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 ofnpoints in R2 ,for each point pi in S, there is the set of points (x, y)in the plane that are closer to pithan any other point in S, calledVoronoi 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/ Voronoiregions
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 ain A, a constant Kain R, c in Rd
Findx in Rd which minimize
Subject to Ka, for all ain A .
where <*, *> is standard inner product inRd.
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 Bwhere d = 2 (or 3).Minkowski Sums/DifferencesMinkowski 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 DifferencesABABMinkowski Difference & TranslationMinkowski-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).PropertiesDistancedistance(A,B) = min a A,b B|| a – b ||2distance(A,B) = min c Minkowski-Diff(A,B)|| c ||2if A and B disjoint, c is a point on boundary of Minkowski differencePenetration 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 differencePracticalityExpensive 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 PolyhedraGJK-DistanceToOrigin ( P ) // dimension is m1. Initialize P0with m+1 or fewer points.2. k= 03. while (TRUE) {4.if origin is within CH( Pk ), return 05.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 + 112.}13.}14. }An Example of GJKRunning Time of GJKEach 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 GJKGiven 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 – bxwhere 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 EnvironmentsFor 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 Schemeamax: an upper bound on relative acceleration between any two points on any pair of objects.alin: relative absolute linear : relative rotational accelerations : relative rotational velocitiesr: vector difference btw CoM of two bodies d: initial separation for two given objectsamax= | alin + x r + x x r |vi= | vlin + xr | Estimated Time to collision:tc = { (vi2 + 2 amax d)1/2 – vi } / amaxCollide System ArchitectureAnalysis & ResponseSweep & PruneSimulationExact CollisionDetectionCollisionTransformOverlapParametersSweep and PruneCompute the axis-aligned bounding box (fixed vs. dynamic) for each objectDimension Reduction by projecting boxes onto each x, y, z- axisSort the endpoints and find overlapping intervalsPossible collision — only if projected intervals overlap in all 3 dimensions Sweep & Pruneb1b2e1e2b3e3b1b2e1b3e2e3T = 1b1b2e1e2b3e3b3b1e3b2e1e2T = 2Updating Bounding BoxesCoherence (greedy algorithm)Convexity properties (geometric properties of convex polytopes)Nearly constant time, if the motion is relatively “small”Use of Sorting MethodsInitial sort — quick sort runs in O(m log m) just as in any ordinary situationUpdating — 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 IssuesCollision matrix –basically adjacency matrixEnlarge bounding volumes with some tolerance thresholdQuick start polyhedral collision test — using bucket sort & look-up tableABP1P2ABP1P2
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