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...

The relative convex hull, or the minimum-perimeter polygon (MPP) of a simple polygon A, contained in a second polygon B, is a unique polygon in the set of nested polygons between A and B. The computation of the minimum-length polygon (MLP), as a special case for isothetic polygons A and B, is useful for various applications in image analysis and robotics. The paper discusses the first recursive...

A simple polyhedron is weakly-monotonic in direction ~ d provided that the intersection of the polyhedron and any plane with normal ~ d is simply-connected (i.e. empty, a point, a line-segment or a simple polygon). Furthermore, if the intersection is a convex set, then the polyhedron is said to be weakly-monotonic in the convex sense. Toussaint [10] introduced these types of polyhedra as genera...

Separating axis theorem is used in a large number of game development, it is mainly used to detect whether two convex polygon product a collision. When the number of sides of a convex polygon is less, separating axis theorem can quickly determine whether it products a collision. However, When the number of edges is increased, the polygon will become more complex and the greater costs will be.A ...

For many software applications, it is sometimes necessary to find the rectangle of largest area inscribed in a polygon, in any possible direction. Thus, given a closed contour C, we consider approximation algorithms for the problem of finding the largest area rectangle of arbitrary orientation that is fully contained in C. Furthermore, we compute the largest area rectangle of arbitrary orientat...

Given a polygon P , a ipturn involves re ecting a pocket p of P through the midpoint of the lid of p. In 1973, Joss and Shannon (published in Gr unbaum (1995)) showed that any polygon on n vertices will become convex after a sequence of at most (n 1)! ipturns. They conjectured that this bound was not tight, and that n=4 ipturns would always be suÆcient. In this work, we show that any polygon o...

We consider the problem of guarding curvilinear art galleries. A closed arc a joining two points, p and q, in the plane is called a convex arc if the curve obtained by joining a with the line segment pq encloses a convex set. A piece-wise convex polygon P with vertices v0, . . . , vn−1 is the region bounded by a set {a0, . . . , an−1} of n convex arcs with pairwise disjoint interiors such that ...

Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be us...

Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12n logn + o(n logn) and the expected number of empty convex four-gons with vertices from S is Θ(n). keywords: random point set, e...

A k-dissection D of a polygon P , is a partition of P into a set of subpolygons {Q1, . . . ,Qm} with disjoint interiors such that these can be reassembled to form k polygons P1, . . . ,Pk all similar to P . D is called non-trivial if none of {Q1, . . . ,Qm} is similar to P . In this paper we show that any convex n-gon has a k-dissection (resp. sequential dissection) with (k − 1)n + 1 pieces, n ...