We show that the visibility graph of a set of non-intersecting translates of the same compact convex object in R always contains a Hamiltonian path. Furthermore, we show that every other edge in the Hamiltonian path can be used to obtain a perfect matching that is realized by a set of non-intersecting lines of sight. ? 2001 Elsevier Science B.V. All rights reserved.

We discuss elliptic Plücker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for “classical” elliptic 3spaces a bijection of its lines is already a Plücker transformation, if related lines go over to related l...

Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in R3 with a total of n edges consists of Θ(n2) connected components in the worst case. In the generic case, each connected component is a ...

In this talk, we discuss the results of investigations that began with a solution to an open problem posed by Schwabhäuser and Szczerba regarding definability (without parameters) in the three dimensional Euclidean geometry of lines, asking whether intersection was definable from perpendicularity (two lines intersecting at a right angle). It is not. The result is a “new” 3-dimensional geometry ...

Abstract We investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field characteristic 3. prove that if contains 112 lines, then is projectively equivalent to Fermat surface; otherwise, at most 67 lines. improve this bound 58 star (ie four distinct intersecting smooth point ). Explicit equations three 1-dimensional families surfaces wi...

The following theorem is proved. The collection of lines of an n-uniform projective Hjelmslev plane is maximal when considered as a collectiion of mutually intersecting sets of equal cardinality.

Suppose one has a line arrangement and one wants to find a shortest path from one point lying on a line of the arrangement to another such point. We look at a special case: the arrangement consists of two intersecting pencils (sets of lines where all intersect in a point), and the path endpoints are at opposite corners of the largest quadrilateral formed. The open problem was to find a shortest...

Mathematics: algebra of intersecting straight lines; trigonometry of 30-60-90-degree triangles; vector addition. Geology: size, shape and gravity of the Earth; nomenclature of sedimentary rocks. Quantitative literacy: visual display of data; National Assessment of Educational Progress.

We consider a vertex model in d dimensions characterized by lines which run in a preferred direction. We show that this vertex model is soluble if the weights of vertices with intersecting lines are given by a free-fermion condition, and that a fugacity −1 is associated to each loop of lines. The solution is obtained by mapping the model into a dimer problem and by evaluating a Pfaffian. We als...