# CS计算机代考程序代写 scheme flex algorithm References

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.

UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin

UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

Simple 2D Example

P2 P1

B

A

UNC Chapel Hill

M. C. Lin

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

UNC Chapel Hill

M. C. Lin

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.

UNC Chapel Hill

M. C. Lin