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Efficient Collision Detection for Animation and Robotics Part 8

Tham khảo tài liệu 'efficient collision detection for animation and robotics part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 94 CO Figure 6.1 A schematized 2-d con guration space and the partition of free space into a1 -channels. 95 another freeway at its endpoints i.e. if c and C b c are two adjacent channels the b endpoint of a freeway of c a b should meet an endpoint of a freeway of C b c Technically since the intervals are open the word endpoint should be replaced by limit point In general when a sped c method of computing freeway curves is chosen there may be several freeways within one channel. For example in the rest of this chapter freeways are de ned using arti cial potential functions which are directly proportional to distance from obstacles. In this case each freeway is the locus of local maxima in potential within slices Fp v of FP as V varies. This locus itself may have some critical points but as we shall see the freeway curves can be extended easily past them. Since there may be several local potential maxima within a slice we may have several disjoint freeway curves within a single channel but with our incremental roadmap construction this is perfectly OK. Now to bridges. A bridge is a one-dimensional set which links freeways from channels that have just joined or are about to split as V increases . Suppose two channels Cl and c 2 have joined into a single channel c 3 as shown in Fig. 6.2. We know that the freeways of Cl and c 2 will continue into two freeway curves in C3. These freeways within c 3 are not guaranteed to connect. However we do know that by de nition c 3 is connected in the slice slice Xi V through the critical point so we add linking curves from the critical point to some freeway point in each of c and c 2. It does not matter which freeway point because the freeway curves inside the channels c and c 2 must be connected within each channel as we show in Sec. 6.5. By adding bridges we guarantee that whenever two channels meet some points on their freeways are connected. Once we can show that whenever channels meet their freeways do also via bridges we have