We study the relationship between the interior angles of a convex polygon and the lengths of external watchman routes.

We introduce the notion of free polygons as combinatorial building blocks for convex integral polygons; that is, polygons with vertices having integer coordinates. In this context, an Euler-type formula is derived for the number of integer points in the interior of an integral polygon. This leads in turn to a formula for the area of an integral polygon P via the enumeration of free integral tri...

We present two algorithms for packing two largest disks in a polygon. The first algorithm locates two disks in a simple polygon in time O(n log2 n) improving the best previous deterministic result (Bespamyatnikh, 1999) by a factor of logn. The second algorithm finds two disks in a convex polygon such that the disks are separated by a diagonal of the polygon. It runs in time O(n log2 n) improvin...

We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then

We consider the problem of nding a corridor (a separating strip) between two polygons, whose intersection with a third (convex) polygon is of maximum area. The application in mind is the interpolation in simple branching cases, where the sought volume branches from one contour in one slice into two polygons in another parallel slice. We present a linear-time plane-sweep algorithm which computes...

In this paper, we present approximation algorithms for the problem of cutting out a convex polygon P with n vertices from another convex polygon Q with m vertices by a sequence of guillotine cuts of smallest total length. Specifically, we give an O(n + m) running time, constant factor approximation algorithm, and an O(n+m) running time, O(log n)-factor approximation algorithm for cutting P out ...

Efficient cutting strategy of a geometric shape, P out of another geometric shape, Q has been extensively studied in recent years. A number of variations of this problem has been addressed including P being convex or concave, cuts being line cut or ray cut, Q being a polygon or a circle etc. In this paper we give a simple linear time O (log n)-approximation algorithm for the problem where Q is ...

We show how to compute the smallest rectangle that can enclose any polygon, from a given set of polygons, in nearly linear time; we also present a PTAS for the problem, as well as a linear-time algorithm for the case when the polygons are rectangles themselves. We prove that finding a smallest convex polygon that encloses any of the given polygons is NP-hard, and give a PTAS for minimizing the ...

Motion planning is studied in a time-varying environment. Each obstacle is a convex polygon that moves in a fixed direction at a constant speed. The robot is a convex polygon that is subject to a speed bound. A method is presented to determine whether or not there is a translational collision-free motion for a polygonal robot from an initial position to a final position, and to plan such a moti...