviour The gyroscope is tipped over but it doesn’t fall

22. Moment of Inertia Said that moment of inertia of a point particle is mr^2 In the general case, I = ∫ ρ r^2 dV where r is the distance perpendicular to the axis of rotation Don't know the axis of rotation beforehand

23. Moment of Inertia I = ∫ρ(x,y,z) dxdydz y^2 + z^2-xy-xz -xyx^2 + z^2-yz -xz-yzx^2+y^2

24. Diagonalized Moment of Inertia Luckily, we can choose axes (principal axes of the body) so that the matrix simplifies: I = where, e.g., Ixx = m(y*y + z*z) Off-diagonal entries called "products of inertia" Ixx00 0Iyy0 00Izz

25. Avoiding products of inertia Do calculations in inertial reference frame whose axes line up with the principal axes of your object Transform the results into worldspace Moment of inertia of a body fixed, so can be precomputed and used at run-time

26. Moment of Inertia In general, the more compact a body is, the smaller the moments of inertia, and the faster it will spin (for the same torque)

27. Fake I Not doing engineering simulation (prediction of how real objects will behave) Can invent I rather than integrating Large values: hard to rotate about this axis Avoid off-diagonal elements

28. Fake constants For that matter, can fake lots of stuff Different gravity for different objects –e.g., slow bullets in FPS –e.g., fast falling in platformer fake forces, approximate bounding geometry

29. Case in 2D In 2D, the vectors T, ω, α become scalars (their direction is known – only magnitude is needed) Moment of inertia becomes a scalar too: I = ∫prdA

30. Single planar rigid body state contains x, y, θ, vx, vy, ω Have –F = ma (2 equations) –T = Iω –x = ∫vx dt –y = ∫vy dt –θ = ∫ω dt Integrate to obtain new state, and proceed

31. Rigid body in 3D Need some way to represent general orientation Need to be able to compose changes in orientation efficiently

32. Quaternions Quaternion: structure for representing rotation –unit vector (axis of rotation) –scalar (amount of rotation) –recall, store (cos(θ/2), v sin(θ/2) ) Can represent orientation as quaternion, by interpreting as rotation from canonical position

33. Quaternions Rotation of θ about axis v: –q = (cos(θ/2), v sin(θ/2)) "Unit quaternion": q.q = 1 (if v is a unit vector) Maintain unit quaternion by normalizing v Arbitrary vector r can be written in quaternion form as (0, r)

34. Quaternion Rotation To rotate a vector r by θ about axis v: –take q = (cos(θ/2), v sin(θ/2) –Let p = (0,r) –obtain p' from the quaternion resulting from qpq -1 –p' = (0, r') –r' is the rotated vector r

35. Note: –q(t) = (s(t), v(t)) –q(t) = [ cos(θ(t)/2), u sin(θ(t)/2) ] –For a body rotating with constant angular velocity ω, it can be shown q’(t) = [0, ½ ω] q(t) Summarize this ½ ω q(t) Rotation Differentiation

36. Using quaternions gives Rigid Body Equations of Motion x(t) q(t) P(t) L(t) v(t) ½ ωq(t) F(t) T(t) d/dt =

37. P and L Note that –v = P/m (from P=mv) –ω = I -1 L (from L = Iω) Often useful to use momentum variables as main variables, and only compute v and ω (auxiliary variables) as needed for the integration

38. Impulse Sudden change in momentum –also, angular momentum (impulsive torque) Collision resolution using impulse –new angular momentum according to conditions of collision –algorithmic means available for resolving