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 nonconvex polygons and other variants such as α-hull, poly hull, r-shape and s-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 Bsplines). We also extend the notion of α-hull of a point set to the set of closed curves and explore the relation between alpha hull (using negative alpha) and the MAE.