computer graphics & visualization collision detection – narrow phase

computer graphics & visualization

Simulation and Animation

Collision Detection – Narrow Phase


Simulation and Animation – SS 07Jens Krüger – Computer Graphics and Visualization Group

Collision DetectionBroad Phase

Narrow Phase

Detection



Contact • Where do bodies hit each other?• Bodies described by polyhedra• Rigid bodies

• 2 cases:– Vertex/Face

example: cube on table, 4 Vertex/Face contacts for 4 cube vertices

– Edge/Edge:example: cube protrudes over table, 2 Vertex/Face contacts and 2 Edge/Edge contacts with desk edge



Contact • Describing the contact:– Involved bodies– Contact point in world coordinates– Collision normal• Face normal in case of Vertex/Face contact • Cross product between edge vectors in case of Edge/Edge

contact



Contact • Discretization problem– Contact detection at simulation time step– Collision has to consider

time constraints 4D problem

t t + 1

Contact occurs, but might be missed due to fix time discretization of simulation



Contact • Basic algorithms for collision detection– Detecting collision occurence – Detecting point of contact– Modelling a response

• Most common in games:– Broad/Narrow phase algorithms

• Broad phase: reject objects that cannot collide– Bsp-Trees, Bounding volume hiearchies, separating axes

• Narrow phase: Apply accurate collision detection– Polyhedron/Polyhedron, closest features



Contact • Narrow phase collision detection– (Convex) polyhedron/polyhedron test

• Any vertex of either object is contained in the other one v: vertex, u: vertex on face, n: outward facing normal

• Penetration between edges and faces– If shortest distance between edge points changes sign– Compute intersection points between edges and infinite planes

– Sort intervals and test their points as above• Check for exactly aligned objects

iii nuv )(

)(

)()(

iji

ji

i

kkjjkkii

vvtvxdd

dt

nuvdnuvd



Narrow Phase between triangles

1

011

10

12

10

111

111

12

11

101

0:

VNd

VVVVN

dXN

VVVT

10V

11V

12V 1

2

Step 1: Compute plane equations




12

21

12

11

12

01

111

22

21

202

12

11

101

22

21

20

0:

dVNd

dVNd

dVNd

dXN

VVVT

VVVT

V

V

V

10V

11V

12V 1

2

Step 2: Compute vertex-to-plane distancesand look for sign change

also compute vice versa distances



Narrow Phase between triangles• Narrow phase collision detection between triangles– Test for overlapping intervals along plane-plane intersection

L L

L=O+tDD=N1×N2




t1 t2K0,1 K1,1

V1,1

V0,1 V2,1

π1

π2

L

pv,1,1=D (V∙ 1,1-O)

pv,0,1=D (V∙ 0,1-O)

d0,1

d1,1

)( 11

01

01

11

01

01

1

vv

vv

v

v

dd

pp

d

pt

11

01

01

01

11

01

1

vv

v

vvv dd

dpppt



Optimizations• Interval shift does not influence overlap

avoid computation of O and replace by

• Projection of L onto an arbitrary axis does not influence overlap

3,2,1,11 iVDp iVi

1iV

p

3,2,1,

,,max if

,,max if

,,max if

1

1

1

1

i

DDDDV

DDDDV

DDDDV

p

zyxziz

zyxyiy

zyxxix

Vi



Contact • Narrow phase collision detection between triangles– Based on axial projections– Triangles have normals N, M and edges Ei, Fj

– Compute plane equation p0,p1 of triangle 0,1• If both triangles are parallel - reject • If both triangles are coplanar – 2D (overlap/in-out)-test

– Compute distances of vertices of 1,0 to p0,p1• If all distances have same sign - reject

– General case: intersection is a line L (o+td, d=NxM)• Compute projections of vertices onto L: (d·(v-o))• Compute projections of vertices onto p0,p1• Compute intervals from similarity observation



Contact • Space-Time constraints– Consider time discretization, e.g. bounding spheres • Size of smallest sphere determined by time step of

simulation and maximal velocity• Spheres are taken from smaller object

– Collisions between dynamic and static objects• Consider velocity of moving objects• Predict first collision within given time interval• Only if collision occurs, tree model has to be processed

recursively



Contact • Dynamically moving sphere and plane

– Sphere of radius r and center C(t) = C(0) + t V– Plane equation: NX=d– Check for collision in time interval t [0,tmax]– Distance(center, plane): |NC(t)-d|– First time of contact: |NC(T)-d| = r

• T<0: sphere moves away• T>0: sphere will intersect at position C(T)

VN

rVNsignCNdT

)()0(



Contact • Dynamically moving box and plane– Project box half-sizes onto normal line of plane– Take projected value as radius of a sphere– Apply sphere-plane test from above– Finding intersection point is more complex• Depends on box orientation - face, line or vertex

computer graphics & visualization collision detection – narrow phase

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