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General Triangle Models:
Representations and
Collision
David Johnson
Triangle Models
► Boundary-
representation (Brep)
 Collection of
surfaces to
represent a volume
► Manifold
 Closed model
 Every edge has two
triangles
Polygon Modeling
►Polygons
are the dominant force in
modeling for real-time graphics
►Why?
Polygons Dominate
► Everything
can be turned into polygons
(almost everything)
 Normally an error associated with the
conversion, but with time and space it may
be possible to reduce this error
► We
know how to render polygons quickly
► Many operations are easy to do with
polygons
► Memory and disk space is cheap
► Simplicity and inertia
What’s Bad About Polygons?
►What
are some disadvantages of
polygonal representations?
Polygons Aren’t Great
► They
are always an approximation to curved
surfaces
 But can be as good as you want, if you are willing to
pay in size
 Normal vectors are approximate
 They throw away information
► They
can be very unstructured
► They are hard to globally parameterize
 How do we parameterize them for texture mapping?
► It
is difficult to perform many geometric
operations
Polygon Meshes
►A
mesh is a set of polygons connected to form an
object
► A mesh has several components, or geometric
entities:
 Faces
 Edges, the boundary between faces
 Vertices, the boundaries between edges, or where three
or more faces meet
 Normals, Texture coordinates, colors, shading
coefficients, etc
► Some
components are implicit, given the others
 For instance, given faces and vertices can determine
edges
Polygonal Data Structures
► Polygon
mesh data structures are application
dependent
► Different applications require different operations
to be fast
 Find the neighbor of a given face
 Find the faces that surround a vertex
► You
typically choose:
 Which features to store explicitly (vertices, faces,
normals, etc)
 Which relationships you want to be explicit (vertices
belonging to faces, neighbors, faces at a vertex, etc)
Triangle Meshes
► We
will look at triangle mesh data
structures
 Can triangulate general polygon
Triangle Soup
► Many
models are just lists of triangles
 What does the model below look like in a
floating point computer?
T4
T1
T2
T5
T3
T6
T1: v1, v2, v3
T2: v4, v5, v6
T3: v7, v8, v9
T4: v10, v11, v12
etc…
Triangle Soup
► No
connection between triangles
 Some models are produced like this, so may
have to work with it
T4
T1
T2
T5
T3
T6
T1: v1, v2, v3
T2: v4, v5, v6
T3: v7, v8, v9
T4: v10, v11, v12
etc…
Triangle Soup Evaluation
► What
are the advantages?
 It’s very simple to read, write, transmit, etc.
 A common output format from CAD modelers
► BIG
disadvantage: No higher order
information
 No information about neighbors
 No open/closed information
 No guarantees on degeneracies/manifoldness
Vertex-Face Mesh
v0
v4
v1
v2
vertices
faces
0
2
1
v0 v1 v2 v3 v4
0
1
4
1
2
3
1
3
4
v3
► There
are reasons not to store the vertices
explicitly at each polygon
 Wastes memory - each vertex repeated many times
 Very messy to find neighboring polygons
 Difficult to ensure that polygons meet correctly
► Solution:
Indirection
 Put all the vertices in a list
 Each face stores the list indices of its vertices
► Advantages?
Disadvantages?
Vertex-Face Evaluation
► Advantages:
 Saving in storage:
►Vertex
index might be only 2 bytes, and a vertex is
probably 12 bytes
►Each vertex gets used at least 3 and generally 4-6
times, but is only stored once
 Normals, texture coordinates, colors etc. can all
be stored the same way
► Disadvantages:
 Connectivity information is not explicit
 How would you find a neighbor face/shared
Add more data
► Face
data structure
 Face:
►Neighbors
= [find1, find2, find3]
►Vertices = [vind1, vind2, vind3]
 Vertex:
►Coords
= [x,y,z]
►Neighborfaces = [find1,..,findn];
Some problems
► Edges
are implicit
► Number of neighbors to vertex is variable
► Use vertex-edge-face structures
 Winged edge
 Half-edge
Half-edge
► Half-edge




leading vertex
Face on left
Edge ahead
Flipped Edge
► Everything
points to one thing
Collision Methods for Triangle Models
► Models
may have millions of triangles
► For two models with m and n triangles
 Brute force: m*n triangle-triangle intersections
 > 1012 intersection computations for two
objects
► Need
an acceleration structure
Bounding Volumes
► Objects
are often not colliding
 Need fast reject for this case
 Surround with some bounding
object
► Like
a sphere
► Why
stop with one layer of
rejection testing?
 Build a bounding volume
hierarchy (BVH)
►A
tree
Bounding Volume Hierarchies
► Model
Hierarchy:
 each node has a simple volume that bounds a set
of triangles
 children contain volumes that each bound a
different portion of the parent’s triangles
 The leaves of the hierarchy usually contain
individual triangles
►A
binary bounding volume hierarchy:
BVH-Based Collision Detection
Type of Bounding Volumes
Spheres
► Ellipsoids
► Axis-Aligned Bounding Boxes (AABB)
► Oriented Bounding Boxes (OBBs)
► Convex Hulls
► k-Discrete Orientation Polytopes (k-dop)
► Spherical Shells
► Swept-Sphere Volumes (SSVs)
►




Point Swetp Spheres (PSS)
Line Swept Spheres (LSS)
Rectangle Swept Spheres (RSS)
Triangle Swept Spheres (TSS)
Observations
► Simple
primitives (spheres, AABBs, etc.) do very well
with respect to the overlap cost. But they cannot fit
some long skinny primitives tightly.
► More
complex primitives (minimal ellipsoids, OBBs,
etc.) provide tight fits, but checking for overlap between
them is relatively expensive.
► Cost
of BV updates needs to be considered.
Trade-off in Choosing BV’s
Sphere
AABB
OBB
6-dop
Convex Hull
increasing complexity & tightness of fit
decreasing cost of (overlap tests + BV update)
Bounding Volume Hierarchies
(Example: Spheres, PSS)
Bounding Volume Hierarchies
(Example: AABB)
Bounding Volume Hierarchies
(Example: k-DOPs, k=6,14,18,26)
Level 0
Note: k=6 is
a AABB.
Bounding Volume Hierarchies
(Example: k-DOPs, k=6,14,18,26)
Level 1
Bounding Volume Hierarchies
(Example: k-DOPs, k=6,14,18,26)
Level 2
Bounding Volume Hierarchies
(Example: k-DOPs, k=6,14,18,26)
Level 5
Bounding Volume Hierarchies
(Example: k-DOPs, k=6,14,18,26)
Level 8
Building Hierarchies
► Choices
of Bounding Volumes
 cost function & constraints
► Top-Down
vs. Bottom-up
 speed vs. fitting
► Depth
vs. breadth
 branching factors
► Splitting
factors
 where & how
Sphere-Trees
sphere-tree is a hierarchy of sets of spheres, used to
approximate an object
►A
► Advantages:
 Simplicity in checking overlaps between two bounding
spheres
 Invariant to rotations and can apply the same transformation
to the centers, if objects are rigid
► Shortcomings:
 Not always the best approximation (esp bad for long, skinny
objects)
 Lack of good methods on building sphere-trees
Methods for Building SphereTrees
► “Tile”
the triangles and build the tree bottom-up
► Covering
each vertex with a sphere and group
them together
► Compute
the medial axis and use it as a skeleton
for multi-res sphere-covering
► Others……
Building an OBBTree
Recursive top-down construction:
partition and refit
Building an OBB Tree
Given some polygons,
consider their vertices...
Building an OBB Tree
… and an arbitrary line
Building an OBB Tree
Project onto the line
Consider variance of
distribution on the line
Building an OBB Tree
Different line,
different variance
Building an OBB Tree
Maximum Variance
Building an OBB Tree
Minimal Variance
Building an OBB Tree
Given by eigenvectors
of covariance matrix
of coordinates
of original points
Building an OBB Tree
Choose bounding box
oriented this way
Building an OBB Tree: Fitting
Covariance matrix of
point coordinates describes
statistical spread of cloud.
OBB is aligned with directions of
greatest and least spread
(which are guaranteed to be orthogonal).
Building an OBB Tree
Good Box
Building an OBB Tree
Add points:
worse Box
Building an OBB Tree
More points:
terrible box
Building an OBB Tree
Compute with extremal points only
Building an OBB Tree
“Even” distribution:
good box
Building an OBB Tree
“Uneven” distribution:
bad box
Building an OBB Tree
Fix: Compute facets of convex hull...
Building an OBB Tree
Better: Integrate over facets
Building an OBB Tree
… and sample them uniformly
Building an OBB Tree: Summary
OBB Fitting algorithm:
► covariance-based
► use
of convex hull
► not foiled by extreme distributions
► O(n log n) fitting time for single BV
► O(n log2 n) fitting time for entire tree
Tree Traversal
Disjoint bounding volumes:
No possible collision
Tree Traversal
Overlapping bounding volumes:
• split one box into children
• test children against other box
Tree Traversal
Tree Traversal
Hierarchy of tests
Separating Axis Theorem
 L is a separating axis for OBBs A & B, since A & B
become disjoint intervals under projection onto L
Separating Axis Theorem
Two polytopes A and B are disjoint iff there
exists a separating axis which is:
perpendicular to a face from either
or
perpendicular to an edge from each
Implications of Theorem
Given two generic polytopes, each with E edges
and F faces, number of candidate axes to test
is:
2F + E2
OBBs have only E = 3 distinct edge directions,
and only F = 3 distinct face normals. OBBs
need at most 15 axis tests.
Because edge directions and normals each form
orthogonal frames, the axis tests are rather
simple.
OBB Overlap Test: An Axis Test
L
s
ha
hb
L is a separating axis iff: s > ha+hb
OBB Overlap Test: Axis Test
Details
Box centers project to interval midpoints, so
midpoint separation is length of vector T’s image.
B
TB
A
TA
T
n
s


s  T  T n
A
B
OBB Overlap Test: Axis Test
Details
►
Half-length of interval is sum of box axis images.
B
rB
n
rB  b1 R1B  n  b2 R 2B  n  b3 R 3B  n
Implementation: RAPID
► Available
at:
http://www.cs.unc.edu/~geom/OBB
► Part
of V-COLLIDE:
http://www.cs.unc.edu/~geom/V_COLLIDE
► Thousands
► Used
of users have ftp’ed the code
for virtual prototyping, dynamic
simulation, robotics & computer animation