Efficient Collision Detection for Animation and Robotics Part 4 pptx

CP E R 1 ’ CP F CP R 2 ’ R 2 R 1 Object A [...].. .41 Proof: The algorithm works by nding a pair of closest features There are two categories: 1 non-degenerate cases vertex-vertex, vertex-edge, and vertex-face 2 degenerate cases edge-edge, edge-face, and face-face For non-degenerate and edge-edge cases, the algorithm rst takes any pair of features fA and fB , then nd the nearest points PA and PB between them Next, it... within the region bounded by the four constraint planes generated by the coboundary the left and right faces and boundary the two end points of E This is the Voronoi region of E If this is the case, the shortest distance between P and the other object is the distance between P and E see Fig.3.3 and Sec 3.2.2 for the construction of pointedge constraint planes. When P fails an applicability constraint,... side of B and on the other side of B from F This corresponds to a local minimum of distance function It requires a linear-time routine to step to the next feature-pair The linear-time routine enumerates all features on the object B and searches for the closest one to the feature fA containing P This necessarily decreases the distance 1 1 44 Voronoi Rgn E P b a b > a Figure 3.9: An Overhead View for Point-Edge... excluding the edges and vertices in its boundary This guarantees that when a switch of features is made, the new features are strictly closer We will rst show that each applicability test returns a pair of candidate features closer in distance than the previous pair when a switch of feature pairs is made Then, we show the closest feature verifying subroutines for both edge-face and 42 face-face cases... non-degenerate cases and the proof of completeness applies here as well The other two degenerate cases must be treated di erently since they may contain in nitely many closest point pairs We would like to recall that edges and faces are treated as open subsets of lines and planes respectively That is, the edge is considered as the set of points between two endpoints of the edge, excluding the head and the tail... distance between P and the corresponding face is shorter than the distance between P and E as in Fig.3.9 If P fails the applicability criterion CV imposed by the head or tail of E , then P is closer to the corresponding endpoint than to E itself as in Fig.3.10 Therefore, the distance between the new pair of features is guaranteed to be shorter than that of the previous pair which fails 43 P Voronoi... between P and the other object is clearly the distance between P and V by the de nition of Voronoi region When P lies outside one of the plane boundaries, say CE1 , the constraint plane of an edge E touching V , then there is at least one point on E closer to P than V itself, i.e the distance between the edge E whose constraint is violated and P is shorter than the distance between V and P This... Figure 3.8: A Side View for Point-Vertex Applicability Criterion Proof the Point-Edge Applicability Criterion, when a switch of feature pairs is made III Point-Face Applicability Criterion If the face F is actually the closest feature to a point P , then P must lie within the prism bounded by the constraint planes which are orthogonal to F and containing the edges in F 's boundary and above F by the... boundary and above F by the de nition of Voronoi region and point-face applicability criterion in Sec 3.2.3 If P fails one applicability constraint, say CE1 imposed by one of F's edges, say E , then E is closer to P than F itself When a point lies beneath a face F and within the prism bounded by other constraint planes, it is possible that P and of course, the object A which contains P  lies inside... is shorter than the distance between V and P This can be seen from Fig.3.8 When a point P lies directly on the bisector CE1 between V and E , then P is equi-distant from both features; else, P is closer to V if P is inside of V 's Voronoi region, and vice versa Therefore, each Point-Vertex Applicability test is guaranteed to generate a pair of features that is closer in distance than the previous pair . h4" alt=""