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Back off time step No collision? Move forward time step Continue until time step too small or exact collision found Slow, may not find exact pointb/Q</Q<   :ODetermining Collision Time  Or just fake it Determine penetration distance Push objects away until they just touch Can use mass, velocities to adjust how much each moves (if n you feel fancy) Problem: can cause a new collision this way *    =  ]fDetermining Collision Time  Alternative: push apart a little, use collision response to do the rest Assumes collision response can handle penetration Apply force (penalty force) to push them apart  ^gDetermining Collision Time  Or use predictive method to determine time of impact (TOI) Integrate up to TOI Do collision response Find next TOI, integrate, respond, repeat until done  _hDetermining Collision Time  2Ex: spheres Spheres sweep out capsules, intersect?3  `iDetermining Collision Time  uOr: relative velocity Compare sphere vs. capsule Sphere centers Pa, Pb; radii ra, rb Relative velocity v = va- vbvD      H                      bjDetermining Collision Time  TIdea: determine point(s) on velocity line distance ra+rb from Pb Or: ||Pb  (Pa+vt)|| = ra+rb Or: (w  vt)" (w  vt)  (ra+rb)2 = 0; w = Pb  Pa Quadratic: solve for tP3          #      ^3                                                    ekDetermining Collision Time  Expand (w  vt)" (w  vt)  (ra+rb)2 = 0 Get (v" v)t2  2(v" w)t + w" w  (ra+rb)2 = 0 Use quadratic equation, geto!    !  !!        $!$((((((((((( (((((( (((( (((((                 &        #  glDetermining Collision Time  Could be 1 solution Just touching, radical = 0 Could be no solutions Never touch, radical is imaginary Pick solution with smallest t 0, 1"'"  >SHave Collision, Will Travel  Have normal n (from object A to object B), collision point P, velocities va, vb Three cases: Separating Relative velocity along normal < 0 Colliding Relative velocity along normal > 0 Resting Relative velocity along normal = 0] # ## -     # #  #FI          hmLinear Collision Response  Have normal n, collision point p, velocities v1, v2 How to respond? Idea: collision is discontinuity in velocity Generate impulse along collision normal  modify velocities How much depends on incoming velocity, masses of objectsZ    Ge  ?TLinear Collision Response  Do simple case  two spheres a & b incoming velocities va,vb collision normal n want to compute impulse magnitude f#Q#  "F7        8  @ULinear Collision Response  !Compute relative velocity vab L       AVLinear Collision Response  FCompute velocity along normal Use dot product to project onto normal >)& G  BWLinear Collision Response  $Outgoing velocity dependant on coefficient of restitution e e = 1: pure elastic collision (superballs) e = 0: pure non-elastic collision (clay):Y*^     +  CXLinear Collision Response  Also need to conserve momentum  DYLinear Collision Response  8Combining last two slides (with a wave of my hand) gives999  EZLinear Collision Response  Compute final velocities  G\Angular Collision Response  MLike linear, but include angular velocity Compute velocity at collision pointN  H]Angular Collision Response  !Need to conserve angular momentum"  I^Angular Collision Response  .Final impulse equation (more waving of hands) /  J_Angular Collision Response  gCompute new angular momentum remember, sim uses angular momentum Then angular velocity from momentum B$$$%*)    <  vMultiple Colliding ContactsT3-body problem (kinda) Do one at a time, will end up with penetration What to do?@}Multiple Colliding ContactsOne solution: Do independently Handle penetration as part of collision resolution Another solution: Generate constraint forces Use relaxation techniquesLD5D5wMultiple Colliding ContactsCrunchy math solution Build systems of equations where , normal component of rel. vel. before & after ai,j describes how body j affects impulse on body i Get matrix equation:Z8 , >XGxMultiple Colliding ContactsResults coupled  want  as close as possible If (separating) want close to If (colliding) want close to Represent difference by new vector b If then If thenb.V%(.V#(yMultiple Colliding ContactsvNow want to minimize length of vector Impulses are non-positive so want Also want Quadratic programming problem Can convert to Linear Complementary Problem (LCP)  use Lemke s AlgorithmResting ContactslMirtich & Canny use micro-impulses, simulate molecular micro-collisions Works with current impulse system flsResting Contacts`Impulses work, but can be kind of jittery Forces generate velocity into object Counteract in next frame  but still move a bit One solution: once close to rest, turn off siminResting ContactVelocity along normal is 0, so look at forces along normal If will push together, counteract with new constraint force C = gn, g < 0 Want acceleration along normal If will eventually push apart, want g = 0 So want: d" 0, g d" 0, g = 0'JF ! !{t~Resting ContactCan build expression for : mtResting ContactExpand out, end up with expression like Ag repr. acceleration due to response b repr. acceleration due to other forces Remember, want d" 0, g d" 0, g = 0 If b > 0, then g = -b/A, = 0 If b d" 0, then g = 0, = bh)%:!! ! ,,"Resting ContactSimple example: assume object on ground, only linear forces If b > 0, then Counteracts force along normal If b d" 0, then C = 0N=!Resting ContactaThings get a lot more complicated if both move, and angular terms See Eberly for the full detailsbaFuMultiple Resting Contacts.Can have stacks of stuff What to do then?Multiple Resting Contacts@Build systems of equations where Get matrix equation where , , A is the same as with colliding contacts! Another LCP problem  use Lemke s<7LUJResting Contacts Resting contact is a hard problem Very susceptible to floating point error, jittering Don t do more than application needs Don t get discouragedContact FrictionGenerate force opposed to velocity For contact: tangential relative velocity Magnitude is relative to normal force If using impulses, can ignore normal force, use constant Putting It TogetherCompute forces, torques on objects Determine collisions Adjust velocities, forces, torques based on collision Predict future collision Step ahead to future collision (if any) or fixed step (if not)" |RecapTry to keep objects non-penetrating Colliding objects get impulses Resting objects either micro-impulses or constraint forces Multiple contacts hard, but manageablePe References   xHecker, Chris,  Behind the Screen, Game Developer, Miller Freeman, San Francisco, Dec. 1996-Jun. 1997. Lander, Jeff,  Graphic Content, Game Developer, Miller Freeman, San Francisco, Jan., Mar., Sep. 1999. Witkin, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modelling, SIGGRAPH 2002.V=$W{               *       { References   Mirtich, Brian and John Canny,  Impulse-Based Simulation of Rigid Bodies , Proceedings of 1995 Symposium on Interactive 3D Graphics, April 1995. (available online) Eberly, David H., Game Physics, Morgan Kaufmann, 2004.@K:1 p          !       0  ` f33` 3f3` ___` 3>?" dd@,?" dd@  " @ `"  n?" dd@   @@``@n?" dd@  @@``PR    @ ` ` p>>D$ 0Essential Math for Games r(    Ngֳgֳ ?P  X Click to edit Master title style!!  2  Hgֳgֳ ?  RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S    Z1 ?``  b* 2    Z1 ?`   X*(2    ZԽ1 ?`   Z*(2  N  6޽h? ? 3  Lecture Template 0 @( ~@K@   BE 1?P%   X* 2     BXI 1? %  Z*(2   p  0 ?*P  ,  BF 1? @2  RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S    HO 1?P   X* 2     HU 1?   Z*(2   H  0h0 ? ̙3380___PPT10.QA9 XPP( \@d@   B4` 1?P%   X* 2     B|f 1? %  Z*(2     HDk 1?P   X* 2     Hn 1?   Z*(2   H  0h0 ? ̙3380___PPT10.Qp {G= TL0 (      Nhgֳgֳ ?0<  ~Collision Response,     Txgֳgֳ ? J  H   0޽h ? 33___PPT10i.#\z+D=' = @B +y  `(0(  (x ( c $uP   x ( c $|v  H ( 0޽h ? 33y___PPT10Y+D=' = @B +y  p,0(  ,x , c $|P   x , c $}  H , 0޽h ? 33y___PPT10Y+D=' = @B +y  00(  0x 0 c $P   x 0 c $d  H 0 0޽h ? 33y___PPT10Y+D=' = @B +y  40(  4x 4 c $P   x 4 c $|  H 4 0޽h ? 33y___PPT10Y+D=' = @B +y  0(  x  c $P   x  c $8  H  0޽h ? 33y___PPT10Y+D=' = @B +y  0(  x  c $P   x  c $L  H  0޽h ? 33y___PPT10Y+D=' = @B +e   (  x  c $P   x  c $`  p2  H 8c?oj p2  H 8c? j / p2  H 8c?o J p2  H 8c? * O B @ ZD8c?/ j B   ZD8c?/ * o vB  @ ND8c? _ vB  @ ND8c?_ * J vB   ND8c?oZ vB   ND8c? z / H  0޽h ? 33y___PPT10Y+D=' = @B +H  .&(  x  c $PP   x  c $   p2  H 8c?] p2  H 8c?]g B @ ZD8c?  ( } B   ZD8c? G ] l b  b ,$D 0x2  H 8c? b~B   ND8c?b' ]~B   ND8c?o B  ND8c?"g & ,$D 0  T jJ?,  bSa& 2     H( jJ?_ Y  bSb& 2   H  0޽h ? 33___PPT10+sD~' = @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+3  ZR (  x  c $P   x  c $h  p2  H 8c?R p2  H 8c? W p2  H 8c?g R vB   ND8c?' vB   ND8c?P o \ vB   ND8c?g    T, jJ?) ,$#  bSa& 2      H jJ?  bSb& 2   |2  T 8c?~s |2  T 8c? /  H  0޽h ? 33y___PPT10Y+D=' = @B +  (  x  c $,?P   x  c $?  r  6A  p?? \  pH  0޽h ? 33y___PPT10Y+D=' = @B +y  0(  x  c $ FP   x  c $F  H  0޽h ? 33y___PPT10Y+D=' = @B +  D0(  Dx D c $LP   x D c $M  v2 D N 8c?db v2 D N 8c?b@ B D ZD8c?b!B D ZD8c?r0cp2  D H 8c? WU{ p2 D H 8c? U3{ |B D TD8c?u Uu |B D TD8c? Vuu p2 D H 8c?S YW?p2 D H 8c?S W5?|B D TD8c?9W9|B D TD8c? WX9H D 0޽h ? 33y___PPT10Y+D=' = @B +y   0(  x  c $VP   x  c $W  H  0޽h ? 33y___PPT10Y+D=' = @B +  0 H}(  Hx H c $DP   x H c $  p2 H H 8c?  p2 H H 8c? |B H TD8c?   H Nt  8c?` 6  Wn$ 2  |B H TD8c? `     H N6 8c? v  tva2 2     |B  H TD8c?P ` @  H N 8c?l @ tvb2 2     H H 0޽h ? 33y___PPT10Y+D=' = @B +   @Lv(  Lx L c $P   x L c $d  r L 6A`  Z??0\ u Zp2 L H 8c?  p2 L H 8c? |B L TD8c?   L N 8c?` 6  Wn$ 2  |B  L TD8c? `     L Nإ 8c? v  tva2 2     |B  L TD8c?P ` @  L NԚ 8c?l @ tvb2 2     |B  L TD8c? `  L N 8c? P  uvab2 2     H L 0޽h ? 33y___PPT10Y+D=' = @B +  P P|(  Px P c $qP   x P c $Pr  p2 P H 8c?  p2 P H 8c? |B P TD8c?   P Nt 8c?` 6  Wn$ 2  |B P TD8c? `   P Ny 8c? P  uvab2 2     r  P 6Aa  [??t  [|B  P TD8c? |B  P TD8c? `   P N, 8c?  tvn2 2     H P 0޽h ? 33y___PPT10Y+D=' = @B +  `T(  Tx T c $P   x T c $|  r T 6A   U??1  U T N 8c?  Xor$ 2   H T 0޽h ? 33y___PPT10Y+D=' = @B +   p X(  Xx X c $P   x X c $t  r X 6A  _??  _ X N8 8c?'  Lor 2   X N| 8c?C   Mand 2    X N 8c?   Lor 2  r  X 6A   ]??  d ]H X 0޽h ? 33y___PPT10Y+D=' = @B +  \(  \x \ c $tP   x \ c $0  r \ 6A  7??q  7H \ 0޽h ? 33y___PPT10Y+D=' = @B +   !  ` (  `x ` c $<P   x ` c $  p2 ` H 8c?  p2 ` H 8c? |B ` TD8c?   ` N 8c?` 6  Wn$ 2  |B ` TD8c? `     ` N  8c?  va-@ 2      |B  ` TD8c?P ` @  ` N 8c?  vb-@ 2      |B ` TD8c? ` P |B ` TD8c?P ` p@ ` NH 8c?@= vb+@ 2       ` NL 8c?P [ T   va+@ 2      |B `@ TD8c? p` |B ` TD8c?P ` P P r ` 6A  h??"  s hr ` 6A  t??;%  tH ` 0޽h ? 33y___PPT10Y+D=' = @B +     hd (  hx h c $P   x h c $  p2 h H 8c?  p2 h H 8c? |B h TD8c? `   |B h TD8c?P ` @ h N 8c?  tvb2 2       h N  8c? v  tva2 2     |B  h TD8c? ` |B  h TD8c? ` P   h N` 8c?  tra2 2       h NH 8c?' f 4  rrb0 2     p2 h H 8c? p2 h@ H 8c?  h ND 8c?U   lwa6 2   h N 8c?0 0   lwb6 2  r h 6Ah  b??@| %  bH h 0޽h ? 33y___PPT10Y+D=' = @B +   ln(  lx l c $P   x l c $T  r l 6A  m??vT"-  m l N$% 8c?C   Mand 2    l N% 8c?'  Lor 2   l N% 8c?C   Mand 2     l Nx % 8c?   Lor 2  r  l 6A  r?? `   rH l 0޽h ? 33y___PPT10Y+D=' = @B +  p(  px p c $%P  % x p c $h% % r p 6A  E??.E  EH p 0޽h ? 33y___PPT10Y+D=' = @B +;  bZt(  tx t c $%P  % x t c $% %  t  `A   v? 8c?Or  v t  `A  x? 8c?  xr t 6A  z?? t  zH t 0޽h ? 33y___PPT10Y+D=' = @B +  (  x  c $ %P  % x  c $d!% % p2  H 8c?"" p2  H 8c?  p2  H 8c?  |B  TD8c?"R b|B @ TD8c?I  H  0޽h ? 33y___PPT10Y+D=' = @B +y  0(  x  c $\(%P  % x  c $)% % H  0޽h ? 33y___PPT10Y+D=' = @B +a   (  x  c $/%P  % x  c $T0% % r  6A  |??@:  |r  6A   ??  r   6A  ?? k  r   6A  ??   H  0޽h ? 33y___PPT10Y+D=' = @B +I  ph (  x  c $08%P  % x  c $8% % r   6A*  ??b a  r   6A  ??J r   6A(  ??e.  r   6A  ??G*  r  6A)  ?? T%# r  6A  ?? o  r  6A!  ?? o  r  6A  ??az H  0޽h ? 33y___PPT10Y+D=' = @B +    (  x  c $H%P  % x  c $I% % r  6A%  ??C U  r  6A   4??  4r   6A#  ??h] 4  H  0޽h ? 33y___PPT10Y+D=' = @B +    8(  pB  HD8c? Q v  N 8c? v  NjJ? x  c $P   x  c ${  |B  TD8c?F + 0 |B  TD8c?I / 4 |B   TD8c?: 5 : |B   TD8c?: _ d |B   TD8c?: ? 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