Given a set S of n points in the plane, an -strongly convex -hull of S is de0ned as a convex polygon P with the vertices taken from S such that no point of S lies farther than outside P and such that even if the vertices of P are perturbed by as much as , P remains convex. This paper presents the 0rst parallel robust method for this generalized convex hull problem (note that the convex hull of ...

A convex hull generated by a sample uniformly distributed on the plane is considered in case when support of distribution polygon. central limit theorem proved for joint number vertices and area using Poisson approximation binomial point processes near boundary distribution. Here we apply results hulls given [6]. From result obtained present paper, particular, follow [3, 7], polygon homogeneous...

We show that the largest similar copy of a convex polygon P with m edges inside a convex polygon Q with n edges can be computed in O mn logn time We also show that the combinatorial complexity of the space of all similar copies of P inside Q is O mn and that it can also be computed in O mn logn time Let P be a convex polygon with m edges and Q a convex polygon with n edges Our goal is to nd the...

In a 1751 letter to Christian Goldbach (1690–1764), Leonhard Euler (1707–1783) discusses the problem of counting the number of triangulations of a convex polygon. Euler, one of the most prolific mathematicians of all times, and Goldbach, who was a Professor of Mathematics and historian at St. Petersburg and later served as a tutor for Tsar Peter II, carried out extensive correspondence, mostly ...

Let P be a polygon with r > 0 reflex vertices and possibly with holes. A subsuming polygon of P is a polygon P ′ such that P ⊆ P ′, each connected component R′ of P ′ subsumes a distinct component R of P , i.e., R ⊆ R′, and the reflex corners ofR coincide with the reflex corners ofR′. A subsuming chain of P ′ is a minimal path on the boundary of P ′ whose two end edges coincide with two edges o...

A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is a convex polygon. A minimum convex partition with respect to S is a convex partition of S such that the number of convex polygons is minimised. In this paper, we wil...

Associated to a convex integral polygon $N$ is cluster integrable system $\mathcal X\_N$ constructed from the dimer model. We compute group $G\_N$ of symmetries X\_N$, called (2-2) modular group, showing that it certain abelian conjectured by Fock and Marshakov. Combinatorially, non-torsion elements are ways shuffling underlying bipartite graph, generalizing domino-shuffling. Algebro-geometrica...

So far, the best result in running time for solving the “floating” watchman route problem (WRP) is O(n log n); the best result in running time for solving the “fixed” safari route problem (SRP) is O(n log n) published in 2003 by M. Dror, A. Efrat, A. Lubiw, and J. Mitchell. The best result in running time for solving the “floating” zookeeper route problem (ZRP) is O(n) published in 2001 by X. T...