Today

1
Today

• Using separating planes/axes for collision
  testing
• Collision detection packages

2
Recall Voronoi Regions

• A Voronoi region is associated with each feature
  of an object each vertex, edge and face
• Each region is the set of points that are closest
  to that feature
• For a triangle, sets are infinite
• Sets are bounded by planes (in 3D)
• How are the planes defined?
• To find the closest feature to a given point,
  find the Voronoi region that contains it
• Can be done with an efficient walk procedure
• Easy to exploit coherence (remember the closest
  feature from the last frame)

Vert 2
Face
Edge 2
Edge 1
Vert 1
Vert 3
Edge 3
3
Triangle Voronoi Example
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Done
Outside Voronoi region planes in error guide us
to next region
Note that regions share planes, so plane tests
can be cached. Also, algorithm cannot infinite
loop if careful.
4
Sphere-Box Testing

• The Voronoi regions method works particularly
  well for boxes of various forms
• The faces of the box cut space into 27 regions
• Membership in each region can be defined by
  setting 6 bits, one for inside/outside each face
  plane
• For AABB, test is even cheaper (inequalities)
• Dont even have to set all bits - some imply
  others
• E.g Outside one face means inside the one
  opposite it
• Determine which region the spheres center lies
  in
• Each region has a closest feature, which will be
  the closest feature to the sphere
• Note that k-dops also have a simple test of this
  form

5
Sphere Test Hack

• Expand objects out by the radius of the sphere,
  and test for point-inside-object using spheres
  center
• Works well for players bounding sphere
• This method is only worth it if the
  point-inside-object test is cheap
• That means that the test is normally
  point-inside-plane for expanded face planes of
  the object
• So, why is this method a hack?
• Hint, think corners. What does the expanded shape
  look like at corners?
• What form of test do you have to use to make the
  method accurate?

6
Triangle-Triangle Testing(Moller and Haines
notes in reader)

• Finding the intersection of two triangles is the
  base case for many general purpose collision
  packages
• Observations
• Each triangle is a subset of the plane it lies
  in, so two triangles only intersect along the
  line where their planes intersect
• Furthermore, they must overlap along the line of
  intersection of their planes
• Hence, for two triangles to collide, each must
  intersect the plane of the other
• Otherwise, they would have zero-size interval
  along the line of intersection
• If the triangle planes are parallel, they can
  only intersect if they are co-planar, and then
  the problem is a 2D one
• Test edges of one triangle against those of the
  other
• Then test for inclusion by testing one point from
  each triangle for containment in the other

7
2D Triangle Testing
Overlap found by testing one vertex from each
triangle for containment in the other
Overlap found by testing edges from one triangle
against those of the other
8
Triangle-Triangle General Case(Moller and Haines
notes in reader)

• Option 1 (fastest or equal fastest)
• Find points where triangle As edges intersect
  the plane of triangle B (at most 2 points) no
  collision if no such points
• Join them with a segment
• Test segment against triangle B test must check
  for containment
• Version in book uses slow segment-triangle test,
  and doesnt test containment
• Option 2 (maybe equal fastest)
• Find the line of intersection of the two planes
• Find the intervals for which the lines lies
  inside each triangle (parameters only)
• Test if the intervals overlap
• Options 3 and 4
• Look for a separating plane, or use Voronoi
  regions

9
Separating Planes

• Two convex objects do not intersect if and only
  if there exists a plane that separates them - a
  separating plane
• Can also use the separating axis - the normal of
  the separating plane
• More convenient for testing
• The number of potential separating plane
  orientations is finite for finite-sided objects
  in 3D
• Must be parallel to one of the face planes from
  either object, or
• Must be simultaneously parallel to an edge from
  each object
• Normal to the cross product of some pair of edges
• In 2D, separating lines must be parallel to an
  edge from one of the objects

10
Using Separating Axes

• Option 1 Use linear programming to find out if
  the point sets of the objects are linearly
  separable
• Expensive, so not used
• Option 2 Use a projection and interval tests
• For each potential separating axis, find the
  extents of the triangle along that axis
• In general case, same as projecting vertices onto
  a line and then taking max/min
• Then look for overlap in the extents
• This test is slower than necessary for triangles,
  but it works great for boxes, as we will see later

11
Separating Axes Example
This one separates
This one doesnt
12
AABB-AABB

• Two AABBs do not intersect if and only if
• Can be seen as a series of separating axes tests
• Three potential separating axes are the x,y and z
  axes
• Each row checks for separation along one of the
  axes
• There is also a positive version of the test,
  using ands
• More like dimension reduction, which can be
  viewed as separating axis tests
• Why use one over the other (in C at least)?

13
OBB-OBB(Gottschalk, Lin, Manocha SIGGRAPH 96)

• The fast OBB-OBB test uses separating axes
• There are 15 possible axes for two OBBs 3 faces
  from each box, and 3x3 edge direction
  combinations
• The test projects the boxes onto each potential
  axis and looks for overlaps
• Finding extents along the axis consists of some
  dot products and multiplications
• Many optimizations make the test possible in
  about 200 operations
• Express one boxs axes in terms of the others
• Re-use common sub-expressions
• Shortcut degenerate cases

14
OBB Test
2D test (3D has three axes per box)
a2
a1
b2
T
b1
D

• OBB consists of three axes and three radii
  along those axes
• In all cases, assume that the separating axis in
  question has its origin at the center of one box
• For separating axes derived from faces, we also
  know that the axis is parallel to one of the box
  axes
• Express the other boxs axes in terms of the
  first box (a rotation)
• Terms disappear

15
Algorithms for 3D Objects

• OBB-Tree
• A hierarchy of OBBs
• Reports all collisions between triangle soup
• Voronoi-Region Methods (Lin-Canny and V-Clip)
• Only works for convex manifold objects, but
  extremely robust
• Actually a closest features algorithm - gives
  closest features even if objects do not intersect

16
OBB-Trees (Gottschalk, Lin, Manocha SIGGRAPH 96)

• Represent an object (collection of triangles)
  with a hierarchy of OBBs
• Leaves are triangles
• Each box bounds its children
• Do collision testing between two objects by doing
  collision testing between two trees
• Test two boxes, one from each tree. If they
  intersect, recursively test 4 combinations of
  child boxes (binary trees)
• Start with root boxes, base case is
  triangle-triangle tests
• Result is a list of triangle pairs that intersect
• Proof that a class project can go a long way

17
Building the Trees

• Assume the objects are rigid, so the tree is
  built as a pre-processing step
• Tree is built top down
• Start with all the triangles, and put an OBB
  around them
• Subdivide the box along the biggest dimension,
  classify triangles, and recurse on the two sets
  (just like building a BSP tree)
• Child boxes dont bear any specific relation to
  the parents different orientations and sizes
• Parents dont even have to bound child boxes
• Basic operation is fitting an OBB to a collection
  of triangles

18
OBB-Tree Construction
19
Fitting OBBs

• The OBB fitting problem requires finding the
  orientation of the box that best fits the data
• Ideally, you want the minimum volume for the box,
  but thats hard
• Instead, turn to an idea from statistics and AI
  (academic AI) principal components
• Point sample the convex hull of the geometry to
  be bound
• Find the mean and covariance matrix of the
  samples
• The mean will be the center of the box
• The eigenvectors of the covariance matrix are the
  principal directions they are used for the axes
  of the box
• The principle directions tend to align along the
  longest axis, then the next longest that is
  orthogonal, and then the other orthogonal axis

20
Principle Components
21
OBB-Tree Evaluation

• OBB-Trees are the method of choice for
  unstructured geometry
• Important analysis Cost of detection, for two
  phases, is NbCbNnCn
• Number of broad tests times cost of broad test
  number of narrow tests times cost of each narrow
  test
• Cb and Nn depend on broad phases scheme -
• OBB-Tree is good at Nn and not bad at Cb, whereas
  AABB is good at Cb but not so good at Nn
• Biggest way to improve it would be to exploit
  coherence, but it is not clear how to do so
• Faster algorithms are restricted to more
  structured geometry
• Available as RAPID (just OBB-Trees) and V-COLLIDE
  (OBB-Trees and dimension reduction broad phase)