Collision Detectionn

8/10/2019 Collision Detectionn

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Collision Detection



Essential Math for G

Collisions

• Up to this point, objects just pass

through each other

• Two parts to handling collisions

Collision detection – uses computational

geometry techniques (useful in other ways,

too Collision response – modifying physical

simulation




Computational !eometry

• "lgorithms for sol#ing geometric

problems

• $bject intersections

• $bject pro%imity

• &ath planning




Distance Testing

• Useful for computing intersection

between simple objects

• 'g sphere intersection boils down to

point)point distance test

• *ust co#er a few e%amples




&oint)&oint Distance

• Compute length of #ector between two

points P 0 and P 1, or

2

10

2

10

2

1010 )()()(),dist( z z y y x x P P −+−+−=




+ine)&oint Distance

• +ine defined by point P and #ector v

• rea- #ector w = Q – P into w⊥ and w||

• w|| = (w • v) v

• ||w⊥||2 = ||w||2 – ||w||||2

^

^ ^

v

Q

P

w

w||

w⊥

^




+ine)&oint Distance

• .inal formula/

• 0f v isn1t normali2ed/

)ˆ(),ˆ,dist( vwwv

−•=Q P

•−•=vv

v

wwv ),,dist( Q P




+ine)+ine Distance

• .rom http://www.geometryalgorithms.com

• 3ector wc perpendicular to u and v or

• Two equations

• Two un-nowns

P 0

u

Q0

v

P ( sc)

Q(t c)

wc0

0

)()(

=•

=•

−=

c

c

ccc t Q s P

wv

wu

w




+ine)+ine Distance

.inal equations/

)(

)(

)()()(

)()()(

00

00

20

2

0

QPv

QPu

vv

vu

uu

vQ

uP

−•=

−•=

•=

•=

•=

⋅−−+=

⋅−−+=

e

d c

b

a

bacbd aet Q

baccd be s P

c

c

P0

u

Q0

v

P(sc)

Q(tc)




4egment)4egment Distance

• Determine closest point between lines

• 0f lies on both segments, done

• $therwise clamp against nearest

endpoint and recompute

• 4ee references for details




ounding $bjects

• Detecting intersections with comple%

objects e%pensi#e

• &ro#ide simple object that surrounds

them to cheaply cull out ob#ious cases

• Use for collision, rendering, pic-ing

• Co#er in increasing order of comple%ity




ounding 4phere

• Tightest sphere that surrounds model

• .or each point, compute distance from

center, sa#e ma% for radius




ounding 4phere (Cont5d

• 6hat to use for center7

+ocal origin of model

Centroid (a#erage of all points

Center of bounding bo%

• 6ant a good fit to cull as much as

possible

• +inear programming gi#es smallest fit




4phere)4phere Collision

• Compute distance d between centers

• 0f d < r 1 + r 2, colliding

• 8ote/ d 2 is not necessarily < r 12 + r 22

want d 2 < (r 1 + r 2)2

d r 1

r 2




ounding o%

• Tightest bo% that surrounds model

• Compare points to min9ma% #ertices

• 0f element less9greater, set element in

min9ma%

(min %, min y

(ma% %, ma% y




"%is)"ligned ounding o%

• o% edges aligned to world a%es

• :ecalc when object changes orientation

• Collision chec-s are cheaper though




"%is)"ligned o%)o%

Collision• Compare % #alues in min,ma% #ertices

• 0f min; < ma%= or min= < ma%;, no

collision (separating plane

• $therwise chec- y and 2 directions

min= ma%= min; ma%;




$bject)$riented ounding o%

• o% edges aligned with local object

coordinate system

• >uch tighter, but collision calcs costly




$ Collision

• 0dea/ determine if separating plane

between bo%es e%ists

• &roject bo% e%tent onto plane #ector,test against projection btwn centers

•!

"•!a•!

a "




$ Collision

• To ensure ma%imum e%tents, ta-e dot

product using only absolute #alues

• Chec- against a%es for both bo%es, plus

cross products of all a%es

• 4ee !ottschal- for more details

|||||| z z y y x x vavava ++




Capsule

• Cylinder with hemispheres on ends

• $ne way to compute

Calc bounding bo%

Use long a%is for length

8e%t largest width for radius

r r




Capsule

• Compact

$nly store radius, endpoints of line

segment

• $riented shape w9faster test than $

• Test path collision




Capsule)Capsule Collision

• ?ey/ swept sphere a%is is line segment

with surrounding radius

• Compute distance between linesegments

• 0f less than r 1 + r 2, collide




Ca#eat

• >ath assumes infinite precision

• .loating point is not to be trusted

• &recision worse farther from @

• Use epsilons

• Careful of operation order

• :e)use computed results

• >ore on floating point on website




6hich To Use7

• "s many as necessary

• 4tart with cheap tests, mo#e up the list

4phere

4wept 4phere

o%

• >ay not need them all




:ecap

• 4phere )) cheap, not a good fit

• "" )) still cheap, but must recalc and

not a tight fit

• 4wept 4phere )) oriented, cheaper than

$ but generally not as good a fit

• $ )) somewhat costly, but a better fit




Collision Detection

• 8aA#e/ n2 chec-sB

• Two part process

road phase

• Cull out non)colliding pairs

8arrow phase

• Determine penetration and contact pointsbetween pairs




road &hase

• $b#ious steps

$nly chec- each pair once

• .lag object if collisions already chec-ed

$nly chec- mo#ing objects

• Chec- against other mo#ing and static

Chec- rough bounding object first• "" or sphere




ierarchical 4ystems

• Can brea- model into hierarchy and

build bounds for each le#el of hierarchy

• .iner le#el of detection

• Test top le#el, cull out lots of lower

le#els




ierarchical 4ystems

• Can use scene graph to maintain

bounding information

• &ropagate transforms down to children

• &ropagate bound changes up to root




4patial 4ubdi#ision

• rea- world into separate areas

• $nly chec- your area and neighbors

• 4implest/ uniform

4labs

!rid

3o%els




4weep and &rune

• 4tore sorted x e%tents of objects

• 4weep from min x to ma% x

• "s object min #alue comes up, ma-e

acti#e, test against acti#e objects

• Can e%tend to more dimensions




4patial 4ubdi#ision

• $ther methods/

uadtrees, octrees

4& trees, k d)trees

:oom)portal

• Choice depends on your game type,

rendering engine, memory a#ailable,etc




Temporal Coherence

• $bjects nearby generally stay nearby

• Chec- those first

• Can ta-e memory to store information




8arrow &hase

• a#e culled object pairs

• 8eed to find

Contact point

8ormal

&enetration (if any




Contact :egion

• Two objects interpenetrate, ha#e one

(or more regions

• " bit messy to deal with

• >any try to a#oid interpenetration




Contact .eatures

• .aceted objects collide at pair of

contact features

• $nly consider ')' and .)3 pairs

• 0nfinite possibilities for normals for

others

• Can generally con#ert to ')' and .)3

• '%/ 3)3, pic- neighboring face for one




Contact .eatures

• .or ')'/

&oint is intersection of edges

8ormal is cross product of edge #ectors

• .or .)3/

&oint is #erte% location

8ormal is face normal




Contact &oints

• Can ha#e multiple contact points

'%/ two conca#e objects

• 4tore as part of collision detection

• Collate as part of collision resolution




'%ample/ 4pheres

• Difference between centers gi#es

normal n (after you normali2e

• &enetration distance p is

p = (r 1+r 2) # ||c2#c1||

1 2




'%ample/ 4pheres

• Collision point/ a#erage of penetration

distance along e%tended normal

• 0f touching, where normal crosses sphere

v = $(c1 + r 1n + c2 # r 2n)^ ^

1

2




+in)Canny

• .or con#e% objects

• 'asy to understand, hard to implement

• Closest features generally same from

frame to frame

• Trac- between frames

• >odify by wal-ing along object




+in)Canny

• .rame @

• .rame =




!*?

• .or Con#e% $bjects

• ard to understand, easy to implement

• .inds point in Configuration 4pace

$bstacle closest to origin Corresponds

to contact point

• 0terati#ely finds points by successi#e

refinement of simplices




!*?

• C4$

• 4imple% :efinement

A

B

A-B




>issing Collision

• 0f time step is too large for object speed,

two objects may pass right through

each other without being detected(tunneling




>issing Collision

• $ne solution/ slice time inter#al

• 4imulate between slices

• 4ame problem, just reduced frequency




>issing Collision

• "nother solution/ use swept #olumes

• 0f #olumes collide, may collide in frame• 6ith more wor- can determine time)of)impact

(T$0, if any




:ecap

• Collision detection comple%

• Combo of math and computing

• rea- into two phases/ broad and

narrow

• e careful of tunneling




:eferences

• &reparata, .ranco & and >ichael 0an 4hamos,

Computational Geometry: An Introduction, 4pringer)

3erlag, 8ew Eor-, =FGH

• $5:our-e, *oseph, Computational Geometry in C ,Cambridge Uni#ersity &ress, 8ew Eor-, =FFI

• 'berly, Da#id , 3D Game Engine Design, >organ

?aufmann, 4an .rancisco, ;@@=

• !ottschal-, 4tephan, >ing +in and Dinesh >anocha,

J$)Tree/ " ierarchical 4tructure for :apid

0nterference Detection,K SIGGRAPH !"#



:eferences

• 3an den ergen, !ino, Collision Detection in

Interacti$e 3D En$ironments, >organ

?aufmann, 4an .rancisco, ;@@L• 'berly, Da#id , Game P%ysics, >organ

?aufmann, 4an .rancisco, ;@@L

• 'ricson, Christer, Real&'ime Collision

Detection, >organ ?aufmann, 4an .rancisco,;@@I

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