A Polynomial Solution For Potato-Peeling And Other Polygon Inclusion And Enclosure Problems

Minimum area circumscribing Polygons

“…The algorithm as presented in this paper depends on Lemma I [4]. However, it can be shown that the same time complexity can also be obtained from the following weakening of DePano's result: There exists a globally minimum k-gon that encloses a given n-gon P such that all its sides are balanced and at least (k-2) sides are flush with P. This result can be deduced from the lemmas in [3]. It is not hard to further show that the two possibly non-flushed edges (called "V-edges" in [3] and "contracting edges" in [4] must be adjacent to each other.…”

“…We will improve the previous solution by using an observation that was first used by Chang and Yap [3]. For all h between 1 and k-3, the above procedure computed all h-sided optimal flush chains.…”

“…For instance, see [1,3,6,7,8,9,10,12]. Potential applications of these algorithms include areas such as robotics [2] and computer aided design [8].…”

“…(Here, and throughout the paper, minimum or maximum is with respect to area.) Chang and Yap [3] used a dynamic programming approach to compute a minimum circumscribing convex k-gon in O(nalogk) time 1. DePano and Aggarwal [5] restricted the problem and provided efficient algorithms for computing a minimum equiangular as well as a regular k-gon that circumscribes a given n-gon.…”

On Finding Large Empty Convex Bodies in 3D Scenes of Polygonal Models

“…For instance, even for the problem of finding the largest area convex polygon inside a simple polygon (the so-called "potato peeling problem"), the best known solution runs in time O(n 7 ) [11], which seems to be useless for any practical application. We are not aware of existing provable results for the analogous problem in three dimensions.…”