animation and games development

242-515 AGD: 13. Collisions 1

• Objectiveo to discuss some of the basic issues related to collision

detection and collision response

Animation and Games

Development242-515, Semester 1, 2014-2015

13. Collision Detection

Overview9. Where do Objects

Touch?10. Collision Response

Algorithms11. Collision Changes12. Momentum and

Collisions13. Complex Collisions14. More Collision Issues15. Summary

1. Collision Detection Uses

2. Collision Algorithms

3. Spatial Partitioning4. Simplifying Shapes5. Terrain Collision6. Collision Time7. The Tunneling

Problem8. More Complex

Tunneling Solutions


• Determine if the player has a hit a wall or obstacle

• Determine if a missile has hit a target• Detect when things should change

o e.g. a hand touches a light switch• Clean up animation

o Make sure a character’s feet touch the floor• To help implementing realistic motion

o E.g. of cloth, water, fire

1. Collision Detection Uses


• Collision Detectiono Check for the intersection of two objects

o Calculate the trajectories of the objects, impact times, and impact points

• Collision Responseo provide a response that fits the game and (customized) laws of

physics

Collision Detection and Response


• Main character and static objects:o terrain, floor, walls, furniture, buildings, etc.

• Main character and dynamic objects:o enemies, bullets, arrows, particles, etc.

• Dynamic game objects (e.g. enemies) and static and other dynamic objects.

Types of Collisions

2. Collision Algorithms• Shapes can be very complex,

potentially requiring expensive testing:o points, line, convex/concave objects

• Shapes move in different ways:o Use different algorithms for fast

or slow moving objectso The algorithms will also depend

on how frequently an object must be updated

Convex Concave

An object is convex if for every pair of points

inside the object, the line joining them is also

inside the object


• Design principles for collision algorithms.

• Fast simple tests first to eliminate most potential collisionso e.g.: test bounding volumes before complex shapes

• Exploit locality to eliminate many potential collisionso e.g. use spatial partitioning to avoid testing two

objects that are far apart

Design Principles


• Use as much geometry information as possibleo e.g. spheres have special properties that make collision

testing faster

• Exploit coherence between successive testso coherence means that things (objects) tend to stay the

same over time


• Compare all game objects against all other objects, one pair at a time

• This approach is easy to understand and to implement.

• But it is slow -- O(n2) running time since every object can potentially collide with every other object.

Collisions in Pairs


• Include non-collidable objects in your game:o background: sky, clouds, grass, etco dead enemies (maybe)o unimportant game objects

• No need to include in collision testing, so speeds up the game.

Non-collidable Objects

Collision Challenges• Question:

o Do objects A and B intersect



• Challenge:o Can be in motion



• Challenge:o Can be in motion

• Bigger Challenge:o Need to know when

they collide.o Realistic physics needs

the exact point of collision

o Necessary for collision response


• Priori: check if something is going to collide before moving it, then make the necessary correction

• Posteri: check if something has collided and make the necessary corrections

Time-based Techniques


• Common types of spatial partitioning:o Octrees (called quadtrees in 2D)

o Uniform Grids

o Others types: portals (used in Quake), BSP trees (Binary Space Partitions), spatial hashing

3. Spatial Partitioning


An Octree

game regions

game regions

http://upload.wikimedia.org/wikipedia/commons/3/35/Octree2.png


• Data stored in each region:o coordinates defining the region

• though a smart implementation can greatly simplify this

o a list of objects inside a region


• Dynamic objects may move between regionso this requires the octree to be updated

• A static (or dynamic) object may be positioned so it spans several regionso collision detection will involve objects from multiple regions

Octree Drawbacks


• Steps:o divide the (2D or 3D) world into a grid of equal sized

cells

o associate each object with the cells it is located inside

o only objects sharing a cell are tested for collisions

Uniform Grids


• What cells does the object currently occupy?

• Possible calculations:o For min/max tile rows/columns:

• sprite.X/tileWidth• sprite.Y/tileHeight• (sprite.X+sprite.Width)/tileWidth• (sprite.Y+sprite.Height)/tileHeight

• For the 'man' object in cells (0,0), (0,1), (1,0), (1,1)o only test against:

• other objects in those cells• also, don’t let an object occupy “collidable cells” (0,2), (1,2), etc

Calculating a Grid Position

0 1 2 3

0

1

2


• Advantageso easy to implemento very fast: O(n)

• Disadvantageso constrains look of game to a grid-like world


• Typically divide an object into convex shapeso Mathematically the easiest

shapes to compute with• Assumed by most algorithms

• What do you do if the shape is not convex?o Break it up into several parts

4. Simplifying Shapes


• A simple geometry used in the collision tests instead of the shape’s real, complicated geometry.

• Properties of good BVs:o inexpensive collision testso tight fit to the original shapeo inexpensive to createo easy to rotate and transformo uses little memory

Bounding Volumes (BVs)


• AABBs: axis-aligned bounding boxes• OBBs: Oriented bounding boxes• DOPs: Discrete oriented polytopes

Common Bounding Volumes


In 3D

Ax is - Alig n ed Bo u n d in g Bo x O r ien ted Bo u n d in g Bo x

AABBs• Axis Aligned Bounding-

Boxes (AABBs)o Intersection is very cheap

• Project rectangle onto axeso Two intervals per box

• Check both intervals overlap

o Poor fit for most objects


• Common solution:o axis separation test

• More sophisticated solution:o Plane Sweep: an extension of axis separation which is more

efficient when dealing with many objects

Do AABB's Collide?


• How do we know if two AABBs are colliding?o their edges overlap on all axes

Axis Separation Test

A

B

C

D

NO COLLISION COLLISION


• Requires sorting of the objects along the axes• Running time: O(n)

C

B

R

A

x

y

A 0 A 1 R 0 B0 R 1 C 0 C 1B1

B0

B1A 1

A 0

R 1

R 0

C 1

C 0

Plane Sweep

OBBs• Oriented Bounding

Boxes (OBBs)o Tighter fit than AABBo Intersection more

expensive• Must check 4 edges

for overlap

DOPs

• Discrete Oriented Polytopes (DOPs)o Captures almost all

convex shapes

o Intersection is more expensive

• O(a2), a = number of edges

Capsule• Capsule

o An OBB with rounded ends

o Intersection cost is roughly the same

o Efficient for swept shapes

Circles/Disks

• Circles/Diskso Not cheaper than AABB

o Poor fit for anything other than a circle


• We can use two AABBs to represent the objecto Don’t test them against each other

How about a Complicated Object?


• Complex objects can have multiple bounding boxeso e.g. a Human object can have one big bounding box

covering the whole body; oro one bounding box per limb, head, etc

• Bounding boxes can be hierarchical:o Test the big 'human' box firsto If there is a possible collision, test the smaller boxes

inside the big one (e.g. the limb, head boxes)

Multiple Bounding Boxes


• Build a tree of boxes:o a parent box surrounds all the boxes of its children

• Intersection test against a bounding box tree by travelling down the tree to a leaf box.

• Generally used to test a moving object against lots of static obstacleso Put all the static obstacles into one large box tree

Bounding Box Hierarchies


• Utilizes the heightmap information, and the fact that the terrain is a mesh of triangles.

5. Terrain Collision


T o p - D o wn Vie w

P e r sp e c t iv e Vie w

T o p - D o wn Vie w ( h e igh t s a dde d)

P e r sp e c t iv e Vie w ( h e igh t s a dde d)

Terrain Mesh and Heights


Locate Triangle on Mesh

Q

R

Q

Q z > Q x

Q z < = Q x

z

x

R z > 1 - R x

R z < = 1 - R x

R

Simplifies to a 2D problem of points inside triangles.


• We can use Barycentric coordinates to test if a point is inside a triangle.

Barycentric Coordinates

P 1

P 2

P 0

Q

P o in t = w 0P 0 + w 1P 1 + w 2P 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


• Given a ball with center at position p0 = (x1, x2, x3), with initial velocity v = (v1, v2, v3), and a constant acceleration a = (a1, a2, a3).

• When will its center hit a plane defined by f(x, y, z) = 0 ?o p = p0 + vt + ½ at2

o Solve the equation f(p) = 0

6. Collision Time


• If the game is running at 30 fps, it only renders 1/30th of the object's stateso sometimes called the Flipbook syndrome

• Many things happen that we don’t see

Time Interval Problem

We don’t see the ball hit the ground


• An object moves through another object because the collision testing is not done at the 'right' time:

7. The Tunneling Problem

time: 5 6 7 8

Tunneling

Tunneling Example

Tunneling


• Things can fall through the floor

• Cars can pass through people or walls

• Players can enter rooms with no doors

• Players can travel over holes

• etc...

Tunneling is a Big Problem


• Small objects can tunnel more easily than large shapes:

Tunneling


• Simple solutionso Minimum size requirement?

• Inadequate; fast objects can still tunnel

Tunneling


• Simple solutionso Minimum size requirement?

• Inadequate; fast objects can still tunnel

o Maximum speed limit?• Inadequate; speed limit is a function of object size, so

small and fast objects (bullets) wouldn’t be allowed

o Smaller time step?• Helpful but inadequate; same problem as a speed

limit

Tunneling


• Movement Bounds• Swept Shapes• Minkowski Sums• Interval Halving

8. More Complex Tunneling Solutions

8.1. Movement Bounds• Disc / Sphere

Movement Bounds• Disc / Sphere• AABB

Movement Bounds• Disc / Sphere• AABB • OBB

Movement Bounds• Question:

o Could A and B have collided during the frame?

• It handles tunneling, but…

Movement Bounds• Question:

o Could A and B have collided during the frame?

• It handles tunneling, but…• … even if the answer is

“yes”, we still don’t know for sure


• Conclusionso Good: it prevent tunnelingo Good: cheap, effective, early rejection test

o Bad: it produces false "yes" answers for collision tests• called a "false positive"

Movement Bounds

8.2. Swept Shapes• Swept disc: capsule

Swept Shapes• Swept disc: capsule• Swept AABB:

polygon


polygon• Swept triangle:

polygon


polygon• Swept triangle:

polygon• Swept polytope:

polygon

Swept Shapes• Better fit than

movement bounds


movement bounds• No false negatives

(tunneling)



(tunneling)• No false positives

either!



(tunneling)• No false positives

either!

• No! False positives still exist


• By taking the Minkowski sum of two complex volumes and creating a new volume, the overlap can be found by testing if a point is within the new volume.

8.3. Minkowski Sums

Y}B and :{ XABAYX

X Y =YX X Y =


Using the Minkowski Sum

t0

t1

t0

t1

The sphere and box objects become a point and the Minkowski sum.

8.4. Interval Halvingt=0

t=1

t=0.5

t=1

t=0.5t=0.75

t=0.5t=0.625

t=0.5625

Sometimes called Bisect Testing


• You know when the objects were not touching(the last frame)

• You know when they are colliding(this frame)• So you have an upper and lower bound on the collision

time

• Bring these bounds closer together• Check for a collision at the midpoint of the current time

intervalo If there is a collision, the midpoint becomes the new upper

boundo If not, the midpoint is the new lower bound

• Keep going until the bounds are the same (or accurate enough)


• What are the contact points on object geometries?

• Most common approach is the Gilbert-Johnson-Keerthi (GJK) algorithm:o a complex algorithm that combines linear algebra,

determinants, and Minkowski sums

9. Where do Objects Touch?


• What happens after a collision?o It depends on the collision type: elastic and inelastic

• Inelastic collisions (no rebound)o No energy preserved in collisiono Easy to implement

• Elastic collisions (there is rebound/bouncing)o 100% energy preservedo More complicated to implement. Must introduce momentum

10. Collision Response Algorithms


• Two billiard balls strike each othero Calculate balls positions at the time of impacto Assign new velocities to the ballso Play a “clinking” sound

• A rocket hits a wallo The rocket disappearso Create an explosion effect, and play an explosion

soundo Change the wall texture to show damage

Collision Response Examples


• Response coding often has three parts:1. Prologue (before)2. Collision Changes3. Epilogue (after)

• Many visual effects can be done in the prologue or epilogue

Collision Response in Parts


• Collision known to have occurred

• Check if collision should be ignored

• Other events might be triggeredo Sound effectso Send collision notification messages

Prologue


• Place objects at point of impact

• Assign new velocitieso Using physics oro Using some other decision logic

Collision Changes


• Propagate post-collision effects

• Possible effectso Destroy one or both objectso Play sound effecto Inflict damage

Epilogue

11. Collision Changes

• What happens to the objects after colliding?

• Two basic approacheso Instantaneous change of objects velocities at

collision time

o Gradual change of their velocities and positions over time, following their collision


• Benefits:o The change can be calculated using fairly simple

equations

• Difficulties:o The precise detection of the time and location of the

collision can be expensive (a frame rate killer)o The coding to manage the necessary objects state is

complex

Instantaneous Change

Gradual Change• Benefits

o Does not require the precise detection of the time and location of the collision

o State management is easyo May look more realistic (e.g. the object meshes can be

deformed)

• Difficultieso Object interpenetration is likely, and hard to calculate (e,g,

may need finite difference methods)

Simple Case: Circles• Simple collisions

o Single point of contact

• Impact coordinateso Point of contact is origin


o Single point of contacto Rigid bodies

• Impact coordinateso Point of contact is origino Perpendicular componento Parallel component




• Change in motion happens along the perpendicular axis




• Exchange in energy happens along the perpendicular axis


• Momentum transfer – if two objects collide:o A perfectly elastic collision: no loss of energy

• Momentum is conserved

o An imperfect elastic collision: some energy is converted into heat, work, and deformation of objects

• Momentum is reduced after collision

12. Momentum and Collisions


• Note: o ignore rotation/angular velocitieso ignore centers of mass (treat as points)

• Two moving balls collide:o ball A: mA and vAi

o ball B: mB and vBi

• If they collide, what should their velocities be immediately after the collision?o vAf and vBf

Calculating New Velocities


• If momentum is conserved after the collision:PAi + PBi = PAf + PBf

(mA * vAi) + (mB * vBi) = (mA * vAf) + (mB * vBf)

• Problem has two unknowns (vaf and vbf)o we need a second equation (Kinetic energy equation)o ke = (m * v2)/2, where ke is never negative Joules (J)o Insert the ke equation into the momentum equation

vAf = ((2 * mB * vBi) + vAi * (mA – mB))/(mA + mB)

vBf = ((2 * mA * vAi) – vBi * (mA – mB))/(mA + mB)

Calculating vAf and vBf


• Ball A: o mass is 10, initial velocity is (4, 0)

• Ball B:o mass is 10, initial velocity is (-4, 1)

• Results:o BallA's final velocity is (-4, 1)o Ball B's final velocity is (4, 0)

Example

13. Complex Collisions• With complex shapes,

the point of contact is more complexo Could be a single pointo Could be an edge

• Solution:o Break object into several pointso Connect the points by

constraintso Apply forces to each of the

pointso Transfer forces to other points

while obeying constraints

• Requires a constraint solver

• With complex shapes, the point of contact is more complexo Could be a single pointo Could be an edge

• Solution:o Break object into several pointso Connect the points by

constraintso Apply forces to each of the

pointso Transfer forces to other points

while obeying constraints

• Requires a constraint solver

• Multiple simultaneous collision points• Articulated bodies, with joints• Friction• Breakable objects• Soft bodies• Smoke, clouds, gases• Water, oil, fluids

14. More Collision Issues


• Accurate collision detection and response is hard

• Tips:o Structure the game to make collision detection easy

• use circles, AABBs, grids, etc.

o Especially in the 3D case, consider using a physics engine (e.g. jBullet) rather than writing your own collision detection and response code

15. Summary

animation and games development

Documents

collision testing

dynamic objects

collision timethe

collision detectioncheck

collision algorithms3

collision algorithmsshapes

collision changes12

collision issues15