Efficient Collision Detection for Animation and Robotics Part 7 ppsx

[...]... the performance of this approach by realizing that we are only interested in one coordinate value of vertices for each update, say x coordinate while updating the minimum or maximum value in x-axis Therefore, there is no need to transform the other coordinates, say y and z values, in updating the x extremal vertices during the comparison with neighboring vertices Therefore, we only need to perform 24... minimum and maximum in x; y; z, coordinates and each update involves 4 vectorvector multiplications, assuming each vertex has 3 neighboring vertices. For concave and articulated bodies, we need to use a hierarchical bounding box structure, i.e a tree of bounding boxes Before the top level bounding boxes collide, there is no need to impose a bounding volume on each subpart or each link Once the collision. .. performing this veri cation process recursively, we can recompute the bounding boxes at expected constant rate Once again, we are exploiting the temporal and geometric coherence and the locality of convex polytopes We only update the bounding volumes for the moving objects Therefore, if all objects are moving around especially in a rather unpredictable way, it's hard to preserve the coherence well and. .. sort and sweep" techniques and a geometric data structure to quickly perform intersection test on the real intervals to reduce the number of pairwise comparison 84 5.2.2 One-Dimensional Sort and Sweep In computational geometry, there are several algorithms which can solve the overlapping problem for d-dimensional bounding boxes in Onlogd, n + s time where s is the number of pairwise overlaps 31, 47, ... pre-sort the minimum and maximum values of each object along the x; y; z,axis as if we are imposing a virtual bounding box hierarchy on each body, sweep through each nearly sorted list every time step and update the list of overlapping intervals as we mentioned before If the environment is sparse and the motions between time frames are smooth", we 1 2 85 expect the extra e ort to check for collision will... make a tree structure dynamic and exible for quick insertion and deletion of a higher dimensional boxes Nevertheless, for more dense or special environments such as a mobile robot moving around in a room cluttered with moving obstacles, such as people, it is more e cient to use an interval tree for 2-dimensional intersection tests to reduce the number of pairwise checks for overlapping We can signi... list and annotate each node if it or any of its descendant contains 1 1 1 1 +1 p+ p 2 86 stored intervals The embellished tree is called the interval tree We can use the interval tree for static query, as well as for the rectangle intersection problem To check for rectangle intersection using the sweep algorithm: we take a sweeping line parallel to the y axis and sweep in increasing x direction, and. .. suitable for an environment where only a few objects are moving while most of objects are stationary, e.g a virtual walk-through environment 5.2.3 Interval Tree for 2D Intersection Tests Another approach is to extend the one-dimensional sorting and sweeping technique to higher dimensional space However, as mentioned earlier, the time bound will be worse than On for two or three-dimensional sort and sweep... coherence Let a one-dimensional bounding box be b; e where b and e are the real numbers representing the beginning and ending points To determine all pairs of overlapping intervals given a list of n intervals, we need to verify for all pairs i and j if bi 2 bj ; ej or bj 2 bi; ei , 1  i j  n This can be solved by rst sorting a list of all bi and ei values, from the lowest to the highest Then, the list... checked by the one-dimensional sort and sweep Here we will brie y describe the data structure of an interval tree and how we use it for intersection test of 2-dimensional rectangular boxes An interval tree is actually a range tree properly annotated at the nodes for fast search of real intervals Assume that n intervals are given, as b ; e ;


; bn; en where bi and ei are the endpoints of the interval . x0 y4 w1 h7" alt=""