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

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I4 C3 I3 P3 P1 C5 C1 T1 (a) PCTs Extended to get first intersection S I5 (b) Regions R1 and R2 for elimination of PCTs/Inner curves Figure 3: Processing PCTs using their extensions boundaries. The other by T IF E and the portion of outer curve (that does not contain E) from IF E to T as boundaries. 2. If CT G is not COC and IF E is on COC, then the region is defined by the boundary curves SIF E and the portion of outer boundary (that does not contain E) from IF E to S (in Figure 3(b), region R1 with boundary SI5S). 3. If CT G is COC and IF E not on COC, then the region is defined by the boundary curves ST and the portion of outer curve (that does not contain E) from T to S . For e.g., in Figure 3(b), region R2 defined by the closed boundary ST6S. It is to be noted that the first intersection of ST6 is at I3 with the curve C3 (Figure 3(a)). A similar search is done in the counter-clockwise direction and CLRP CT (clockwise region PCT) is obtained. Also corresponding regions are identified. 4.1.2 Interior region identification Interior regions are formed when footpoints of PCTs and/or extended PCTs first intersection points are falling on the same curve, but not on COC. The curve portion between the mutually farthest points on the curve and their respective PCT/extended PCT form a region. For example, in Figure 4(a), several extended PCTs (a different starting point W for the PCTs is used for illustrative purpose) have IF E’s on the same curve CSM. The interior region is formed by W I1I2W , where I1 and I2 are the mutually farthest points along the curve CSM. In this particular case, I1 and I2 are the first intersections of the extended PCTs of the curves C1 and C2 respectively. 7 R1 T6 T5 R2 S
C1 I1 I2 CSM (a) An example of interior region elimination C2 W 4.1.3 Region-based elimination E E (b) PCTs from S after elimination Figure 4: Valid PCTs from S, E and interior region S (c) Termination list from E Once the regions are identified, all PCTs and curves that are contained in the regions are eliminated (Lemma 4). In order to get the PCTs within the regions, the values counterclockwise angle with respect to the SE line or parameter values of IOC are used. Since the PCTs are from the same starting point, the angle and the parameter of IOC correlate. Firstly a range of parametric values is identified using the PCTs that contribute to the identification of regions. Then tangents falling within or outside the range are eliminated accordingly. For curve elimination within a region, typical intersection check followed by ray shooting is done. Figure 4(b) and 4(c) show the remaining PCTs from S and E after employing region-based elimination. Inner loops that are eliminated using exterior regions R1 and R2 are shown as gray in Figure 3(b). Algorithm 1 describes the pseudocode for obtaining PCTs and processing them. 4.2 Processing BTs BTs are computed from each PCT in the starting list (which contains thus far processed PCTs, Figure 4(b)). A similar region elimination strategy is followed for BTs with some modifications to the region identification. Consider a path ζ, shown in Figure 5(a) from S to a point T on a curve CT G, which is an incomplete potential SIP (i.e., the path has not yet reached the end point E). At and near T , CT G is concave. Let φ (clockwise or counterclockwise) be the direction induced by ζ on CT G. Let NIF be the closest inflection point to T on CT G in the direction φ. Now T and NIF identify a concave portion of CT G. To identify the next set of potential paths for ζ, the set of bitangents from concave portion T NIF is computed. The bitangents that are not completely contained in MCD and those that are not consistent with direction φ are removed. Let the starting foot point of each bitangent be denoted by FST . The starting curve, CST , for all bitangents is CT G. Let the end foot point and corresponding end curve be denoted by FEN and CEN. Each bitangent is extended from 8
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: E S (a) Sample set of curves (b) PC
Page 11 and 12: (a) Rejected path (in blue) (b) Rej
Page 13 and 14: (a) All potential SIPs (b) SIP (c)
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
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