Minimum area enclosure and alpha hull of a set of freeform planar closed curves

Of late, researchers appear to be intrigued with the question; Given a set of points, what is the region occupied by them? The answer appears to be neither straight forward nor unique. Convex hull, which gives a convex enclosure of the given set, concave hull  , which generates non-convex polygons and other variants such as αα-hull, poly hull, rr-shape and ss-shape etc. have been proposed. In this paper, we extend the question of finding a minimum area enclosure (MAE) to a set of closed planar freeform curves, not resorting to sampling them. An algorithm to compute MAE has also been presented. The curves are represented as NURBS (non-uniform rational B-splines). We also extend the notion of αα-hull of a point set to the set of closed curves and explore the relation between alpha hull (using negative alpha) and the MAE.

► This paper addresses region occupation to a set of closed planar freeform curves.
► Minimum area enclosure (MAE) has been proposed and implemented using curves exactly.
► MAE’s relation to αα-hull of a set of curves has also been explored.
► Comparison with discretization-based approach has been performed.
► Applications of MAE are also discussed.