Shortest curves in planar regions with curved boundary

AbstractA general framework is presented for describing shortest curve algorithms and their time complexity in regions of the plane whose boundaries may be curved. An algorithm that accepts curved boundary Jordan regions along with given start and end points and produces the shortest curve between them is presented. Its time complexity is bounded by the product of the complexity of the region's boundary and that of the output shortest curve. (When the region is a simple polygon with N vertices, the time bound is O(Nk), where k is the number of vertices in the shortest curve.) A second algorithm produces shortest curves in multiply connected regions with possibly curved boundary