Notes_10_04
1 of 17
Collision Detection for Polygonal Objects
given global location ri  and attitude i for all bodies at time t
given s i ' P for all vertices on all bodies (constants)
polygons must not be self-crossing
polygons must not have holes
ri P 4
ri 
P3
yi '
xi '
ri P1
y j'
x j'
r 
P2
ri P 2
j
r 
P1
j
r 
P3
j
Notes_10_04
2 of 17
Bounding Circle - quick check


before the simulation, find maximum radius  i  max norm s i ' P over all vertices for each
body (one constant scalar value per body that may be computed a priori)
at each time t, compute center distance c ij  norm ri   r j  between all pairs of bodies
if c ij   i   j then no collision can occur between bodies i and j
if c ij   i   j then bodies i and j are candidates for collision detection
does not require calculation of ri  for any vertices on any bodies
P
works best for centroidal origins and objects with low aspect ratios
i
yi '
xi '
c ij
y j'
j
x j'
Notes_10_04
3 of 17
Axis Aligned Bounding Box (AABB) - quick check
at each time t, calculate ri  for all vertices on all bodies
P
find AABB for each body
xi
MAX
xi
MIN
yi
MAX
yi
MIN
test between all pairs of bodies
x
MAX
i
 xj
MIN
 AND x
MAX
j
 xi
MIN
 AND y
MAX
i
 yj
MIN
 AND y
 yi
MAX
j
MIN

if true, then AABBs intersect and bodies i and j are candidates for collision detection
if false, then AABBs do not intersect and collision cannot occur between bodies i and j
works best for objects with low aspect ratios
yi
yi '
xi '
yj
MAX
 yi
MIN
y j'
x j'
MAX
 yj
MIN
Notes_10_04
4 of 17
Point in Polygon
must check if any vertices on body j are inside body i AND
are inside body j
check if any vertices on body i
computational complexity O( ni x nj ) for convex objects and O( 2 ni x nj ) for ray casting
point in polygon does not always work for thin bodies (may need edge intersection)
Notes_10_04
5 of 17
Point in Polygon – Convex Objects
vertices must be ordered CW around polygon
at each time t, calculate ri  for all points on all bodies that are candidates for collision
P
select one point r j  on body j and test if it is to the right or to the left of edge ri   ri  on
body i
P
Q
 
external normal n̂  unit R  ri   ri 

P

P

P
Q
P
P
 to the left of current edge

if rj   ri  n̂  0 , the point is to the left of the edge, is outside the polygon and the search
across edges may be terminated for that point
P T
if rj   ri 
 n̂  0 , the point is on the edge and may be inside the polygon
P T
if rj   ri 
 n̂  0 , the point is to the right of the edge and may be inside the polygon
P T

expand rj   ri 
P
 Rr 
P T
Q
i
 
 ri   y j  y i
P
P
P
x
Q
i
 
 xi  x j  xi
P
P
P
y
point r j  must be to the right of ALL edges to be inside body i
P
computational complexity O( ni x nj /2 ) because may terminate search early
does not work for bodies with concavities or holes
Q
i
 yi
P

Notes_10_04
6 of 17
Point in Polygon – Ray Casting
at each time t, calculate ri  for all points on all bodies that are candidates for collision
P
select one point r j  on body j and test if it is inside body i
P
cast an infinite ray in any direction from r j  and compute intersections with all edges on body i
P
an infinite ray to the right from r j  intersects edge ri   ri 
P
P
Q
0  d i  1 AND d j  0
y j  yi
P
yi  yi
P
P
di 
Q

 
d j  xi  xi di  x j  xi
Q
P
P
P
if

if there are an even number of intersections, r j  is not inside body i
P
if there are an odd number of intersections, r j  is inside body i
P
computational complexity O( ni x nj )
works for concavities, holes and self-crossing and is independent of CW versus CCW boundary
special case when r j  is on an edge or vertex of body i
P
algorithm used by MATLAB “inpolygon”
0
1
2
3
4
2
4
3
2
Notes_10_04
7 of 17
% t_inpolygon.m - test inpolygon
% HJSIII, 11.4.13
clear
% vertices for body i - bounded 0 to 1
ni = 5;
ri = rand(2,ni);
ri = [ ri ri(:,1) ]; % close the boundary
% points for j
nj = 500;
rj = rand(2,nj);
% which are inside?
%
points j
vertices i
in = inpolygon( rj(1,:),rj(2,:),
ri(1,:),ri(2,:) );
figure( 1 )
clf
plot( ri(1,:),ri(2,:),'b', rj(1,in),rj(2,in),'.r', rj(1,~in),rj(2,~in),'.g' )
% bottom of t_inpolygon
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Notes_10_04
8 of 17
Edge Intersection
must check every edge on body i for intersection with every edge on body j
ri P
r 
dj
Q
j
di
r 
P
ri Q
j
ri P  d i ri Q  ri P   rj P  d j rj Q  rj P 
intersect if 0  d i  1 and
r 

Q
i
di
dj
 ri 
x

x
x

x
P
Q
j
Q
j
Q
i
Q
j




Q
P d i 
P
P
 r j   r j     rj   ri 
d
 j
xj
P
xj
P
 xi
P
xj
P
y
y
P
j
Q
i
y
y
P
j
Q
i
 yi
P
 yi
P
 yi
P
 yi
P
  y
  x
  y
  x
Q
j
Q
i
Q
i
Q
i
 yj
P
 xi
P
 yi
P
 xi
P
x
y
x
y
P
 xi
P
Q
 yj
P
P
 xi
Q
 yj
j
j
j
j
P
P
0  dj 1




if denominator = 0, segments are parallel

expand y j  y i
P
P
x
Q
i
 xi
P
  x
r 
 ri 
 xi
y
P
must check if they are coincident
j
P
j
P
 Rr 
P T
Q
i
 yi
Q
i
P

 ri   0
P
 0
computational complexity O( ni x nj )
edge intersection does not always work if one body is completely inside the other (may need
point in polygon)
Notes_10_04
9 of 17
Separating Axis Theorem (SAT)
vertices must be ordered CW around polygon
before the simulation, select one edge s i 'Q s i ' P on body i and compute external normal
n̂  unit R  s i 'Q s i ' P
 

compute the projection p i for all points s i ' R onto the normal and store the minimum value
R
pi
MIN
(one scalar constant per edge that may be computed a priori)


 
p i  s i ' R s i ' P unit R  s i 'Q s i ' P
R
T

at each time t, calculate ri  for all points on all bodies that are candidates for collision
P
 
select edge ri   ri  on body i and compute external normal n̂  unit R  ri   ri 
called the separating axis
Q
P
Q
P

select point r j  on body j and compute its projection p j onto the separating axis
P

P
p j  rj   ri 
P
 n̂
P T
P
P
compute the projection p j onto the separating axis for all other points on body j and store the
maximum p j
MAX
and minimum p j
MIN
values
check for overlap of projections onto the separating axis
0  p  AND p
MIN
j
MAX
j
 pi
MIN

if false, the projections do not overlap, there can be no collision and the test may be terminated
if true, the projections overlap and must continue checking all other edges ri   ri 
Q
P
does not work for bodies with concavities or holes
very similar to convex body point in polygon however do not need to check (i in j) and (j in i)
computational complexity O( ni x nj / 2) because may terminate search early AND do not need to
check body i against body j
Notes_10_04
n̂
10 of 17
ri Q
separating
axis
ri R
ri P
piMIN
pjMAX
pjMIN
r 
P
j
Notes_10_04
11 of 17
Classification of Contacts
i
i
j
edge-edge
j
vertex-edge
i
i
j
j
multiple vertex-edge
vertex-edge
edge-edge
multiple vertex-edge
multiple edge-edge
Body i
vertices
in j
0
1
0
1
multiple edge-edge
Body i
edge
intersections
1
2
1
2
Body j
vertices
in i
1
1
≥2
≥2
Body j
edge
intersections
2
2
2
2
Contact
points
3
4
5
5
Notes_10_04
12 of 17
Interpolating Time of Collision (vertex-edge)
r 
P
j
ri Q
ri P
given candidate vertex r j t on body j that collides with body i at time t
P
given candidate edge ri t  ri t on body i that has two edge intersections with body j at time t
Q
P
must know position and velocity for all three points at time t-h (time step h)
Constant velocity predictor
predict position of all three points at local time 0  t  h using vales from t-h
ri P  ri Pt h  ri Pt h t 
ri Q  ri Qt h  ri Qt h t 
r   r 
P
P
j t h
j
P
 r j t  h t 
when vertex r j  intersects edge ri   ri 
P
Q
P
r 
P
j
 ri 
 Rr 
P T
Q
i
substituting position equations and collecting similar terms
a t   bt   c  0
2

P
P
a  rj t h  ri t h

P

P
b  rj t h  ri t h
P
c  rj t h  ri t h
P
 R r 
 ri t h
 R r 
P
P
P
 ri t h  rj t h  ri t h
 R r 
 ri t h
T
T
T
Q
i t h
Q
i t h
Q
i t h
P

 
P

 R r 
T
Q
i t h
 ri t h
P


 ri   0
P
Notes_10_04
t 
13 of 17
 b  b 2  4ac
2a
must check 0  t  h and it must not be imaginary
if not true, then that vertex did not actually collide with that edge
r 
P
j
ri P
ri Q
relative motion of body j with
respect to body i
Constant acceleration predictor
predict position of all three points at local time 0  t  h measured from t-h
ri P  ri Pt h  ri Pt h t   12 ri Pt h t 2
ri Q  ri Qt h  ri Qt h t   12 ri Qt h t 2
r   r 
P
P
j t h
j
 rj t  h t   12 rj t  h t 
P
P
2
substituting position equations and collecting similar terms
c 4 t   c 3 t   c 2 t   c1 t   c 0  0
4
3
2
c4 
1
4
r 
 ri t h
 R r 
 ri t h
c3 
1
2
r 
P
 ri t h
 R r 
P
P
P
 ri t h  12 rj t h  ri t h
c2 
P
j t h
P
j t h
P
T
T
P
i t h
Q
i t h
P

 
r   r   R r   r    r 
 r   r   R r   r  
P
j t h
1
2
c1  r j t h  ri t h
P
Q
i t h
T
P
i t h
P
j t h
1
2

T
P
i t h
P
Q
i t h
 R r 
T
Q
i t h
P
i t h
P
j t h
 ri t h
P
 R r 
T
Q
i t h
 R r 
T
Q
i t h
P
 ri t h
 ri t h
P


P
i t h
 
P
P
P
 ri t h  rj t h  ri t h
 R r 
T
Q
i t h
 ri t h
P

(same as b above)
Notes_10_04

c 0  r j t h  ri t h
P
P
 R r 
T
Q
i t h
 ri t h
P

(same as c above)
use root finding algorithm for t (e.g. MATLAB “roots”)
must check 0  t  h and it must not be imaginary
14 of 17
Notes_10_04
15 of 17
Three-dimensional Collision Detection
Quick checks
Bounding spheres and AABB both work well
Point in polyhedron – convex bodies
at each time t, calculate ri  for all points on all bodies that are candidates for collision
P
select one point r j  on body j and test if it is outside or inside a facet on body i defined by
P
external normal n̂ and any vertex ri  on the facet
P

P

P

P

if rj   ri  n̂  0 , the point is outside the polyhedron and the search across facets may be
terminated for that point
P T
if rj   ri 
 n̂  0 , the point is on the facet and may be inside the polyhedron
P T
if rj   ri 
 n̂  0 , the point is inside the facet and may be inside the polyhedron
P T
point r j  must be inside of ALL facets to be inside body i
P
point in polygon does not always work for thin bodies (may need edge intersection)
Point in polyhedron – ray casting
at each time t, calculate ri  for all points on all bodies that are candidates for collision
P
select one point r j  on body j and test if it is inside body i
P
cast an infinite ray in any direction from r j  and compute intersections with all facets on body i
P
if there are an even number of intersections, r j  is not inside body i
P
if there are an odd number of intersections, r j  is inside body i
P
works for concavities, holes and self-crossing and is independent of CW versus CCW boundary
Notes_10_04
16 of 17
ray-facet intersection is similar to edge-facet intersection described below
point in polygon does not always work for thin bodies (may need edge intersection)
Facet intersection
check if edges for a given facet on body j intersect facet on body i
establish local coordinate frame with origin at the first vertex for facet on i, local x axis along
first edge and local y defined by second edge (CCW vertex sequence around facet i)
local z axis will be parallel to external normal
transform vertices for facet i and facet j into local coordinates and perform all following
computations in local coordinates
test where edge r j   r j  intersects facet i
Q

use d j  z j / z j  z j
P
P
Q
P
P
 and check 0  d
P
j
1
if false, that edge does not intersect the facet and proceed to check next edge on j for same facet i
if true that edge intersects the plane of the facet at r  r j   d j
P
P
r   r  
Q
j
P
j
use planar point in polygon test if x,y components of r are inside x,y components of facet i
if false, the edge intersects the plane of the facet outside its boundary
if true, the edge intersects the facet inside its boundary
must also check if edges for same facet on body i intersect same facet on body j
facet intersection does not always work if one body is completely inside the other (may need
point in polygon)
Notes_10_04
17 of 17
Handling Collision
Physical simulators differ in the way they react to a collision. Some use the softness of the
material to calculate a force, which will resolve the collision in the following time steps like it is
in reality. Due to the low softness of some materials this is very CPU intensive.
Other simulators estimate the time of collision by interpolation, roll back the simulation, and
calculate the exact time of collision. This may require several iterations.
After an inelastic collision, special states of sliding are often imposed. For example, an open
pin-in-slot constraint is added for a vertex-edge intersection to force that vertex to slide on the
edge rather than penetrate it. The Open Dynamics Engine uses this approach.