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

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paper, termed as generalized region lemma (GRL) (a generalized version of the region check lemma in [18]) has been proposed. • Exterior and interior regions are formulated based on Lemma 4 to eliminate PCTs/BTs/ inner curves that do not further play a role in path computation. • Merging of paths (which was not required in SIP of SCD) to avoid redundant computation has also been proposed. • A tree data structure has been proposed and implemented for storing and manipulation of identified paths. • A methodology has been proposed and implemented for curves having C 1 discontinuities (which was not handled in SCD). In this paper, the curved boundary is assumed to be defined by a set of non-self-intersecting closed parametric curves without discontinuities (an extension to curves with discontinuities is also discussed with implementation results) and also they do not contain straight line portions. The domain is assumed to be multiply-connected, with one outer boundary (bounding box replacing the outer boundary has also been discussed) and at least one inner boundary. It is assumed that, when traveled along the increasing direction of paramaterization of the curve, its interior lies to its left.The curves are typically defined by Non-Uniform Rational B-Spline (NURBS) [15]. The degree of the curves are assumed such that the inflection points on the curve can be captured (i.e. minimum degree of the curve used is 3). 2 Definitions Let the set of parametric curves be Q = {C0(r0), C1(r1), ... , Coc = Cn(rn)} for which SIP has to be determined. Let S and E lie on the outer boundary curve Coc. All definitions from Section 2 of [18] are directly applicable and hence not repeated here. Definition 1 A path is a set of PCTs/BTs which are completely contained and curve portions forming a contiguous segment. Definition 2 An extended PCT/BT is the extension (along the direction of PCT/BT) of a completely contained PCT/BT till its first intersection with one of the input curves. By virtue of this definition, the extended PCT/BT is also completely contained. Definition 3 SIP is the shortest among all available paths from S to E. 3 Preliminaries/Observations Lemma 1 SIP consists of only concave portions of outer/inner loops and straight lines that are completely contained in the MCD (adopted from [18]). Lemma 2 The SIP preserve the direction of traversal (clockwise or counter-clockwise) in the input curves [18]. 3
It is to be noted that when both S and E lie on the same concave portion, then the SIP is the portion of the curve C(t) between the two points [[3], Theorem 2.7, p220]. Also, straight portions of SIP are tangential to the concave portions [18]. Definition 4 A BT is said to be consistent with respect to a path if it preserves the direction of traversal of the path as stated in Lemma 2 . It is also to be noted that, only one SIP is possible for a SCD, joining two distinct points, lying on the boundary or interior [[21], Theorem 6.1, p147], which does not hold good for a MCD. Suppose that there is an SIP for a set curves constituting MCD. If an inner loop is added, it should be observed that the entire SIP can change altogether. This suggests that an incremental construction may not be possible and one has to consider all possible inner loops at the same time. Moreover, local shortest path cannot be fine tuned to get the global path. For example, a shortest path between two inner loops may not play a role at all in the global shortest path between S and E (even-though those two curves may contribute to the final path). Hence, all possible paths have to be detected to compute SIP. Lemma 3 indicates the interaction that a path can have and the cases for which the path can be eliminated, reducing the number of candidates and also computation of further BTs for identifying SIP. Lemma 3 (Path Elimination): Let there be a path. Let there be extended PCTs from S or extended BTs consistent with the direction of traversal of the path. The path is eliminated if it intersects any extended PCT from S or extended BT originating from the same concave portion through which the path has passed. Proof Let Fi represent the starting footpoint of BTi. Let η consist of the PCT, concave portion between T and F1, and BT1 itself (refer to Figure 1(a)). Let the the bitangents BT2, and BT3 arise out of the same concave portion as that of BT1. Let EBT2 and EBT3 represent the intersection point of a path with their respective extended BTs. Case 1: If η intersects any extended PCT from S at some point, say X, then the portion of η till X can be replaced by the straight line portion SX which would make the path shorter and hence η cannot be a potential SIP. Case 2: Let the footpoint of a bitangent lies between T and F1 (such as F2 of BT2). Let the path ζ = η + ζa and ζa intersects extended BT2 at EBT2. Then the cumulative portion formed by concave portion of F2F1, BT1 and ζa can be replaced by straight line portion from F2 to EBT2 to get a shorter path. Hence ζ can be removed. Case 3a: Let the footpoint of a BT lies outside of T and F1 (such as F3 of BT3). Let the path ζ = η + ζc. BT1 + ζc is longer than the path (F1F3 + F3 to EBT3) by the convex hull property. Hence ζ can be removed. Case 3b: Let the path ζ = η + ζb. The path ζ has self-intersection and hence it can be eliminated. Lemma 4 Let ζ be a path from S to E. Let G ∈ ζ be a point on a concave portion of one of the input curves. Let SG and GE denote the portions of ζ before and after G respectively. 4
Page 1 and 2: Accepted Manuscript Shortest path i
Page 3: non-intersecting polygonal obstacle
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 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
Page 21 and 22: S. Bharath Ram is currently a resea

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