Collision handling:
detection and response
Collision handling overview
Detection
Discrete collision detection
Convex polygon intersection test
General polygon intersection test
Continuous collision detection
Vertex‐triangle test
Edge‐edge test
R
Response
Kinematic response
Penalty method
Discrete Collision Handlingg
• Run
Run the simulation from t to t+1, without the simulation from t to t+1 without
considering collision
• Check if any intersection happens at t+1
Check if any intersection happens at t+1
• If so, remove it
Geometric Modeling Primitives
g
• Point
• Line segment
• Polygon
Terminology – convex and concave polygons
l
Edge Test Decides convex/concave
Always on one side
Always on one side
Not always on one side
Not always on one side
Works same for convex/concave polyhedra (always on on side of the polygonal face?)
(always on on side of the polygonal face?)
An important property of Convex polygons!
l
!
Initial Pair
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
No more improvement, done!
You can always find the closest vertex pair by gradually moving one vertex at each time!
An important property of Convex polygons!
l
!
• The computational cost depends on the initial pair
– Worst case O(N)
– Best case O(1)!
B t
O(1)!
• In animation, since we can keep the pair over time and the object movement within a time step is
and the object movement within a time step is small, the initial pair is often good!
– O(1)
• Problem: not everything is convex
– Solution: decompose them into convex components
p
p
– Doesn’t always work
General Polygon Intersection Tests
• To
To test whether two polygons intersect, we can test whether two polygons intersect, we can
test whether any vertex is inside of another Inside/Outside Test
• To test whether a point is within a polygon/polyhedron shoot a ray and count the
polygon/polyhedron, shoot a ray and count the intersections!
• Odd => inside; even =>outside
Odd => inside; even =>outside
3 intersections ‐> inside
d
Inside/Outside Test
• To test whether a point is within a polygon/polyhedron shoot a ray and count the
polygon/polyhedron, shoot a ray and count the intersections!
• Odd => inside; even =>outside
Odd => inside; even =>outside
1 intersection ‐> inside
Inside/Outside Test
• To test whether a point is within a polygon/polyhedron shoot a ray and count the
polygon/polyhedron, shoot a ray and count the intersections!
• Odd => inside; even =>outside
Odd => inside; even =>outside
4 intersection > outside
4 intersection ‐> outside
Inside/Outside Test
• To test whether a point is within a polygon/polyhedron shoot a ray and count the
polygon/polyhedron, shoot a ray and count the intersections!
• Odd => inside; even =>outside
Odd => inside; even =>outside
0 intersection ‐> outside
Inside/Outside Test
•
•
•
•
Doesn’t always work…
intersections but still inside?
2 intersections but still inside?
This is called a degenerate case. Solutions? Try some other directions
Solutions? Try some other directions
2 intersections???
Problem 1 (in 3D)?
(
)
Collision can happen even there is no vertex inside of pp
the other polyhedron. Called edge‐edge colllision
The solution requires complex intersection tests. We skip it here…
Discrete Collision Handlingg
• Run
Run the simulation from t to t+1, without the simulation from t to t+1 without
considering collision
• Check if any intersection happens at t+1
Check if any intersection happens at t+1
• If so, remove it
Problem 2?
• Discrete collision handling has a tunneling artifact.
No intersection at t or t , but intersection happens in
• No intersection at t or t+1, but intersection happens in‐
between.
Time t
Time t
Time t+1
Time t+1
Problem 3?
• To do inside/outside test, the object must be closed.
• In other words, it has an area (or volume).
• How do I know if two curves/surfaces are intersecting?
– Knowing the location of the intersection is not sufficient! Solution
• We don’t just check whether intersection happens at t or t+1.
• We check if any intersection happens between t and t+1.
( pp
g
• This is called continuous collision detection (opposing to discrete collision detection).
For example: Vertex‐Triangle
For example: Vertex
Triangle Test
Test
• Four points
– P0(t)=P0+tV0; P1(t)=P1+tV1; P2(t)=P2+tV2; P3(t)=P3+tV3
• Intersection happens when they are co‐planar
pp
y
p
–
(P1(t)‐P0(t))(P2(t)‐P0(t))×(P3(t)‐P0(t))=0
• Once you find t, verify if P
y
,
y 0((t) is within the triangle.
)
g
P2
P1
P0
P3
For example: Edge‐EdgeTest
For example: Edge
EdgeTest
• Four points
– P0(t)=P0+tV0; P1(t)=P1+tV1; P2(t)=P2+tV2; P3(t)=P3+tV3
• Intersection happens when they are co‐planar
–
(P1(t)‐P0(t))(P2(t)‐P0(t))×(P3(t)‐P0(t))=0
• Once you find t, verify if they do intersect!
Once you find t, verify if they do intersect!
P2
P1
P0
P3
Polygon and polyhedra complexity
Polygon and polyhedra complexity
How many edges?
How many points?
Bounding objects
Bounding objects
• Sphere
• Axis aligned bounding box
Axis aligned bounding box
• Oriented bounding box
Bounding volume construction
Bounding volume construction
• Given
Given a set of points as input, can we a set of points as input can we
automatically create bounding volumes
– Axis aligned bounding box
– Sphere
– Convex hull
Axis aligned bounding box
Axis aligned bounding box
Axis aligned bounding box (AABB)
Axis aligned bounding box (AABB)
Bounding sphere
Bounding sphere
Bounding sphere
Bounding sphere
Bounding sphere
Bounding sphere
Bounding sphere
Bounding sphere
Convex Hull
Convex Hull
Best fit convex polyhedron to concave
polyhedron but takes some (one-time)
(one time)
computation
1. Find highest vertex, V1
2. Find remaining vertex that minimizes angle
with horizontal plane through point. Call
C
edge L
3. Form plane with this edge and horizontal
line perpendicular to L at V1
4. Find remaining vertex that for triangle that
minimizes angle with this plane. Add this
triangle to convex hull, mark edges as
unmatched
t h d
5. For each unmatched edge, find remaining
vertex that minimizes angle with the plane
g
triangle
g
of the edge’s
6. 3D convex hull is more complex…
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Convex hull
Bounding Slabs
For better fit bounding polyhedron: use arbitrary
(user-specified) collection of bounding planepairs
i
Is a vertex between each pair?
d 2  N  P  d1
Sphere vs. Sphere
Sphere vs. Sphere
Compare distance (p1,p2) to r1+r2
distance(p1,p2)2 to (r1 + r2)2
AABB vs. AABB
AABB vs. AABB
AABB vs. AABB
AABB vs. AABB
Oriented bounding boxes and arbitrary polygons
l
• Separating axis theorem
Separating axis theorem
– http://gamedev.tutsplus.com/tutorials/implement
ation/collision detection with the separating axis
ation/collision‐detection‐with‐the‐separating‐axis‐
theorem/
– http://www.sevenson.com.au/actionscript/sat/
– http://www.metanetsoftware.com/technique/tut
orialA.html
Hierarchical bounding volumes
Hierarchical bounding volumes
Approximating polyhedra with spheres for time‐critical collision detection, by Philip M. Hubbard
HBV example
HBV example
HBV example
HBV example
HBV example
HBV example
Spatial subdivision – grid (
(quadtrees and octrees)
d
d
)
Spatial subdivision – binary space partitioning
ii i
Collision response
Collision response
Collisions: physics review
Collisions: physics review
• Momentum p  mv
• In a closed system, momentum is conserved
p  mv  m'v'
• After a collision, the sum of all momentums is the same as the sum of all momentum before the collision
Collision types
Collision types
Elastic collisions – no loss of kinetic energy (e g deformations heat)
energy (e.g. deformations, heat)
Kinetic energy
Inelastic collisions – loss of kinetic energy
kinetic energy
1 2
E   mv
2
Elastic collision with stationary object: h i
horizontal and vertical walls/planes
l d
i l ll / l
• For vertical wall/boundary
o e ca a /bou da y
– Negate x component of velocity
– Preserve y component of P
f
velocity
• Ex: v0 = (3,‐3) ; v = (‐3,‐3) ( , );
( , )
• For horizontal walls/boundaries
– Preserve x
– Negate y
Elastic collision with stationary object: angled walls/planes
l d ll / l
• Still split velocity vector into components
– Now with respect to the normal (n) of the wall
• u = (v * n / n * n) n
– Note: * is dot product
– Vector/direction is parallel to n
Vector/direction is parallel to n
– Scalar/magnitude is in opposite direction of n; (v * n) < 0
• w
w = v v–u
• Reflected velocity:
v’ = w – u
Inelastic Collisions
Coefficient of restitution
ratio of velocities before and
after collision
(dampen the resulting
velocity)
l it )
Objects stick together,
momentum is conserved
Collision response – penalty method