View
224
Download
1
Category
Preview:
Citation preview
19 September 2001 Sunghwan Min
The Type of Intersection
Three categories Linear component versus object (picking)
Chapter 5 already described
Object versus plane (culling) Chapter 4 described
Object versus object (general collision)
19 September 2001 Sunghwan Min
Design Issues
Hierarchical representation of objects for
collision purposes
Should the hierarchy be built top-down or
bottom-up? Top-down : decomposition of complex object
Bottom-up : construction the world from small models
Should the bounding volumes be built manually
or automatically? Automatic : not always generate a good set of volumes
Manual : can be time consuming
The best approach : mixture of the two
19 September 2001 Sunghwan Min
Design Issues (cont.)
How should the intersection information be
reported? To use callbacks
How should the propagation of the test collision
calls be controlled?
How much information should be retained about
the current collision state to support future test
collision calls?
19 September 2001 Sunghwan Min
Dynamic Objects and Lines
The line
The Object constant linear velocity
time interval
If Moving parallel to the line (Static test)
Presented here Determine only if the line and object will intersect on the time interval
DsP
W
],0[ maxtt0
WD
19 September 2001 Sunghwan Min
Spheres
The moving sphere has center
The distance C to the line is , where
If , then the line intersects the sphere
WtC
2
)()()(
DDD
PCDPCtD
DD
WDWtQ
2)( rtQ
cbtat 2: 2
|)(| 0DtPC
DD
PCDt
)(
0
19 September 2001 Sunghwan Min
Spheres
The problem is now one of determining the minimum of
Q on the interval Solve T=-b/a
If : the minimum is Q(T)
If T<0 : the minimum is Q(0)
If T>tmax: the minimum is Q(tmax)
Then compared to
0)( TQ],0[ maxtT
2r
19 September 2001 Sunghwan Min
Oriented Boxes
Static oriented box Rd>Rb+Rs : non intersection
Rb
Rs
LR
19 September 2001 Sunghwan Min
Oriented Boxes (cont.)
line the only potential separating axes for i=
0,1,2
For the motion case , is replaced by
If any of these tests are true do not intersect
12210 )( UDeUDePCDU
02201 )( UDeUDePCDU
01102 )( UDeUDePCDU
WtC
C
iUD
19 September 2001 Sunghwan Min
Capsules
The moving capsule is WtEuC
2
),( tutuQ
DDD
WDW
DDD
EDE
DDD
PCDPC
)(
)(
E
19 September 2001 Sunghwan Min
Lozenges
The moving lozenges is WtEuEuC
10
2),,( tvutvuQ
DDD
WDW
DDD
EDE
00
DDD
PCDPC
)(
)(DDD
EDE
11
E0
E1
19 September 2001 Sunghwan Min
Cylinders
An extremely complicated and some what expensive Not recommended for use as bounding volumes
19 September 2001 Sunghwan Min
Ellipsoids
Static ellipsoid
The line
The quadratic equation has a real
-valued root
1)()( CXMCX T
DsP
022 cbsas042 acb
1,,),(
McPCMDbDMDa TTT
19 September 2001 Sunghwan Min
Ellipsoids (cont.)
For a moving ellipsoid
The center is
01 btbb
0)2()()( 012
22
01 ctctcabtbtQ
WMWcWMcMc
WMDbMDbTTT
TT
210
10
,,1
,
WtC
012
2 2 ctctcc
19 September 2001 Sunghwan Min
Triangles
The plane of the triangles at time t
The line be
100 ,)( EEwhereNWNtVXN
DsP
19 September 2001 Sunghwan Min
Dynamic Objects and Planes
The Plane
The Object constant linear velocity
time interval
Presented here Determine only report an intersection time of t=0 when the object and plane
are initially intersecting
W
],0[ maxtt
dXN
19 September 2001 Sunghwan Min
Spheres
The moving center
The distance between center and plane
If initially intersecting
Else
The first time of contact T of the sphere
WtCtC
)(dtCN )(
rdCN 0
rdTCN )(
19 September 2001 Sunghwan Min
Oriented Boxes
The radius of the interval of the projected
box
Computation of the first time of contact T
is identical to that of a sphere versus a
plane
If initially intersecting
Else The first time of contact T of the sphere
2
0iii ANar
rdCN 0
rdTCN )(
19 September 2001 Sunghwan Min
Capsules
Line segment
And where
Define the signed distances
If initially intersecting
DstP
)(
DPP
01
dPN 00
dPN 11
010
P0 P1
D
19 September 2001 Sunghwan Min
Capsules (cont.)
The sign of decide which of
and is closer
Apply the intersection testing algorithm
between a sphere and a plane
DN
0P
1P
19 September 2001 Sunghwan Min
Lozenges
Lozenges is
The four corners
The signed distance
Not all positive or not all negative initially
intersecting
Applied to the sphere corresponding to
that corner
10 EvEuP
1011101
01000
,
,,
EEPPEPP
EPPPP
dPN ijij
P00 P10 E0
E1
P01 P11
19 September 2001 Sunghwan Min
Triangles
Let the three vertices be
Three signed distances for 0<= i <=2
Initially intersecting Not all positive or not all negative
The closest vertex use signed distance
iV
dVN ii
19 September 2001 Sunghwan Min
Static Object-Object
In this section determine if two of the same type or stati
onary objects
19 September 2001 Sunghwan Min
Spheres, Capsules, And Lozenges
Intersection dist < rsum*rsum
Sphere Capsule Lozenge
Sphere Dist(pnt,pnt) Dist(pnt,seg) Dist(pnt,rct)
Capsule Dist(seg,pnt) Dist(seg,seg) Dist(seg,rct)
Lozenge Dist(rct,pnt) Dist(rct,seg) Dist(rct,rct)
19 September 2001 Sunghwan Min
Oriented Boxes
Let first box have axes and extents
second box have axes and extents
The potential separating axe
2,1,0 AAA
2,1,0 aaa
2,1,0 BBB
2,1,0 bbb
L
19 September 2001 Sunghwan Min
Oriented Boxes (cont.)
15 separating axis
D
R0
R1
LR
a1A1a2A2
b1B1
b2B2
2
0
2
010
iii
iii BLbALaRRRDL
19 September 2001 Sunghwan Min
Oriented Boxes and Triangles
Let Box have axes and extents
Triangles have vertices and the edges of the triangles
are
2,1,0 AAA
2,1,0 aaa
2,1,0 UUU
012,021,100 EEEUUEUUE
19 September 2001 Sunghwan Min
Oriented Boxes and Triangles (cont.)
Non-intersection test
Min(p0,p1,p2) > R , max(p0,p1,p2)<-R
D
L
-R
a1A1a2A2
R
E1
E0p0
p1
p2
Min(ui) Max(ui)
Recommended