physics for games programmers tutorial motion and collision – its all relative squirrel eiserloh...

Physics for Games Programmers Tutorial

Motion and Collision – It’s All Relative

Squirrel Eiserlohsquirrel@eiserloh.net

Lead ProgrammerRitual Entertainment

www.ritual.comwww.algds.org

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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

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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.

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Overview

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

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Also, I promise...

No math

Simulation(Sucks)

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Problems with Simulation

Flipbook syndrome

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Problems with Simulation

Flipbook syndrome Things can happen

in-between snapshots

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Problems with Simulation

Flipbook syndrome Things mostly

happen in-between snapshots

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Problems with Simulation

Flipbook syndrome Things mostly

happen in-between snapshots

Curved trajectories treated as piecewise linear

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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

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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

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Problems with Simulation (cont’d)

Rotations are often assumed to happen instantaneously at frame boundaries

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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!

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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!)

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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).

Tunneling(Sucks)

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Tunneling

Small objects tunnel more easily

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Tunneling (cont’d)

Possible solutions Minimum size requirement?

Inadequate; fast objects still tunnel

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Tunneling (cont’d)

Fast-moving objects tunnel more easily

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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

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Tunneling (cont’d)

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

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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?

Movement Bounds

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Movement Bounds

Disc / Sphere

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Movement Bounds

Disc / Sphere

AABB (Axis-Aligned Bounding Box)

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Movement Bounds

Disc / Sphere

AABB (Axis-Aligned Bounding Box)

OBB (Oriented Bounding Box)

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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 don’t know for sure (false positives).

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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.

Swept Shapes

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Swept Shapes

Swept disc / sphere (n-sphere): capsule

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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

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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

Swept polytope: convex polytope

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Swept Shapes (cont’d)

Like movement bounds, only with a perfect fit!

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Swept Shapes (cont’d)

Like movement bounds, only with a perfect fit!

Still no false negatives (tunneling).

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Swept Shapes (cont’d)

Like movement bounds, only with a perfect fit!

Still no false negatives (tunneling).

Finally, no false positives, either!

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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.

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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.

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Einstein Says...

Coordinate systems are relative

Relative Coordinate Systems

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Relative Coordinate Systems

World coordinates

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Relative Coordinate Systems

World coordinates A’s local

coordinates

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Relative Coordinate Systems

World coordinates A’s local

coordinates B’s local

coordinates

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Relative Coordinate Systems

x2 + y2 = r2(x-h)2 + (y-k)2 = r2

Math is often nicer at the origin.

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Einstein Says...

Coordinate systems are relative Motion is relative

Relative Motion

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Relative Motion

"Frames of Reference"

World frame

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Relative Motion

"Frames of Reference"

World frame A's frame

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Relative Motion

"Frames of Reference"

World frame A's frame B's frame

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Relative Motion

"Frames of Reference"

World frame A's frame B's frame Inertial frame

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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)

Relative Collision Bodies

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Relative Collision Bodies

Collision check equivalencies (disc)

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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 body’s origin

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Relative Collision Bodies (cont’d)

Disc + disc

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Relative Collision Bodies (cont’d)

Disc + disc AABB + AABB

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Relative Collision Bodies (cont’d)

Disc + disc AABB + AABB Triangle + AABB

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Relative Collision Bodies (cont’d)

Disc + disc AABB + AABB Triangle + AABB AABB + triangle

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Relative Collision Bodies (cont’d)

Disc + disc AABB + AABB Triangle + AABB AABB + triangle Polytope +

polytope

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Relative Collision Bodies (cont’d)

Disc + disc AABB + AABB Triangle + AABB AABB + triangle Polytope +

polytope Polytope + disc

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Relative Collision Bodies (cont’d)

Things start to get messy when combining bodies explicitly / manually.

(Especially in 3d.) General solution?

Minkowski Arithmetic

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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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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

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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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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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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)

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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”

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Minkowski Differences (cont’d)

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

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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.

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Minkowski Differences (cont’d)

Aorigin vs. Borigin

-Borigin -Borigin

___ ___(A-B)origin vs. 0

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Minkowski Differences (cont’d)

In world space, A-B is “near” the origin

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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.

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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)

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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)

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Minkowski Differences (cont’d)

Translations in A or B simply translate the CSO

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Minkowski Differences (cont’d)

Rotations in A or B mutate the CSO

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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)!

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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)

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(Does Not Suck)

Relative Everything

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Relative Everything

Let’s combine: Relative Coordinate Systems Relative Motion Relative Collision Bodies

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Relative Everything (cont’d)

A vs. B in world frame

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Relative Everything (cont’d)

A vs. B in world frame

A vs. B, inertial frame

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Relative Everything (cont’d)

A vs. B in world frame

A vs. B, inertial frame

A is moving, B is still

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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

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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

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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

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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...

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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

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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”.

Determining Collision Time

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Determining Collision Time

Method #1: Frame Subdivision

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Subdividing Movement Frame

If a swept-shape (or movement bounds) test says “yes”:

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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.

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Subdividing Movement Frame (cont’d)

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

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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

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Determining Collision Time

Method #1: Frame Subdivision Method #2: 4D* Continuous Collision

Detection *(N+1 dimensions; 3D for 2D physics,

etc.)

Spacetime

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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)

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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.

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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)

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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!

?

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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

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Spacetime Diagrams (cont’d)

2D spacetime still image

1D animation

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Spacetime Diagrams (cont’d)

3D spacetime still image

2D animation

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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!)

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Spacetime Diagrams (cont’d)

Invented by Hermann Minkowski Also called “Minkowski Diagrams”

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(Rules)

Time-Swept Shapes

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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:

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Time-Swept Shapes (cont’d)

A vs. B in world frame

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Time-Swept Shapes (cont’d)

A vs. B in world frame

A is moving, B is still

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Time-Swept Shapes (cont’d)

A vs. B in world frame

A is moving, B is still

A is CSO, B is point

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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

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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?)

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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...

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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.

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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).

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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!

Quality vs. Quantity

or

“You Get What You Pay For”

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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?

Rotations(Suck)

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Rotations

Continuous rotational collision detection sucks

Rotational tunneling alone is problematic

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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

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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

Summary

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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.

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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.

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Questions?

Feel free to reach me by email at:

squirrel@eiserloh.net or

squirrel@ritual.com