Efficient Collision Detection for Animation and Robotics Part 4 doc

[...].. .44 Voronoi Rgn E P b a b > a Figure 3.9: An Overhead View for Point-Edge Applicability Criterion Proof Voronoi Region P TE E HE Figure 3.10: A Side View for Point-Edge Applicability Criterion Proof 45 Voronoi Region P NF EF F Figure 3.11: A Side View for Point-Face Applicability Criterion Proof between the closest feature candidates base upon the minimum distance information during... and a face F , the algorithm rst checks IF E AND F ARE PARALLEL, i.e the two endpoints HE and TE are equi-distant from the face F i.e their signed distance is the same The signed distance between HE and F can be computed easily by HE
NF where NF is F 's unit outward normal and similarly for the signed distance between F and HT THEN, when the head HE and tail TE of the edge E are equi-distant from... feature veri cation for the other non-degenerate cases and edge-edge solely depends on point-vertex, point-edge, and point-face applicability tests which we have already studied Now we will examine each of these two cases closely Edge-Face E , F  The indentation corresponds to the level of nested loops in the pseudo code Appendix B Please cross-reference if necessary 46 Given an edge E and a face F ,... , fB  as the next candidate feature pair 48 ELSE, if the closer endpoint VE of E to F lies outside of F 's prism region, then there is one edge on F 's boundary which is closer to E than the interior of F So, the algorithm will nd the closest edge or vertex fB , which has the minimum distance to E among all edges and vertices in F 's boundary, and return E , fB  as the next candidate pair V Verifying... containing F to E by enumerating all the features on B Then the algorithm will return E and this new feature fB as the next candidates for the closest feature pair veri cation ELSE, E does not intersect the prism de ned by F 's boundary edges, then there is at least one edge of F 's boundary which is closer to E 47 than the interior of F is to E by the de nition of Voronoi region So, when the test... boundary and returns the edge E and the closest edge or vertex, say fF , on the F 's boundary to E ELSE, the algorithm checks IF ONE END POINT LIES ABOVE THE FACE F AND THE OTHER LIES BENEATH F THEN, if so there is one edge on F 's boundary which is closer to E than the interior of F So, the algorithm will nd the closest edge or vertex fB , which has the minimum distance to E among all edges and vertices... NORMALS OF THE EDGE'S LEFT AND RIGHT FACES, say NL and NR respectively, to verify that the face F lies inside the Voronoi region of the edge E , not in the Voronoi region of E 's left or right ~ face FL or FR This is done by checking if NL  E 
NF 0 ~ and E  NR
NF 0, where NF is the outward normal of the ~ face F , E is the edge vector as described in Sec 2.1.2, and NR, NL denote the outward... parallel LIES ABOVE FB , where PA is one of the points on FA and its projection down to the FB is in the overlap region 3 THEN, if PA thus FA lies above FB, the algorithm next checks IF PB THUS FB  ALSO LIES ABOVE FA, where PB is one of the points and its projection down to the FB is in the overlap region 4 THEN, if PB thus FB  is above FA and vice versa, we have a pair of closest points PA ,... the overlap region 4 THEN, if PB thus FB  is above FA and vice versa, we have a pair of closest points PA , PB  satisfy the respective applicability constraints of FB and FA So, PA and PB are the closest points on A and B ; and the ... information during the search Therefore, we can conclude that the distance decreases when a switch of feature pairs is made as the result of Point-Face Applicability Test IV Verifying Subroutines in Edge-Face Case With the exception of edge-edge as mentioned earlier in the beginning of the proof, only edge-face and face-face may have multiple pairs of closest points and require a more complex treatments