Physics%20for%20Games%20Programmers%20Tutorial%20Motion%20and%20Collision%20

1
Physics for Games Programmers TutorialMotion
and Collision Its All Relative

• Squirrel Eiserloh
• squirrel_at_eiserloh.net
• Lead Programmer
• Ritual Entertainment
• www.ritual.com
• www.algds.org

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 youre
  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 (contd)

• Rotations are often assumed to happen
  instantaneously at frame boundaries

14
Problems with Simulation (contd)

• 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 (contd)

• 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)
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Tunneling

• Small objects tunnel more easily

19
Tunneling (contd)

• Possible solutions
• Minimum size requirement?
• Inadequate fast objects still tunnel

20
Tunneling (contd)

• Fast-moving objects tunnel more easily

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Tunneling (contd)

• 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 (contd)

• Besides, even with min. size requirements and
  speed limits and a small timestep, you still have
  degenerate cases that cause tunneling

23
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
• Okay, so tunneling really sucks. What can we do
  about it?

24
Movement Bounds
25
Movement Bounds

• Disc / Sphere

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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...

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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 positives).

30
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.

31
Swept Shapes
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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)

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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 (contd)

• Like movement bounds, only with a perfect fit!

37
Swept Shapes (contd)

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

38
Swept Shapes (contd)

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

39
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.

40
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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Einstein Says...

• Coordinate systems are relative

43
Relative Coordinate Systems
44
Relative Coordinate Systems

• World coordinates

45
Relative Coordinate Systems

• World coordinates
• As local coordinates

46
Relative Coordinate Systems

• World coordinates
• As local coordinates
• Bs local coordinates

47
Relative Coordinate Systems
Math is often nicer at the origin.
x2 y2 r2
(x-h)2 (y-k)2 r2
48
Einstein Says...

• Coordinate systems are relative
• Motion is relative

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Relative Motion
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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)

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

• Collision check equivalencies (disc)

58
Relative Collision Bodies

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

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Relative Collision Bodies

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

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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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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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Relative Collision Bodies (contd)

• Disc disc

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Relative Collision Bodies (contd)

• Disc disc
• AABB AABB

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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

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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

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Relative Collision Bodies (contd)

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

69
Minkowski Arithmetic
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Minkowski Sums

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

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Minkowski Sums

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

74
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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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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Minkowski Differences (contd)

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

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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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Minkowski Differences (contd)

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

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Minkowski Differences (contd)

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

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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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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)

82
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)

83
Minkowski Differences (contd)
Translations in A or B simply translate the CSO
84
Minkowski Differences (contd)
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 (contd)

• 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

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

90
Relative Everything (contd)

• A vs. B in world frame

91
Relative Everything (contd)

• A vs. B in world frame
• A vs. B, inertial frame

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Relative Everything (contd)

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

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Relative Everything (contd)

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

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Relative Everything (contd)

• 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 (contd)

• 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 (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...

97
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

98
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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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 (contd)

• Can recurse (1/2, 1/4, 1/8...) to the desired
  level of granularity

104
Subdividing Movement Frame (contd)

• 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
• (N1 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 N1
  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!

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Spacetime Diagrams

• 1D space time 2D
• Just an X vs. T graph!
• 2D space time 3D
• No problem.

110
Spacetime Diagrams

• 1D space time 2D
• Just an X vs. T graph!
• 2D space time 3D
• No problem.
• Another example (2d space time 3D)

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Spacetime Diagrams

• 1D space time 2D
• 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 (contd)

• Note that an N-dimensional system in motion is
  the same as a still snapshot in N1 dimensions
• 1D animation 2D spacetime still image
• 2D animation 3D spacetime still image
• 3D animation 4D spacetime still image

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Spacetime Diagrams (contd)
2D spacetime still image
1D animation
114
Spacetime Diagrams (contd)
3D spacetime still image
2D animation
115
Spacetime Diagrams (contd)

• How do you envision a 4D object?
• Use 2D animation -gt 3D spacetime diagram as a
  mental analogy.
• Fun reading Flatland by Edwin Abbott
• Think about a 4D object that youre already
  familiar with.
• (The universe in motion!)

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Spacetime Diagrams (contd)

• Invented by Hermann Minkowski
• Also called Minkowski Diagrams

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(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 (contd)

• A vs. B in world frame

121
Time-Swept Shapes (contd)

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

122
Time-Swept Shapes (contd)

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

123
Time-Swept Shapes (contd)

• 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 (contd)

• 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 N1 dimensions
• Which means, in a 3D game, we collide a 4D ray
  vs. a 4D body! (What?)

125
Time-Swept Shapes (contd)

• 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 (contd)

• We can reduce this back down to N dimensions
  (from N1) 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.

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128
Time-Swept Shapes (contd)

• 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 (contd)

• 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
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 weve discussed here often dont work on
  rotations, or their rotational analogue is quite
  complex

135
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

136
Summary
137
Summary

• The nature of simulation causes us real
  problems... problems which cant 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 (contd)

• Einstein and Minkowski are cool.
• Rotations suck.
• Doing real-time collision detection in 4D
  spacetime doesnt have to be hard.
• Or expensive.
• Or confusing.

139
Questions?

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