3D Polyhedral Morphing
References
Additional lecture notes for 2/18/02.
I-COLLIDE: Interactive and Exact Collision Detection for Large-Scale Environments, by Cohen, Lin, Manocha & Ponamgi, Proc. of ACM Symposium on Interactive 3D Graphics, 1995.
(More details in Chapter 3 of M. Lin’s Thesis)
A Fast Procedure for Computing the Distance between Objects in Three-Dimensional Space, by E. G. Gilbert, D. W. Johnson, and S. S. Keerthi, In IEEE Transaction of Robotics and Automation, Vol. RA-4:193–203, 1988.
Geometric Proximity Queries
Given two object, how would you check:
If they intersect with each other while moving?
If they do not interpenetrate each other, how far are they apart?
If they overlap, how much is the amount of penetration
Collision Detection
Update configurations w/ TXF matrices
Check for edge-edge intersection in 2D
(Check for edge-face intersection in 3D)
Check every point of A inside of B &
every point of B inside of A
Check for pair-wise edge-edge intersections
Imagine larger input size: N = 1000+ ……
Classes of Objects & Problems
2D vs. 3D
Convex vs. Non-Convex
Polygonal vs. Non-Polygonal
Open surfaces vs. Closed volumes
Geometric vs. Volumetric
Rigid vs. Non-rigid (deformable/flexible)
Pairwise vs. Multiple (N-Body)
CSG vs. B-Rep
Static vs. Dynamic
And so on… This may include other geometric representation schemata, etc.
Some Possible Approaches
Geometric methods
Algebraic Techniques
Hierarchical Bounding Volumes
Spatial Partitioning
Others (e.g. optimization)
Voronoi Diagrams
Given a set S of n points in R2 , for each point pi in S, there is the set of points (x, y) in the plane that are closer to pi than any other point in S, called Voronoi polygons. The collection of n Voronoi polygons given the n points in the set S is the “Voronoi diagram”, Vor(S), of the point set S.
Intuition: To partition the plane into regions, each of these is the set of points that are closer to a point pi in S than any other. The partition is based on the set of closest points, e.g. bisectors that have 2 or 3 closest points.
Generalized Voronoi Diagrams
The extension of the Voronoi diagram to higher dimensional features (such as edges and facets, instead of points); i.e. the set of points closest to a feature, e.g. that of a polyhedron.
FACTS:
In general, the generalized Voronoi diagram has quadratic surface boundaries in it.
If the polyhedron is convex, then its generalized Voronoi diagram has planar boundaries.
Voronoi Regions
A Voronoi region associated with a feature is a set of points that are closer to that feature than any other.
FACTS:
The Voronoi regions form a partition of space outside of the polyhedron according to the closest feature.
The collection of Voronoi regions of each polyhedron is the generalized Voronoi diagram of the polyhedron.
The generalized Voronoi diagram of a convex polyhedron has linear size and consists of polyhedral regions. And, all Voronoi regions are convex.
Voronoi Marching
Basic Ideas:
Coherence: local geometry does not change much, when computations repetitively performed over successive small time intervals
Locality: to “track” the pair of closest features between 2 moving convex polygons(polyhedra) w/ Voronoi regions
Performance: expected constant running time, independent of the geometric complexity
Simple 2D Example
Objects A & B and their Voronoi regions: P1 and P2 are the pair of closest points between A and B. Note P1 and P2 lie within the Voronoi regions of each other.
Basic Idea for Voronoi Marching
Linear Programming
In general, a d-dimensional linear program-ming (or linear optimization) problem may be posed as follows:
Given a finite set A in Rd
For each a in A, a constant Ka in R, c in Rd
Find x in Rd which minimize
Subject to Ka, for all a in A .
where <*, *> is standard inner product in Rd.
LP for Collision Detection
Given two finite sets A, B in Rd
For each a in A and b in B,
Find x in Rd which minimize whatever
Subject to > 0, for all a in A
And < 0, for all b in B
where d = 2 (or 3).
Minkowski Sums/Differences
Minkowski Sum (A, B) = { a + b | a A, b B }
Minkowski Diff (A, B) = { a - b | a A, b B }
A and B collide iff Minkowski Difference(A,B) contains the point 0.
Some Minkowski Differences
A
B
A
B
Minkowski Difference & Translation
Minkowski-Diff(Trans(A, t1), Trans(B, t2)) = Trans(Minkowski-Diff(A,B), t1 - t2)
Trans(A, t1) and Trans(B, t2) intersect iff Minkowski-Diff(A,B) contains point (t2 - t1).
Properties
Distance
distance(A,B) = min a A, b B || a - b ||2
distance(A,B) = min c Minkowski-Diff(A,B) || c ||2
if A and B disjoint, c is a point on boundary of Minkowski difference
Penetration Depth
pd(A,B) = min{ || t ||2 | A Translated(B,t) = }
pd(A,B) = mint Minkowski-Diff(A,B) || t ||2
if A and B intersect, t is a point on boundary of Minkowski difference
Practicality
Expensive to compute boundary of Minkowski difference:
For convex polyhedra, Minkowski difference may take O(n2)
For general polyhedra, no known algorithm of complexity less than O(n6) is known
GJK for Computing Distance between Convex Polyhedra
GJK-DistanceToOrigin ( P ) // dimension is m
1. Initialize P0 with m+1 or fewer points.
2. k = 0
3. while (TRUE) {
4. if origin is within CH( Pk ), return 0
5. else {
6. find x CH(Pk) closest to origin, and Sk Pk s.t. x CH(Sk)
7. see if any point p-x in P more extremal in direction -x
8. if no such point is found, return |x|
9. else {
10. Pk+1 = Sk {p-x}
11. k = k + 1
12. }
13. }
14. }
An Example of GJK
Running Time of GJK
Each iteration of the while loop requires O(n) time.
O(n) iterations possible. The authors claimed between 3 to 6 iterations on average for any problem size, making this “expected” linear.
Trivial O(n) algorithms exist if we are given the boundary representation of a convex object, but GJK will work on point sets - computes CH lazily.
More on GJK
Given A = CH(A’) A’ = { a1, a2, ... , an } and
B = CH(B’) B’ = { b1, b2, ... , bm }
Minkowski-Diff(A,B) = CH(P), P = {a - b | a A’, b B’}
Can compute points of P on demand:
p-x = a-x - bx where a-x is the point of A’ extremal in direction -x, and bx is the point of B’ extremal in direction x.
The loop body would take O(n + m) time, producing the “expected” linear performance overall.
Large, Dynamic Environments
For dynamic simulation where the velocity and acceleration of all objects are known at each step, use the scheduling scheme (implemented as heap) to prioritize “critical events” to be processed.
Each object pair is tagged with the estimated time to next collision. Then, each pair of objects is processed accordingly. The heap is updated when a collision occurs.
Scheduling Scheme
amax: an upper bound on relative acceleration between any two points on any pair of objects.
alin: relative absolute linear
: relative rotational accelerations
: relative rotational velocities
r: vector difference btw CoM of two bodies
d: initial separation for two given objects
amax = | alin + x r + x x r |
vi = | vlin + x r |
Estimated Time to collision:
tc = { (vi2 + 2 amax d)1/2 - vi } / amax
Collide System Architecture
Analysis &
Response
Sweep & Prune
Simulation
Exact
Collision
Detection
Collision
Transform
Overlap
Parameters
Sweep and Prune
Compute the axis-aligned bounding box (fixed vs. dynamic) for each object
Dimension Reduction by projecting boxes onto each x, y, z- axis
Sort the endpoints and find overlapping intervals
Possible collision -- only if projected intervals overlap in all 3 dimensions
Sweep & Prune
b1
b2
e1
e2
b3
e3
b1
b2
e1
b3
e2
e3
T = 1
b1
b2
e1
e2
b3
e3
b3
b1
e3
b2
e1
e2
T = 2
Updating Bounding Boxes
Coherence (greedy algorithm)
Convexity properties (geometric properties of convex polytopes)
Nearly constant time, if the motion is relatively “small”
Use of Sorting Methods
Initial sort -- quick sort runs in O(m log m) just as in any ordinary situation
Updating -- insertion sort runs in O(m) due to coherence. We sort an almost sorted list from last stimulation step. In fact, we look for “swap” of positions in all 3 dimension.
Implementation Issues
Collision matrix -- basically adjacency matrix
Enlarge bounding volumes with some tolerance threshold
Quick start polyhedral collision test -- using bucket sort & look-up table
A
B
P1
P2
A
B
P1
P2
