Shortest paths help solve geometric optimization problems in planar regions
SIAM Journal on Computing
Shortest curves in planar regions with curved boundary
Theoretical Computer Science
Shortest paths among obstacles in the plane
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
The NURBS book
Computing the visibility graph via pseudo-triangulations
Proceedings of the eleventh annual symposium on Computational geometry
Geometric constraint solver using multivariate rational spline functions
Proceedings of the sixth ACM symposium on Solid modeling and applications
Rapid and accurate computation of the distance function using grids
Journal of Computational Physics
Shortest path amidst disc obstacles is computable
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Shape Classification Using the Inner-Distance
IEEE Transactions on Pattern Analysis and Machine Intelligence
Industrial geometry: recent advances and applications in CAD
Computer-Aided Design
Computing shortest paths amid pseudodisks
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Computing the shortest path, overcoming obstacles in a plane, is a well-known geometric problem. However, widely assumed obstacles are polygonal in nature. Very few papers have focused on curved obstacles, and in particular, for curved multiply-connected domains (domains having holes). Given a set of parametric curves forming a multiply-connected domain (MCD), with one closed curve acting as an outer boundary and several non-intersecting inner curves (loops) as representing holes and two distinct points S and E lying on the outer boundary, this paper provides an algorithm to find the shortest interior path (SIP) between the two points in the domain. SIP consists of portions of curves along with straight line segments that are tangential to the curve. The algorithm initially computes point-curve tangents (PCTs) and bitangents (BTs) using their respective constraints. A generalized region lemma is proposed, which is then employed to remove PCTs/BTs that will not contribute to the potential path and subsequently aiding in removing a few inner loops. The algorithm is designed to explore all potential paths. Merging of paths is also proposed to avoid redundant computations. A final SIP is chosen from potential paths using the length of each path. The algorithm also has the potential to give all paths of equal lengths contributing to shortest paths (within a tolerance level). As the algorithm computes all the potential paths on the fly, there is no need to employ a visibility graph to compute the shortest path. Curves are represented using non-uniform rational B-splines (NURBS). The algorithm uses the curves as such and does not discretize into point-sets or polygons. Extensions to handle curves with C^1 discontinuities and S and E not on the outer boundary have also been described. Results of the implementation are provided and complexity of the algorithm is also discussed. This paper follows up the one presented for simply-connected domains (SCDs) in Bharath Ram and Ramanathan (2011) [4].