Collisions%20using%20separating-axis%20tests

1
Collisions using separating-axis tests
Christer EricsonSony Computer EntertainmentSlid
es _at_ http//realtimecollisiondetection.net/pubs/
2
Problem statement

• Determine if two (convex) objects are
  intersecting.
• Possibly also obtain contact information.

?
!
3
Underlying theory

• Set C is convex if and only if the line segment
  between any two points in C lies in C.

4
Underlying theory

• Separating Hyperplane Theorem
• States two disjoint convex sets are separable by
  a hyperplane.

5
Underlying theory

• Nonintersecting concave sets not generally
  separable by hyperplane (only by hypersurfaces).
• Concave objects not covered here.

6
Underlying theory

• Separation w.r.t a plane P ? separation of the
  orthogonal projections onto any line L parallel
  to plane normal.

7
Underlying theory

• A line for which the projection intervals do not
  overlap we call a separating axis.

8
Testing separation

• Compare absolute intervals

Separated if or
9
Testing separation

• For centrally symmetric objects compare using
  projected radii

Separated if
10
Code fragment

• void GetInterval(Object o, Vector axis,
• float min, float max)
• min max Dot(axis, o.getVertex(0))
• for (int i 1, n o.NumVertices() i lt n
  i)
• float value Dot(axis, o.getVertex(i))
• min Min(min, value)
• max Max(max, value)

11
Axes to test

• But which axes to test?
• Simplification
• Deal only with polytopes
• Convex hulls of finite point sets
• Planar faces

12
Axes to test

• Handwavingly
• Look at the ways A and B can come into contact.
• In 3D, reduces to vertex-face and edge-edge
  contacts.
• Vertex-face
• a face normal from either polytope will serve as
  a separating axis.
• Edge-edge
• the cross product of an edge from each will
  suffice.

13
Axes to test

• Theoretically
• Consider the Minkowski difference C of A and B.
• When A and B disjoint, origin outside C,
  specifically outside some face F.
• Faces of C come from A, from B, or from sweeping
  the faces of either along the edges of the other.
• Therefore the face normal of F is either from A,
  from B, or the cross product of an edge from
  either.

14
Axes to test

• Four axes for two 2D OBBs

15
Axes to test
3D Objects Face dirs (A) Face dirs (B) Edge dirs (AxB) Total
SegmentTri 0 1 1x3 4
SegmentOBB 0 3 1x3 6
AABBAABB 3 0(3) 0(3x0) 3
OBBOBB 3 3 3x3 15
TriTri 1 1 3x3 11
TriOBB 1 3 3x3 13
16
Code fragment

• bool TestIntersection(Object o1, Object o2)
• float min1, max1, min2, max2
• for (int i 0, n o1.NumFaceDirs(), i lt n
  i)
• GetInterval(o1, o1.GetFaceDir(i), min1,
  max1)
• GetInterval(o2, o1.GetFaceDir(i), min2,
  max2)
• if (max1 lt min2 max2 lt min1) return
  false
• for (int i 0, n o2.NumFaceDirs(), i lt n
  i)
• GetInterval(o1, o2.GetFaceDir(i), min1,
  max1)
• GetInterval(o2, o2.GetFaceDir(i), min2,
  max2)
• if (max1 lt min2 max2 lt min1) return
  false
• for (int i 0, m o1.NumEdgeDirs(), i lt m
  i)
• for (int j 0, n o2.NumEdgeDirs(), j lt
  n j)
• Vector axis Cross(o1.GetEdgeDir(i),
  o2.GetEdgeDir(j))
• GetInterval(o1, axis, min1, max1)
• GetInterval(o2, axis, min2, max2)
• if (max1 lt min2 max2 lt min1)
  return false

17
Moving objects

• When objects move, projected intervals move

18
Moving objects

• Objects intersect when projections overlap on all
  axes.
• Objects are in contact over the interval maxi
  tifirst, mini tilast, where tifirst and
  tilast are time of first and last contact on axis
  i.
• No contact if maxi tifirst gt mini tilast

19
Moving objects

• Optimization 1
• Consider relative movement only.
• Shrink interval A to point, growing interval B by
  original width of A.
• Becomes moving point vs. stationary interval.
• Optimization 2
• Exit as soon as maxi tifirst gt mini tilast

20
Nonpolyhedral objects

• Spheres, capsules, cylinders, cones, etc.
• E.g. sphere vs. OBB

Closest point on OBB to sphere center
21
Nonpolyhedral objects

• Capsule tests
• Split into three features
• Test axes that can separate features of capsule
  and features of second object.

22
Robustness warning

• Cross product of edges can result in zero-vector.

Typical test
if (Dot(foo, axis) gt Dot(bar, axis)) return false
Becomes, due to floating-point errors
if (epsilon1 gt epsilon2) return false
Results in Chaos!
23
Contact determination

• Covered by Erin Catto (later)

24
References

• Ericson, Christer. Real-Time Collision Detection.
  Morgan Kaufmann 2005. http//realtimecollisiondete
  ction.net/
• Levine, Ron. Collisions of moving objects.
  gdalgorithms-list mailing list article, November
  14, 2000. http//realtimecollisiondetection.net/fi
  les/levine_swept_sat.txt
• Boyd, Stephen. Lieven Vandenberghe. Convex
  Optimization. Cambridge University Press, 2004.
  http//www.stanford.edu/boyd/cvxbook/
• Rockafellar, R. Tyrrell. Convex Analysis.
  Princeton University Press, 1996.