Algorithm for computing positive α-hull for a set of planar closed curves

•Computation for planar closed curves without sampling the curves into point-sets or polylines.•No Voronoi diagram or Delaunay triangulation have been employed.•Results indicate that the algorithm is amenable for implementation.•Complexity, running time, etc. have been discussed.

In this paper, the computation of positive α-hull for a set of planar closed C1-continuous curves has been addressed without sampling the curves into point-sets or polylines. Positive α-hull, so far, has been computed only for a set of points, using the farthest Delaunay triangulation, a dual of farthest Voronoi diagram. However, Delaunay triangulation does not exist for a set of curved boundaries and the computation of Voronoi diagram for such a set is still a topic of active research. The key insight behind our algorithm is to merge adjacent pairs of curves on the convex hull into a set of triplets. Along with a directed-cyclic graph and a R-List (list of radii), α-neighbours are derived. Using the constraint equations, α  -discs are then computed. The algorithm is first provided for convex non-intersecting closed curves, but later explained how it can be generalized for non-convex curves. We show that the algorithm has time complexity of O(n2)O(n2) time where n is the number of curves, which leads to a practical implementation with a reasonable running time in seconds for a few dozen curves. By directly operating on the curves, our method is both robust and accurate thus avoiding the problems that arise on polyline/point-set approximations of the curve networks.

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