Chapter 4.2Collision Detection and Resolution

2

Collision Detection

Complicated for two reasons1. Geometry is typically very complex,

potentially requiring expensive testing2. Naïve solution is O(n2) time complexity,

since every object can potentially collide with every other object

Collision Detection

Two basic techniques1. Overlap testing

Detects whether a collision has already occurred

2. Intersection testing Predicts whether a collision will occur in the

future

4

Overlap Testing Facts

Most common technique used in games Exhibits more error than intersection

testing Concept

For every simulation step, test every pair of objects to see if they overlap

Easy for simple volumes like spheres, harder for polygonal models

5

Overlap Testing:Useful Results Useful results of detected collision

Time collision took place Collision normal vector

6

Overlap Testing:Collision Time Collision time calculated by moving object

back in time until right before collision Bisection is an effective technique

B B

t 1

t 0 . 3 7 5

t 0 . 2 5

B

t 0

I t e r a t i o n 1F o r w a r d 1 / 2

I t e r a t i o n 2B a c k w a r d 1 / 4

I t e r a t i o n 3F o r w a r d 1 / 8

I t e r a t i o n 4F o r w a r d 1 / 1 6

I t e r a t i o n 5B a c k w a r d 1 / 3 2

I n i t i a l O v e r l a pT e s t

t 0 . 5t 0 . 4 3 7 5 t 0 . 4 0 6 2 5

BB B

A

A

A

AA A

7

Overlap Testing:Limitations Fails with objects that move too fast

Unlikely to catch time slice during overlap Possible solutions

Design constraint on speed of objects Reduce simulation step size

t 0t - 1 t 1 t 2

b u l l e t

w i n d o w

8

Intersection Testing Predict future collisions When predicted:

Move simulation to time of collision Resolve collision Simulate remaining time step

9

Intersection Testing:Swept Geometry Extrude geometry in direction of movement Swept sphere turns into a “capsule” shape

t 0

t 1

10

Intersection Testing:Sphere-Sphere Collision

Q 1

Q 2

P 1

P 2

P

Q

t = 0

t = 0

t = 1

t = 1

t

( ) ( ) ( )( ),

2

2222

B

rrΑBt

QP +−−⋅−⋅−=

BAΒΑ( ) ( ).QQPPB

QPA

1212

11

−−−=−=

11

Intersection Testing:Sphere-Sphere Collision Smallest distance ever separating two

spheres:

If there is a collision

( )2

222

BAd

BA ⋅−=

( ) 22QP rrd +>

12

Intersection Testing:Limitations Issue with networked games

Future predictions rely on exact state of world at present time

Due to packet latency, current state not always coherent

Assumes constant velocity and zero acceleration over simulation step Has implications for physics model and choice of

integrator

13

Dealing with Complexity

Two issues1. Complex geometry must be simplified2. Reduce number of object pair tests

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Dealing with Complexity:Simplified Geometry Approximate complex objects with

simpler geometry, like this ellipsoid

15

Dealing with Complexity:Minkowski Sum By taking the Minkowski Sum of two

complex volumes and creating a new volume, overlap can be found by testing if a single point is within the new volume

16

Dealing with Complexity:Minkowski Sum

Y}B and :{ ∈∈+=⊕ XABAYX

X ⊕ Y⊕ =YX X ⊕ Y =

17

Dealing with Complexity:Minkowski Sum

t 0

t 1

t 0

t 1

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Dealing with Complexity:Bounding Volumes Bounding volume is a simple geometric

shape Completely encapsulates object If no collision with bounding volume, no

more testing is required Common bounding volumes

Sphere Box

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Dealing with Complexity:Box Bounding Volumes

A x i s - A l i g n e d B o u n d i n g B o x O r i e n t e d B o u n d i n g B o x

20

Dealing with Complexity:Achieving O(n) Time Complexity

One solution is to partition space

21

Dealing with Complexity:Achieving O(n) Time Complexity

Another solution is the plane sweep algorithm

C

B

R

A

x

y

A 0 A 1 R 0 B 0 R 1 C 0 C 1B 1

B 0

B 1A 1

A 0

R 1

R 0

C 1

C 0

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Terrain Collision Detection:Height Field Landscape

T o p - D o w n V i e w

P e r s p e c t i v e V i e w

T o p - D o w n V i e w ( h e i g h t s a d d e d )

P e r s p e c t i v e V i e w ( h e i g h t s a d d e d )

23

Terrain Collision Detection:Locate Triangle on Height Field

Q

R

Q

Q z > Q x

Q z < = Q x

z

x

R z > 1 - R x

R z < = 1 - R x

R

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Terrain Collision Detection:Locate Point on Triangle Plane equation: A, B, C are the x, y, z components of

the plane’s normal vector Where

with one of the triangles vertices being

Giving:

0=+++ DCzByAx

0PN ⋅−=D

0P

( ) ( ) ( ) ( ) 00 =⋅−+++ PNNNN zyx zyx

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Terrain Collision Detection:Locate Point on Triangle The normal can be constructed by taking the

cross product of two sides:

Solve for y and insert the x and z components of Q, giving the final equation for point within triangle:

( ) ( )0201 PPPPN −×−=

( )y

zzxxy N

PNQNQNQ 0⋅+−−

=

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Terrain Collision Detection:Locate Point on Triangle Triangulated Irregular Networks (TINs)

Non-uniform polygonal mesh Barycentric Coordinates

P 1

P 2

P 0

Q

P o i n t = w 0 P 0 + w 1 P 1 + w 2 P 2

Q = ( 0 ) P 0 + ( 0 . 5 ) P 1 + ( 0 . 5 ) P 2

R = ( 0 . 3 3 ) P 0 + ( 0 . 3 3 ) P 1 + ( 0 . 3 3 ) P 2R

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Terrain Collision Detection:Locate Point on Triangle Calculate barycentric coordinates for point

Q in a triangle’s plane

If any of the weights (w0, w1, w2) are negative, then the point Q does not lie in the triangle

( )

⋅⋅

⋅−

⋅−⋅−

=

2

1

2121

212

22

212

22

12

1 1

VS

VS

VV

VV

VV V

V

VVw

w

022

011

0

PPV

PPV

PQS

−=−=

−=

210 1 www −−=

28

Collision Resolution:Examples Two billiard balls strike

Calculate ball positions at time of impact Impart new velocities on balls Play “clinking” sound effect

Rocket slams into wall Rocket disappears Explosion spawned and explosion sound effect Wall charred and area damage inflicted on nearby

characters Character walks through wall

Magical sound effect triggered No trajectories or velocities affected

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Collision Resolution:Parts Resolution has three parts

1. Prologue2. Collision3. Epilogue

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Collision Resolution:Prologue Collision known to have occurred Check if collision should be ignored Other events might be triggered

Sound effects Send collision notification messages

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Collision Resolution:Collision Place objects at point of impact Assign new velocities

Using physics or Using some other decision logic

32

Collision Resolution:Epilogue Propagate post-collision effects Possible effects

Destroy one or both objects Play sound effect Inflict damage

Many effects can be done either in the prologue or epilogue

33

Collision Resolution:Resolving Overlap Testing

1. Extract collision normal2. Extract penetration depth3. Move the two objects apart4. Compute new velocities

34

Collision Resolution:Extract Collision Normal Find position of objects before impact Use two closest points to construct the

collision normal vector

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Collision Resolution:Extract Collision Normal Sphere collision normal vector

Difference between centers at point of collision

t 0

t 0

t 0 . 2 5

t 0 . 5

t 0 . 2 5

t 0 . 5

t 0 . 7 5

t 0 . 7 5

t 1

t 1

C o l l i s i o n

N o r m a l

36

Collision Resolution:Resolving Intersection Testing Simpler than resolving overlap testing

No need to find penetration depth or move objects apart

Simply1. Extract collision normal2. Compute new velocities