1. Collisions Collision Avoidance - force fields, steer to avoid, etc.
- as in flocking
Collision Detection
- calculating when objects overlap
- during a time interval
Collision Response
- computation v. accuracy
- back up time v. after-the-fact

2. Point-Plane Collision - Detect Collision
Every time step, see if point is one “wrong” side of plane
Detect collision by planar equation
if a*x+b*y+c*z +d < 0 => collision
Either back time up to point of collision and go from there Or
Respond to collision only at time steps
d1+d2 d1 d2
T = t + Dt*d1/(d1+d2)

3. Kinematic Response Negate velocity component in direction of normal
v’ = v - 2*(v.N) N v N
(v.N) N

4. Penalty Method
F = k*d d
Spring with zero rest length

5. Testing Planar Polyhedra
Point inside a convex/concave polyhedron test
Edge-plane intersection
Temporal aliasing

6. Swept Volumes Compute motion of one relative to other
Compute swept volume
Determine if other object intersects swept volume
Tractable only for simple motion

7. Impulse Force of Collision
J = F * Dt
J = F * Dt = M*a*Dt = M*Dv = D (M*v) = DP
(Notation: J = j*N)
v+rel = -e*v -rel
Effect of collision has linear and angular components

8. Relative velocity
Determine contact normal
AxB A B
Velocity of a point (pA (t)) = linear + angular velocities
= vA(t) + wA (t) X rA
Relative velocity is difference of velocity of 2 points of contact

9. Update Velocities after Collision
v+A = v - A + j*n/M A
v+B = v - B + j*n/M B
w + A = w - A + I -1A(t)*(r A x j*n)
w + B = w - B + I -1 B(t)*(r B x j*n)

10. Computing Impulse v+rel = -e*v -rel pA (t)) = vA(t) + wA (t) X rA
v+rel = n* (p+A(t) - p+B(t))
v+rel = n * (v+A(t) + w+A(t) x r A -v+B(t) - w+B(t) x rB)

11. Computing Impulse Forces
Given: two points of contact and normal of contact
Compute velocities of points of contact: va, vb
Compute relative velocities in direction of normal
If Vrelative > threshold // actually colliding
Compute j
PA += J
PB -= J
LA += rA x J
LB -= rB x J
Else if Vrelative < -threshold, then resting contact
Else they are moving away from each other

12. Other Forces
Gravity
Friction
Viscosity
Springs (Hooke’s law)

13. Friction Supporting object Static friction Fs = us * FN Fs
Resting contact F
Normal force FN

14. Friction Supporting object Fk = uk * FN Kinetic friction Fk v
Resting contact F FN
Normal force
Static friction
Fs = us * FN

15. Gravity Fgravity = G*m1*m2/r2
a object= F/m object = G*mearth/r2earth = 32 ft/s2 = 9.8 m/s2

16. Viscosity Fvis = -K*n*v K - depends on shape of object
n - depends on properties of liquid
K = 6*p*r
For spherical object:
Terminal velocity - viscosity and gravity balance
m*g = 6*p*r*n*v

17. Springs (Hooke’s law) Spring’s rest length: exerts zero force
Fspring = kspring * (x - xrest) x
xrest F x F

18. Spring Mesh
Edges => springs
Internal springs to stabilize shape

19. Damping Calm down spring oscillations Fdamping = - kdamping * v
F = (x’ - xrest) * kspring - v * kdamper

20. Forces - Recap Based on position: springs, gravity, wind,
Based on velocity: viscosity, dampers,
Based on object interactions: collisions, friction
Give rise to accelerations