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Physics for Games Programmers Reframing the Problem Squirrel Eiserloh Technical Director MumboJumbo Games squirrel@eiserloh.net www.algds.org squirrel@eiserloh.net www.algds.org

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Physics for Games Programmers Reframing the Problem a.k.a. Its All Relative Squirrel Eiserloh Technical Director MumboJumbo Games squirrel@eiserloh.net www.algds.org squirrel@eiserloh.net www.algds.org

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3 Overview Tunneling Movement Bounds Swept Shapes Einstein Says... Minkowski Says... Rotation

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Tunneling (Sucks)

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5 Tunneling Question #1: Do objects A and B overlap? Plenty of reference material to help solve this, but......this is often the wrong question to ask (begs tunneling).

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6 Tunneling

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10 Tunneling (contd) Tunneling is very, very bad – this is not a mundane detail Things falling through world Bullets passing through people or walls Players getting places they shouldnt Players missing a trigger boundary Tunneling is a false negative Okay, so tunneling really sucks. What can we do about it?

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Movement Bounds

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12 Movement Bounds Disc / Sphere

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13 Movement Bounds Disc / Sphere AABB (Axis-Aligned Bounding Box)

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14 Movement Bounds Disc / Sphere AABB (Axis-Aligned Bounding Box) OBB (Oriented Bounding Box)

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15 Movement Bounds Question #2: Could A and B have collided during the frame? Better than Question #1 (solves tunneling!), but...

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16 Movement Bounds Question #2: Could A and B have collided during the frame? Better than Question #1 (solves tunneling!), but......even if the answer is yes, we still dont know for sure (false positive).

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17 Movement Bounds Conclusion Good: They prevent tunneling! (i.e. no false negatives) Bad: They dont actually tell us whether A and B collided (still have false positives). Good: They can be used as a cheap, effective early rejection test.

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Swept Shapes

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19 Swept Shapes Swept disc / sphere (n-sphere): capsule

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20 Swept Shapes Swept disc / sphere (n-sphere): capsule Swept AABB: convex polytope (polygon in 2d, polyhedron in 3d)

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21 Swept Shapes Swept disc / sphere (n-sphere): capsule Swept AABB: convex polytope (polygon in 2d, polyhedron in 3d) Swept triangle / tetrahedron (simplex): convex polytope

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22 Swept Shapes Swept disc / sphere (n-sphere): capsule Swept AABB: convex polytope (polygon in 2d, polyhedron in 3d) Swept triangle / tetrahedron (simplex): convex polytope Swept polytope: convex polytope

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23 Swept Shapes (contd) Like movement bounds, only with a perfect fit!

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24 Swept Shapes (contd) Like movement bounds, only with a perfect fit! Still no false negatives (tunneling).

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25 Swept Shapes (contd) Like movement bounds, only with a perfect fit! Still no false negatives (tunneling). Finally, no false positives, either!

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26 Swept Shapes (contd) Like movement bounds, only with a perfect fit! Still no false negatives (tunneling). Finally, no false positives, either! No, wait, nevermind. Still have em. Rats.

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27 Swept Shapes (contd) Like movement bounds, only with a perfect fit! Still no false negatives (tunneling). Finally, no false positives, either! No, wait, nevermind. Still have em. Rats.

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28 Swept Shapes (contd) Like movement bounds, only with a perfect fit! Still no false negatives (tunneling). Finally, no false positives, either! No, wait, nevermind. Still have em. Rats.

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29 Swept Shapes (contd) Conclusion Suck? Can be used as early rejection test, but......movement bounds are better for that. If youre not too picky......they DO solve a large number of nasty problems (especially tunneling)...and can serve as a poor mans continuous collision detection for a basic engine.

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31 Einstein Says... Coordinate systems are relative

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Relative Coordinate Systems

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33 Relative Coordinate Systems World coordinates

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34 Relative Coordinate Systems World coordinates As local coordinates

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35 Relative Coordinate Systems World coordinates As local coordinates Bs local coordinates

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36 Relative Coordinate Systems World coordinates As local coordinates Bs local coordinates Many others (e.g. origin at point of impact)

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37 Relative Coordinate Systems (contd) Ball vs. world...

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38 Relative Coordinate Systems (contd) Ball vs. world... in world coordinates

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39 Relative Coordinate Systems (contd) Ball vs. world... in world coordinates x component y component

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40 Relative Coordinate Systems (contd) Ball vs. world... in world coordinates x component y component in impact coordinates

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41 Relative Coordinate Systems (contd) Ball vs. world... in world coordinates x component y component in impact coordinates parallel component perpendicular component

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42 Relative Coordinate Systems (contd) Ball vs. world... in world coordinates x component y component in impact coordinates parallel component perpendicular component Change in motion happens along the perpendicular axis

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43 Relative Coordinate Systems (contd) Ball vs. ball...

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44 Relative Coordinate Systems (contd) Ball vs. ball... in world coordinates

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45 Relative Coordinate Systems (contd) Ball vs. ball... in world coordinates x component y component

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46 Relative Coordinate Systems (contd) Ball vs. ball... in world coordinates x component y component in impact coordinates

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47 Relative Coordinate Systems (contd) Ball vs. ball... in world coordinates x component y component in impact coordinates parallel component perpendicular component Energy is exchanged along the perpendicular axis

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48 Relative Coordinate Systems (contd) x 2 + y 2 = r 2 x 2 - 2xh + h 2 + y 2 - 2yk + k 2 = r 2 Also, math is often nicer at the origin.

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49 Einstein Says... Coordinate systems are relative Motion is relative

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Relative Motion

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51 Relative Motion "Frames of Reference" World frame

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52 Relative Motion "Frames of Reference" World frame A's frame

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53 Relative Motion "Frames of Reference" World frame A's frame B's frame

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54 Relative Motion "Frames of Reference" World frame A's frame B's frame Inertial frame

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55 Relative Motion (contd) A Rule of Relativistic Collision Detection: It is always possible to reduce a collision check between two moving objects to a collision check between a moving object and a stationary object (by reframing)

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56 (Does Not Suck)

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Relative Collision Bodies

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58 Relative Collision Bodies Collision check equivalencies (disc)

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59 Relative Collision Bodies Collision check equivalencies (disc)...AABB

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60 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity

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61 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together

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62 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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63 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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64 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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65 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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66 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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67 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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68 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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69 Relative Collision Bodies Collision check equivalencies (disc)...AABB Can even reduce one body to a singularity Tracing or Rubbing collision bodies together Spirograph-out the reduced bodys origin

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70 Relative Collision Bodies (contd) Disc + disc

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71 Relative Collision Bodies (contd) Disc + disc AABB + AABB

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72 Relative Collision Bodies (contd) Disc + disc AABB + AABB Triangle + AABB

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73 Relative Collision Bodies (contd) Disc + disc AABB + AABB Triangle + AABB AABB + triangle

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74 Relative Collision Bodies (contd) Disc + disc AABB + AABB Triangle + AABB AABB + triangle Polytope + polytope

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75 Relative Collision Bodies (contd) Disc + disc AABB + AABB Triangle + AABB AABB + triangle Polytope + polytope Polytope + disc

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76 Relative Collision Bodies (contd) Things start to get messy when combining bodies explicitly / manually. (Especially in 3d.) General solution?

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Minkowski Arithmetic

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78 Minkowski Sums The Minkowski Sum (A+B) of A and B is the result of adding every point in A to every point in B.

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79 Minkowski Differences The Minkowski Difference (A-B) of A and B is the result of subtracting every point in B from every point in A (or A + -B)

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80 Minkowski Differences The Minkowski Difference (A-B) of A and B is the result of subtracting every point in B from every point in A Resulting shape is different from A+B.

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81 Minkowski Differences (contd) Minkowski Differences are not commutative: A-B != B-A Minkowski Difference of convex objects is convex (since A-B = A+ -B)

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82 Minkowski Differences (contd) Minkowski Differences are not commutative: A-B != B-A Minkowski Difference of convex objects is convex (since A-B = A+ -B) Minkowski Difference produces the same shape as Spirograph

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83 Minkowski Differences (contd) If the singularity is outside the combined body, A and B do not overlap.

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84 Minkowski Differences (contd) If the singularity is outside the combined body, A and B do not overlap. If the singularity is inside the combined body (A-B), then A and B overlap.

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85 Minkowski Differences (contd) In world space, A-B is near the origin

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86 Minkowski Differences (contd) A origin vs. B origin -B origin ___ (A-B) origin vs. 0

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87 Minkowski Differences (contd) Since the singularity point is always at the origin (B-B), we can say... If (A-B) does not contain the origin, A and B do not overlap.

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88 Minkowski Differences (contd) If (A-B) contains the origin, A and B overlap. In other words, we reduce A vs. B to: combined body (A-B) vs. point (B-B, or origin)

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89 Minkowski Differences (contd) If A and B are in the same coordinate system, the comparison between A-B and the origin is said to happen in configuration space...in which case A-B is said to be a configuration space obstacle (CSO)

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90 Minkowski Differences (contd) Translations in A or B simply translate the CSO

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91 Minkowski Differences (contd) Rotations in A or B mutate the CSO

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92 (Does Not Suck)

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Relative Everything

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94 Relative Everything Lets combine: Relative Coordinate Systems Relative Motion Relative Collision Bodies

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95 Relative Everything (contd) A vs. B in world frame

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96 Relative Everything (contd) A vs. B in world frame A is CSO, B is point

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97 Relative Everything (contd) A vs. B in world frame A is CSO, B is point A is moving CSO, B is still point

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98 Relative Everything (contd) A vs. B in world frame A is CSO, B is point A is moving CSO, B is still point A is still CSO, B is moving point This is the one we want!

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99 Relative Everything (contd) Question #3: Did A and B collide during the frame? Yes! We can now get an exact answer. No false negatives, no false positives! However, we still dont know WHEN they collided...

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100 Relative Everything (contd) Why does the exact collision time matter? Outcomes can be different Order of events (e.g. multiple collisions) is relevant Collision response is easier when you can reconstruct the exact moment of impact

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101 Relative Everything (contd) Question #4: When, during the frame, did A and B collide? The time at which the ray intersects the CSO is the time at which the collision occurred. Finally, the right question - and we have a complete answer!

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102 Relative Everything (contd) The Minkowski Difference (A-B) / CSO can also be thought of as the set of all translations [from the origin] that would cause a collision. A.K.A. the set of inadmissible translations.

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Quality vs. Quantity or You Get What You Pay For

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104 Quality vs. Quantity The more you ask, the more you pay. Question #1: Do A and B overlap? Question #2: Could A and B have collided during the frame? Question #3: Did A and B collide during the frame? Question #4: When, during the frame, did A and B collide?

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Rotations (Suck)

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106 Rotations Continuous rotational collision detection sucks Rotational tunneling alone is problematic

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107 Rotations Continuous rotational collision detection sucks Rotational tunneling alone is problematic Methods weve discussed here often dont work on rotations, or their rotational analogue is quite complex

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108 Rotations (contd) However: Rotational tunneling is usually not as jarring as translational tunneling Rotational speed limits are actually feasible Can do linear approximations of swept rotations Can use bounding shapes to contain pre- and post-rotated positions This is something that many engines never solve robustly

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Summary

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110 Summary Have to worry about false negatives (tunneling!) as well as false positives. Knowing when a collision event took place can be very important (especially when resolving it). Sometimes a problem (and math) looks easier when we look at it from a different viewpoint. Can combine bodies in cheaty ways to simplify things even further.

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111 Summary (contd) Einstein and Minkowski are cool. Rotations suck. Doing real-time collision detection doesnt have to be hard. Or expensive. Or confusing.

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112 Questions? Feel free to reach me by email at: squirrel@eiserloh.net

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