In this thesis we study various combinatorial problems relating to the geometry of point sets in the Euclidean plane. The unifying theme is that all the problems involve point sets that are not in general position, but have some collinearities. As well as giving rise to natural and interesting problems, the study of point sets with collinearities has important connections to other areas of math...

Given a collection of objects in the plane along with a viewpoint !, the visibility problem involves determining the portion of each object that is visible to an observer positioned at !. The visibility problem is central to various application areas including computer graphics , image processing, VLSI design, and robot navigation, among many others. The main contribution of this work is to pro...

An obstacle representation of a graph G is a set of points on the plane together with a set of polygonal obstacles that determine a visibility graph isomorphic to G. The obstacle number of G is the minimum number of obstacles over all obstacle representations of G. Alpert, Koch, and Laison [1] gave a 12-vertex bipartite graph and proved that its obstacle number is two. We show that a 10-vertex ...

In this paper, a method for matching symbols in line-drawings is presented. Facing both segmentation and recognition of symbols is a difficult challenge. Starting from the results of a vectorization procedure, a visibility graph is built to enhance the main geometric constraints which were specified during the construction of the initial document. The cliques detection, which correspond to a pe...

This paper gives an introduction to the problem of mapping simple polygons with autonomous agents. We focus on minimalistic agents that move from vertex to vertex along straight lines inside a polygon, using their sensors to gather local observations at each vertex. Our attention revolves around the question whether a given configuration of sensors and movement capabilities of the agents allows...

The problem of generating \random" geometric objects is motivated by the need to generate test instances for geometric algorithms. We examine the speciic problem of generating a random x-monotone polygon on a given set of n vertices. Here, \random" is taken to mean that we select uniformly at random a polygon, from among all those x-monotone polygons having the given n vertices. We give an algo...

Let S be a collection of n non-intersecting line segments in the plane in general position. Two segments u and v are deened as visible if a line segment could be drawn from some point of u to some point of v that intersects no other segment in S. The full visibility graph associated with S is denoted as G(S), and deened to be the graph whose n vertices correspond to the line segments of S and w...

Imaging the vicinity of black hole is one of the ultimate goals of VLBI astronomy. The closest massive black hole, SgrA*, located at Galactic center is the leading candidate for such observations. Combined with recent VLBI recording technique and sub-mm radio engineering, we now have the sufficient sensitivity for the observations. We here show performance simulations of sub-mm VLBI arrays for ...

The challenge of estimating the crystalline-fraction of dense randomly-oriented nano-particles, contained in a (less-dense) amorphous support, is increasingly important in fields as diverse as heterogeneous-catalyst design and electronic-data storage. Lattice-fringe visibility-analysis [1,2] is a potential tool in this context, but of course interfering contrast from the amorphous support (as w...

We discuss rectangle and box visibility representations of graphs in 3-dimensional space. In these representations, vertices are represented by axis-aligned disjoint rectangles or boxes. Two vertices are adjacent if and only if their corresponding boxes see each other along a small axis-parallel cylinder. We concentrate on lower and upper bounds for the size of the largest complete graph that c...