Let $$C_{k_1}, \ldots , C_{k_n}$$ be cycles with $$k_i\ge 2$$ vertices ( $$1\le i\le n$$ ). By attaching these n together in a linear order, we obtain graph called polygon chain. cyclic graph, which is ring if it can embedded on the plane; and twisted Möbius band. It known that sandpile group of chain always cyclic. Furthermore, there exist edge generators. In this paper, not only show any (twi...

Visibility computation is a classical problem in computer graphics. A wide variety of algorithms provides solutions with a different accuracy. However, the four dimensional nature of the 3D visibility has prevented for a long time from leading to exact from-polygon visibility algorithms. Recently, the two first tractable solutions were presented by Nirenstein, then Bittner. Their works give the...

Triangulation is one of the most popular methods for decomposing a planar polygon into primitive cells. Often trapezoidation is performed as a first step in triangulation. That is, a polygon is decomposed into a set of trapezoids; a trapezoid being a four sided polygon with two parallel sides. Although much work has gone into fast triangulation methods, there has been little work on trapezoidat...

Given a small polygon S, a big simple polygon B and a positive integer k, it is shown to be NP-hard to determine whether k copies of the small polygon (allowing translation and rotation) can be placed in the big polygon without overlap. Previous NP-hardness results were only known in the case where the big polygon is allowed to be non-simple. A novel reduction from Planar-Circuit-SAT is present...

In this paper we introduce the notion of general position. We also show some auxiliary theorems for proving Jordan curve theorem. The following main theorems are proved: 1. End points of a polygon are in the same component of a complement of another polygon if number of common points of these polygons is even; 2. Two points of polygon L are in the same component of a complement of polygon M if ...