[...]... Vectors ~ and ~ are e e de ned as ~ = H , T and ~ = H , T We can nd for the nearest point pair P e e and P on E and E by the following: 1 2 1 1 2 1 2 2 1 1 1 2 2 1 2 2 1 2 P = H + sT , H  = H , s~ e P = H + uT , H  = H , u~ e 1 1 2 1 1 2 2 A.3 1 2 2 1 2 where s and u are scalar values parameterized between 0 and 1 to indicate the relative location of P and P on the edges E and E Let ~ = P , P and. .. magnitude of 1 and NF = n; ,d = a; b; c; ,d and ,d is the signed distance of the face F from the origin, and the vertex V de ned as above We de ne a new vector quantity ~ F by ~ F = n; 0 The nearest point PF on F to V can be simply expressed n n as: 137 PF = V , V
NF ~ F n A:2 IV EDGE-EDGE: Let H and T be the head and tail of the edge E respectively And H and T be the head and tail of the... dynamics for robotic operations In IEEE Int Conf on Robotics and Automation, pages 678 685, May 1987 87 C G Gibson K Wirthmuller and A A du Plessis E J N Looijenga Topological Stability of Smooth Mappings Springer-Verlag, Berlin Heidelberg New York, 1976 88 Andrew Witkin, Michael Gleicher, and William Welch Interactive dynamics Computer Graphics, 242:11 22, March 1990 89 Andrew Witkin and William... Witkin and William Welch Fast animation and control of nonrigid structures Computer Graphics, 244:243 252, August 1990 90 P Wolfe Finding the nearest points in a polytope Math Programming, 11:pp 128 149, 1976 91 K Zikan and W D Curtis Intersection and separations of polytopes Boeing Computer Services Technology, BCSTECH-93-031, 1993 A note on collision and interference detection 136 Appendix A Calculating... polyhedra using binary space partitioning trees Computer Graphics SIGGRAPH'87, 4, 1987 135 84 R Thom Sur l'homologie des varietes algebriques reelles Di erential and Combinatorial Topology, pages 255 265, 1965 85 G Turk Interactive collision detection for molecular graphics Master's thesis, Computer Science Department, University of North Carolina at Chapel Hill, 1989 86 Yu Wang and Matthew T Mason Modeling... K Fleischer, B Currin, and A Barr Interval methods for multi-point collisions between time-dependent curved surfaces Computer Graphics, Proceedings of ACM SIGGRAPH'93, pages 321 334, August 1993 82 D Sturman A discussion on the development of motion control systems In SigGraph Course Notes: Computer Animation: 3-D Motion Speci cation and Control, number 10, 1987 83 W Thibault and B Naylor Set operations... J.T Schwartz and M Sharir On the `Piano Movers' Problem, II General Techniques for Computing Topological Properties of Real Algebraic Manifolds, chapter 5, pages 154 186 Ablex publishing corp., New Jersey, 1987 79 R Seidel Linear programming and convex hulls made easy In Proc 6th Ann ACM Conf on Computational Geometry, pages 211 215, Berkeley, California, 1990 80 H W Six and D Wood Counting and reporting... Let ~ = P , P and j ~ j is the shortest n n distance between the two edges E and E Since ~ must be orthogonal to the vectors n ~ and ~ , we have: e e 1 2 1 2 1 1 1 2 2 2 ~
~ = P , P 
~ = 0 n e e ~
~ = P , P 
~ = 0 n e e 1 1 2 2 1 2 A.4 1 2 By substituting Eqn. A.3 into these equations A.4, we can solve for s and u: e e e , e e e A.5 s = ~
~  H , H 
~ det~
~  H , H 
... , ~
~   ~
~  However, to make sure P and P e e e e e e e e lie on the edges E and E , s and u are truncated to the range 0,1 which gives the correct nearest point pair P ; P  1 2 1 1 2 2 2 2 1 2 1 1 1 2 1 1 2 2 1 2 1 2 1 1 2 1 1 2 2 1 2 2 1 2 V EDGE-FACE: Degenerate, we don't compute them explicitly Please see the pseudo code in Appendix B for the detailed treatment 138 VI FACE-FACE:...134 74 D F Rogers Procedural Elements for Computer Graphics McGraw-Hill Book Company, 1985 75 Edward J Routh Elementary Rigid Dynamics 1905 76 J Ruppert and R Seidel On the di culty of tetrahedralizing 3-dimensional nonconvex polyhedra In Proc of the Fifth Annual Symposium on Computational Geometry, pages 380 392 ACM, 1989 77 N Sancheti and S Keerthi Computation of certain measures of proximity