We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show th...

We study the problem of constructing convex polygons and convex polyhedra given the number of visible edges and visible faces from some orthogonal projections. In 2D, we find necessary and sufficient conditions for the existence of a feasible polygon of size N and give an algorithm to construct one, if it exists. When N is not known, we give an algorithm to find the maximum and minimum size of ...

Let II be a convex lattice polygon with b boundary points and c (5 1) interior points. We show that for any given a , the number b satisfies b 5 2e + 7 , and identify the polygons for which equality holds. A lattice polygon II is a simple polygon whose vertices are points of the integral lattice. We let A = 4(11) denote the area of II , b{U) the number of lattice points on the boundary of II , ...

In this paper, an O ( n log log n ) algorithm is proposed for minimally rectangular partitioning a simple rectilinear polygon. For any simple rectilinear polygon P , a vertex-edge visible pair is a vertex and an edge that can be connected by a horizontal or vertical line segment that lies entirely inside P . We show that, if the vertex-edge visible pairs are found, the maximum matching and the ...

The minimum number of mutually non-overlapping congruent copies of a convex body K so that they can touch K and prevent any other congruent copy of K from touching K without overlapping each other is called the protecting number of K. In this paper we prove that the 1 2 Arnfried Kemnitz, LL aszll o Szabb o, Zoltt an Ujvv ary-Menyhh art protecting number of any regular polygon is three or four, ...

In 2020, Cameron et al. introduced the restricted numerical range of a digraph (directed graph) as tool for characterizing digraphs and studying their algebraic connectivity. Notably, with degenerate polygon (that is, point or line segment) were completely described. this article, we extend those results to include whose is non-degenerate convex polygon. general, refer polygonal. We provide com...