Collisions using separatingaxis tests

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

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

?
!
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Underlying theory

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

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Underlying theory

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

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Underlying theory

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

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Underlying theory

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

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Underlying theory

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

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Testing separation

• Compare absolute intervals

Separated if or
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Testing separation

• For centrally symmetric objects compare using
  projected radii

Separated if
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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)

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Axes to test

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

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Axes to test

• Handwavingly
• Look at the ways features of A and B can come
  into contact.
• Features are vertices, edges, faces.
• 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.

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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.

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Axes to test

• Four axes for two 2D OBBs

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Axes to test
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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

Note here objects assumed to be in the same
space.
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Moving objects

• When objects move, projected intervals move

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Moving objects

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

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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

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Nonpolyhedral objects

• What about
• Spheres, capsules, cylinders, cones, etc?
• Same idea
• Identify all features
• Test all axes that can possibly separate feature
  pairs!

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Nonpolyhedral objects

• Sphere tests
• Has single feature its center
• Test axes going through center
• (Radius is accounted for during overlap test on
  axis.)

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Nonpolyhedral objects

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

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Nonpolyhedral objects

• Sphere vs. OBB test
• sphere center vs. box vertex
• pick candidate axis parallel to line thorugh both
  points.
• sphere center vs. box edge
• pick candidate axis parallel to line
  perpendicular to edge and goes through sphere
  center
• sphere center vs. box face
• pick candidate axis parallel to line through
  sphere center and perpendicular to face
• Use logic to reduce tests where possible.

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Nonpolyhedral objects

• For sphere-OBB, all tests can be subsumed by a
  single axis test

Closest point on OBB to sphere center
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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! (Address using means discussed
earlier.)
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Contact determination

• Covered by Erin Catto (later)

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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.