Shortest path in a multiply-connected domain having curved ...

More documents

Recommendations

Info

4.5 Termination of a path In the case of SCD, the algorithm terminates, when the latest footpoint of the PCT/BT of the SIP algorithm thus far and the footpoint of the PCT from E lie on a same concave portion. However, as the SIP is not unique in MCD, a similar check is used to compute only the termination of a path. For completeness, the following corollary (Corollary 1) from [18] has been modified for a path termination. Corollary 1 Let CP1 be the latest footpoint of the PCT/BT, which will lie in a concave portion of the curve of the SIP algorithm thus far. Let CP2 be the footpoint of the PCT from E. If CP1 and CP2 lie on a same concave portion, then the path is terminated. The algorithm terminates when Corollary 1 is satisfied for all the path. SIP is then computed using the integration of each path and identifying the least of them. Algorithm 3 gives the pseudo-code for computing the shortest interior path, given two points S and E on the set of curves CurveList (please note that the pseudocode does not include handling C 1 discontinuities). 5 Results and Discussion Fig. 7 show the implementation results for a few test examples, with all potential SIPs (in blue) and SIP (in black) and green indicating the boundaries (outer and inner curves) forming MCD with different start and end points. Implementation in this paper have been carried out using IRIT [8], a solid modeling kernel. The constraint equations for PCT and BT were solved using the geometric constraint solver [9] in IRIT. The algorithm does not limit the number of holes that can be present in the MCD. Also, local spiral-shapes such as the one in Fig. 7(p) are supported. As the algorithm generates all potential SIPs (for eg., all blue paths in Fig. 7(a)), it has the potential to give all path of approximately equal lengths (within a tolerance level, specified by the user). Moreover, using generalized lemma (Lemma 4) eliminates inner loops effectively reducing the number of curves involved in the computation of BTs and inflection points. Fig. 7(o) shows an example where SIP did not pass through any of the inner loops. For the input points S and E not on the outer boundary, a bounding box has been employed to identify regions for eliminating curves and tangents (PCTs/BTs). Figure 8 shows a few results with outer boundary replaced by a box. 5.1 Handling C 1 discontinuities Points of C 1 discontinuity on the curves are identified [15]. From these points and the other curves, PCTs are obtained and processed (as described in Section 4.1). Lines between C 1 discontinuity points are considered as pseudo-Bitangents (pBTs). Initially, pBTs that cuts across the MCD in the neighbourhood of its C 1 discontinuity points are removed. Other processing are done similar to that of BTs. Figure 9 shows a few test results for SIP with curves having C 1 discontinuities. 11
(a) All potential SIPs (b) SIP (c) All potential SIPs (d) SIP (e) All potential SIPs (f) SIP (g) All potential SIPs (h) SIP (i) All potential SIPs (j) SIP (k) All potential SIPs (l) SIP (m) All potential SIPs (n) SIP (o) SIP (p) SIP Figure 7: Test results of the algorithm showing 12all potential shortest paths (in blue) and SIP (in black).
Page 1 and 2: Accepted Manuscript Shortest path i
Page 3 and 4: non-intersecting polygonal obstacle
Page 5 and 6: It is to be noted that when both S
Page 7 and 8: E S (a) Sample set of curves (b) PC
Page 9 and 10: C1 I1 I2 CSM (a) An example of inte
Page 11: (a) Rejected path (in blue) (b) Rej
Page 15 and 16: (a) All potential SIPs (b) SIP (c)
Page 17 and 18: compute the tangency points to use
Page 19 and 20: 7 Appendix: Pseudocode of the algor
Page 21 and 22: S. Bharath Ram is currently a resea

Shortest path in a multiply-connected domain having curved ...

Create successful ePaper yourself

Delete template?

Save as template?