Continuous Collision Detection of General Convex Objects Under Translation

1
Physics for Games Programmers Reframing the
Problem
Squirrel EiserlohTechnical DirectorMumboJumbo
Gamessquirrel_at_eiserloh.netwww.algds.org
2
Physics for Games Programmers Reframing the
Problema.k.a. Its All Relative
Squirrel EiserlohTechnical DirectorMumboJumbo
Gamessquirrel_at_eiserloh.netwww.algds.org
3
Overview

• Tunneling
• Movement Bounds
• Swept Shapes
• Einstein Says...
• Minkowski Says...
• Rotation

4
Tunneling
(Sucks)
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).

6
Tunneling
7
Tunneling
8
Tunneling
9
Tunneling
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?

11
Movement Bounds
12
Movement Bounds

• Disc / Sphere

13
Movement Bounds

• Disc / Sphere
• AABB (Axis-Aligned Bounding Box)

14
Movement Bounds

• Disc / Sphere
• AABB (Axis-Aligned Bounding Box)
• OBB (Oriented Bounding Box)

15
Movement Bounds

• Question 2 Could A and B have collided during
  the frame?
• Better than Question 1 (solves tunneling!),
  but...

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).

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.

18
Swept Shapes
19
Swept Shapes

• Swept disc / sphere (n-sphere) capsule

20
Swept Shapes

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

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

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

23
Swept Shapes (contd)

• Like movement bounds, only with a perfect fit!

24
Swept Shapes (contd)

• Like movement bounds, only with a perfect fit!
• Still no false negatives (tunneling).

25
Swept Shapes (contd)

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

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.

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.

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.

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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(No Transcript)
31
Einstein Says...

• Coordinate systems are relative

32
Relative Coordinate Systems
33
Relative Coordinate Systems

• World coordinates

34
Relative Coordinate Systems

• World coordinates
• As local coordinates

35
Relative Coordinate Systems

• World coordinates
• As local coordinates
• Bs local coordinates

36
Relative Coordinate Systems

• World coordinates
• As local coordinates
• Bs local coordinates
• Many others (e.g. origin at point of impact)

37
Relative Coordinate Systems (contd)

• Ball vs. world...

38
Relative Coordinate Systems (contd)

• Ball vs. world...
• in world coordinates

39
Relative Coordinate Systems (contd)

• Ball vs. world...
• in world coordinates
• x component
• y component

40
Relative Coordinate Systems (contd)

• Ball vs. world...
• in world coordinates
• x component
• y component
• in impact coordinates

41
Relative Coordinate Systems (contd)

• Ball vs. world...
• in world coordinates
• x component
• y component
• in impact coordinates
• parallel component
• perpendicular component

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

43
Relative Coordinate Systems (contd)

• Ball vs. ball...

44
Relative Coordinate Systems (contd)

• Ball vs. ball...
• in world coordinates

45
Relative Coordinate Systems (contd)

• Ball vs. ball...
• in world coordinates
• x component
• y component

46
Relative Coordinate Systems (contd)

• Ball vs. ball...
• in world coordinates
• x component
• y component
• in impact coordinates

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

48
Relative Coordinate Systems (contd)
Also, math is often nicer at the origin.
x2 y2 r2
x2 - 2xh h2 y2 - 2yk k2 r2
49
Einstein Says...

• Coordinate systems are relative
• Motion is relative

50
Relative Motion
51
Relative Motion

• "Frames of Reference"
• World frame

52
Relative Motion

• "Frames of Reference"
• World frame
• A's frame

53
Relative Motion

• "Frames of Reference"
• World frame
• A's frame
• B's frame

54
Relative Motion

• "Frames of Reference"
• World frame
• A's frame
• B's frame
• Inertial frame

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)

56
(Does Not Suck)
57
Relative Collision Bodies
58
Relative Collision Bodies

• Collision check equivalencies (disc)

59
Relative Collision Bodies

• Collision check equivalencies (disc)
• ...AABB

60
Relative Collision Bodies

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

61
Relative Collision Bodies

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

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

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

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

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

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

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

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

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

70
Relative Collision Bodies (contd)

• Disc disc

71
Relative Collision Bodies (contd)

• Disc disc
• AABB AABB

72
Relative Collision Bodies (contd)

• Disc disc
• AABB AABB
• Triangle AABB

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Relative Collision Bodies (contd)

• Disc disc
• AABB AABB
• Triangle AABB
• AABB triangle

74
Relative Collision Bodies (contd)

• Disc disc
• AABB AABB
• Triangle AABB
• AABB triangle
• Polytope polytope

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Relative Collision Bodies (contd)

• Disc disc
• AABB AABB
• Triangle AABB
• AABB triangle
• Polytope polytope
• Polytope disc

76
Relative Collision Bodies (contd)

• Things start to get messy when combining bodies
  explicitly / manually.
• (Especially in 3d.)
• General solution?

77
Minkowski Arithmetic
78
Minkowski Sums

• The Minkowski Sum (AB) of A and B is the result
  of adding every point in A to every point in B.

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)

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 AB.

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)

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

83
Minkowski Differences (contd)

• If the singularity is outside the combined body,
  A and B do not overlap.

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.

85
Minkowski Differences (contd)

• In world space, A-B is near the origin

86
Minkowski Differences (contd)

• Aorigin vs. Borigin
• -Borigin -Borigin
• ___ ___
• (A-B)origin vs. 0

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.

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)

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)

90
Minkowski Differences (contd)
Translations in A or B simply translate the CSO
91
Minkowski Differences (contd)
Rotations in A or B mutate the CSO
92
(Does Not Suck)
93
Relative Everything
94
Relative Everything

• Lets combine
• Relative Coordinate Systems
• Relative Motion
• Relative Collision Bodies

95
Relative Everything (contd)

• A vs. B in world frame

96
Relative Everything (contd)

• A vs. B in world frame
• A is CSO, B is point

97
Relative Everything (contd)

• A vs. B in world frame
• A is CSO, B is point
• A is moving CSO, B is still point

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!

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...

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

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!

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.

103
Quality vs. Quantity

• or
• You Get What You Pay For

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?

105
Rotations
(Suck)
106
Rotations

• Continuous rotational collision detection sucks
• Rotational tunneling alone is problematic

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

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

109
Summary
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.

111
Summary (contd)

• Einstein and Minkowski are cool.
• Rotations suck.
• Doing real-time collision detection doesnt have
  to be hard.
• Or expensive.
• Or confusing.

112
Questions?

• Feel free to reach me by email at
• squirrel_at_eiserloh.net