1. Physics for Games Programmers Tutorial Motion and Collision – It’s All Relative
Squirrel Eiserloh
Lead Programmer
Ritual Entertainment

2. Takeaway A comfortable, intuitive understanding of:
The Problems of Discrete Simulation
Continuous Collision Detection
Applying Relativity to Game Physics
Configuration Space
Collisions in Four Dimensions
The Problems of Rotation
Why this is all really important even if you’re doing simple cheesy 2d games at home in your underwear in your spare time

3. The Problem
Discrete physics simulation falls embarrassingly short of reality.
“Real” physics is prohibitively expensive...
...so we cheat.
We need to cheat enough to be able to run in real time.
We need to not cheat so much that things break in a jarring and unrecoverable way.
Much of the challenge is knowing how and when to cheat.

4. Overview Simulation Tunneling Movement Bounds Swept Shapes
Einstein Says...
Minkowski Says...
Rotation

5. Also, I promise...
No math

6. Simulation
(Sucks)

7. Problems with Simulation
Flipbook syndrome

8. Problems with Simulation
Flipbook syndrome
Things can happen in-between snapshots

9. Problems with Simulation
Flipbook syndrome
Things mostly happen in-between snapshots

10. Problems with Simulation
Flipbook syndrome
Things mostly happen in-between snapshots
Curved trajectories treated as piecewise linear

11. Problems with Simulation
Flipbook syndrome
Things mostly happen in-between snapshots
Curved trajectories treated as piecewise linear
Terms often assumed to be constant throughout the frame

12. Problems with Simulation
Flipbook syndrome
Things mostly happen in-between snapshots
Curved trajectories treated as piecewise linear
Terms often assumed to be constant throughout the frame
Error accumulates

13. Problems with Simulation (cont’d)
Rotations are often assumed to happen instantaneously at frame boundaries

14. Problems with Simulation (cont’d)
Rotations are often assumed to happen instantaneously at frame boundaries
Energy is not always conserved
Energy loss can be undesirable
Energy gain is evil
Simulations explode!

15. Problems with Simulation (cont’d)
Rotations are often assumed to happen instantaneously at frame boundaries
Energy is not always conserved
Energy loss can be undesirable
Energy gain is evil
Simulations explode!
Tunneling
(Also evil!)

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

17. Tunneling
(Sucks)

18. Tunneling
Small objects tunnel more easily

19. Tunneling (cont’d) Possible solutions Minimum size requirement?
Inadequate; fast objects still tunnel

20. Tunneling (cont’d)
Fast-moving objects tunnel more easily

21. Tunneling (cont’d) Possible solutions Minimum size requirement?
Inadequate; fast objects still tunnel
Maximum speed limit?
Inadequate; since speed limit is a function of object size, this would mean small & fast objects (bullets) would not be allowed
Smaller time step?
Helpful, but inadequate; this is essentially the same as a speed limit

22. Tunneling (cont’d)
Besides, even with min. size requirements and speed limits and a small timestep, you still have degenerate cases that cause tunneling

23. Tunneling (cont’d)
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 shouldn’t
Players missing a trigger boundary
Okay, so tunneling really sucks. What can we do about it?

24. Movement Bounds

25. Movement Bounds
Disc / Sphere

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

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

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

29. 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 don’t know for sure (false positives).

30. Movement Bounds Conclusion
Good: They prevent tunneling! (i.e. no false negatives)
Bad: They don’t actually tell us whether A and B collided (still have false positives).
Good: They can be used as a cheap, effective early rejection test.

31. Swept Shapes

32. Swept Shapes
Swept disc / sphere (n-sphere): capsule

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

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

35. 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

36. Swept Shapes (cont’d)
Like movement bounds, only with a perfect fit!

37. Swept Shapes (cont’d) Like movement bounds, only with a perfect fit!
Still no false negatives (tunneling).

38. Swept Shapes (cont’d) Like movement bounds, only with a perfect fit!
Still no false negatives (tunneling).
Finally, no false positives, either!

39. Swept Shapes (cont’d) 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.

40. Swept Shapes (cont’d) Conclusion Suck?
Can be used as early rejection test, but...
...movement bounds are better for that.
If you’re not too picky...
...they DO solve a large number of nasty problems (especially tunneling)
...and can serve as a poor man’s continuous collision detection for a basic engine.

42. Einstein Says...
Coordinate systems are relative

43. Relative Coordinate Systems

44. Relative Coordinate Systems
World coordinates

45. Relative Coordinate Systems
World coordinates
A’s local coordinates

46. Relative Coordinate Systems
World coordinates
A’s local coordinates
B’s local coordinates

47. Relative Coordinate Systems
Math is often nicer at the origin.
(x-h)2 + (y-k)2 = r2
x2 + y2 = r2

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

49. Relative Motion

50. Relative Motion
"Frames of Reference"
World frame

51. Relative Motion
"Frames of Reference"
World frame
A's frame

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

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

54. Relative Motion 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)

55. (Does Not Suck)

56. Relative Collision Bodies

57. Relative Collision Bodies
Collision check equivalencies (disc)

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

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

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

61. 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 body’s origin

62. Relative Collision Bodies (cont’d)
Disc + disc

63. Relative Collision Bodies (cont’d)
Disc + disc
AABB + AABB

64. Relative Collision Bodies (cont’d)
Disc + disc
AABB + AABB
Triangle + AABB

65. Relative Collision Bodies (cont’d)
Disc + disc
AABB + AABB
Triangle + AABB
AABB + triangle

66. Relative Collision Bodies (cont’d)
Disc + disc
AABB + AABB
Triangle + AABB
AABB + triangle
Polytope + polytope

67. Relative Collision Bodies (cont’d)
Disc + disc
AABB + AABB
Triangle + AABB
AABB + triangle
Polytope + polytope
Polytope + disc

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

69. Minkowski Arithmetic

70. 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.

71. 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.
Minkowski Sums are commutative:
A+B = B+A
Minkowski Sum of convex objects is convex

72. 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)

73. 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.

74. Minkowski Differences (cont’d)
Minkowski Differences are not commutative:
A-B != B-A
Minkowski Difference of convex objects is convex (since A-B = A+ -B)

75. Minkowski Differences (cont’d)
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”

76. Minkowski Differences (cont’d)
If the singularity is outside the combined body, A and B do not overlap.

77. Minkowski Differences (cont’d)
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.

78. Minkowski Differences (cont’d)
Aorigin vs. Borigin
-Borigin -Borigin
___ ___
(A-B)origin vs. 0

79. Minkowski Differences (cont’d)
In world space, A-B is “near” the origin

80. Minkowski Differences (cont’d)
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.

81. Minkowski Differences (cont’d)
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)

82. Minkowski Differences (cont’d)
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)

83. Minkowski Differences (cont’d)
Translations in A or B simply translate the CSO

84. Minkowski Differences (cont’d)
Rotations in A or B mutate the CSO

85. Minkowski Sum vs. Difference
Lots of confusion over Minkowski “Sum” vs. “Difference”.
Sum is used to “fatten” an object by “adding” another object (in local coordinates) to it
Difference is used to put the objects in configuration space, i.e. A-B vs. origin.
Difference sometimes called Sum since A-B can be expressed as A+(-B)!

86. Minkowski Sum vs. Difference (cont’d)
Difference is the same as “Spirograph” or “rubbing”
Difference is not commutative!
A-B != B-A
Difference and sum produce different-shaped results
Difference produces CSO (configuration space obstacle)

87. (Does Not Suck)

88. Relative Everything

89. Relative Everything Let’s combine: Relative Coordinate Systems
Relative Motion
Relative Collision Bodies

90. Relative Everything (cont’d)
A vs. B in world frame

91. Relative Everything (cont’d)
A vs. B in world frame
A vs. B, inertial frame

92. Relative Everything (cont’d)
A vs. B in world frame
A vs. B, inertial frame
A is moving, B is still

93. Relative Everything (cont’d)
A vs. B in world frame
A vs. B, inertial frame
A is moving, B is still
A is CSO, B is point

94. Relative Everything (cont’d)
A vs. B in world frame
A vs. B, inertial frame
A is moving, B is still
A is CSO, B is point
A is moving CSO, B is still point

95. Relative Everything (cont’d)
A vs. B in world frame
A vs. B, inertial frame
A is moving, B is still
A is CSO, B is point
A is moving CSO, B is still point
A is still CSO, B is moving point

96. Relative Everything (cont’d)
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 don’t know WHEN they collided...

97. Relative Everything (cont’d)
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

98. Relative Everything (cont’d)
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”.

99. Determining Collision Time

100. Determining Collision Time
Method #1: Frame Subdivision

101. Subdividing Movement Frame
If a swept-shape (or movement bounds) test says “yes”:

102. Subdividing Movement Frame
If a swept-shape (or movement bounds) test says “yes”:
Cut the frame in half; perform two separate tests (first half first, second half second).
First positive test is when the collision occurred.

103. Subdividing Movement Frame (cont’d)
Can recurse (1/2, 1/4, 1/8...) to the desired level of granularity

104. Subdividing Movement Frame (cont’d)
Can recurse (1/2, 1/4, 1/8...) to the desired level of granularity
If both tests negative, no collision (was a false positive).
Still inexact (minimizing, not eliminating, false positives)
Gets expensive

105. Determining Collision Time
Method #1: Frame Subdivision
Method #2: 4D* Continuous Collision Detection
*(N+1 dimensions; 3D for 2D physics, etc.)

106. Spacetime

107. Spacetime
Spacetime is a Physics construct which combines N-dimensional space with an extra dimension for time, yielding a unified model with N+1 dimensions.
Space (1D) + time (1D) = spacetime (2D)
Space (2D) + time (1D) = spacetime (3D)
Space (3D) + time (1D) = spacetime (4D)

108. Spacetime Diagrams
1D space + time = 2D
Just an X vs. T graph!

109. Spacetime Diagrams 1D space + time = 2D 2D space + time = 3D
Just an X vs. T graph!
2D space + time = 3D
No problem.

110. Spacetime Diagrams 1D space + time = 2D 2D space + time = 3D
Just an X vs. T graph!
2D space + time = 3D
No problem.
Another example (2d space + time = 3D)

111. Spacetime Diagrams 1D space + time = 2D 2D space + time = 3D ?
Just an X vs. T graph!
2D space + time = 3D
No problem.
Another example (2d space + time = 3D)
3D space + time = 4D
Brainbuster! ?

112. Spacetime Diagrams (cont’d)
Note that an N-dimensional system in motion is the same as a still snapshot in N+1 dimensions
1D animation = 2D spacetime still image
2D animation = 3D spacetime still image
3D animation = 4D spacetime still image

113. Spacetime Diagrams (cont’d)
1D animation
2D spacetime still image

114. Spacetime Diagrams (cont’d)
2D animation
3D spacetime still image

115. Spacetime Diagrams (cont’d)
How do you envision a 4D object?
Use 2D animation -> 3D spacetime diagram as a mental analogy.
Fun reading: Flatland by Edwin Abbott
Think about a 4D object that you’re already familiar with.
(The universe in motion!)

116. Spacetime Diagrams (cont’d)
Invented by Hermann Minkowski
Also called “Minkowski Diagrams”

117. (Rules)

118. Time-Swept Shapes

119. Time-Swept Shapes
Sweep out shapes, but do it over time in a spacetime diagram
Define time over frame as being in the interval [0,1]
As before, we can play around with lots of relativistic variations:

120. Time-Swept Shapes (cont’d)
A vs. B in world frame

121. Time-Swept Shapes (cont’d)
A vs. B in world frame
A is moving, B is still

122. Time-Swept Shapes (cont’d)
A vs. B in world frame
A is moving, B is still
A is CSO, B is point

123. Time-Swept Shapes (cont’d)
A vs. B in world frame
A is moving, B is still
A is CSO, B is point
A is still CSO, B is moving (swept) point

124. Time-Swept Shapes (cont’d)
To solve for collision time, we intersect the point-swept ray against the CSO
The ‘t’ coordinate at the intersection point is the time [0,1] of collision
Collision check is done in N+1 dimensions
Which means, in a 3D game, we collide a 4D ray vs. a 4D body! (What?)

125. Time-Swept Shapes (cont’d)
Wait, it gets easier...
When we view this diagram (CSO vs moving point) down the time axis, i.e. from “overhead”:
Since CSO is not moving, it looks 2D from overhead...

126. Time-Swept Shapes (cont’d)
We can reduce this back down to N dimensions (from N+1) since we are looking down the time axis!
So it becomes an N-dimesional ray vs. N-dimensional body again.
Which means, in a 3D game, we collide a 3D ray vs. a 3D body.

128. Time-Swept Shapes (cont’d)
Question #4: When, during the frame, did A and B collide?
Finally, the right question - and we have a complete answer!
With fixed cost, and with exact results (no false anything).

129. Time-Swept Shapes (cont’d)
BTW, this is essentially the same as solving for the fraction of the singularity-translation ray from our original Minkowski Difference “inadmissible translations” picture!

130. or “You Get What You Pay For”
Quality vs. Quantity or
“You Get What You Pay For”

131. 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?

132. Rotations
(Suck)

133. Rotations Continuous rotational collision detection sucks
Rotational tunneling alone is problematic

134. Rotations Continuous rotational collision detection sucks
Rotational tunneling alone is problematic
Methods we’ve discussed here often don’t work on rotations, or their rotational analogue is quite complex

135. Rotations (cont’d) 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

136. Summary

137. Summary
The nature of simulation causes us real problems... problems which can’t be ignored.
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.

138. Summary (cont’d) Einstein and Minkowski are cool. Rotations suck.
Doing real-time collision detection in 4D spacetime doesn’t have to be hard.
Or expensive.
Or confusing.

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