Crossing-free segments and triangles in point configurations

Napoleon-like Configurations and Sequences of Triangles

We consider the sequences of triangles where each triangle is formed out of the apices of three similar triangles built on the sides of its predecessor. We show under what conditions such sequences converge in shape, or are periodic.

Crossing Patterns of Segments

Given a system S of simple continuous curves (``strings'') in the plane, we can define a graph GS as follows. Assign a vertex to each curve, and connect two vertices by an edge if and only if the corresponding two curves intersect. GS is called the intersection graph of S. Not every graph is an intersection graph of a system of curves [EET76] (see Fig. 1 for a simple example). This implies that...

A Point in Many Triangles

We give a simpler proof of the result of Boros and Füredi that for any finite set of points in the plane in general position there is a point lying in 2/9 of all the triangles determined by these points. Introduction Every set P of n points in R in general position determines ( n d+1 ) d-simplices. Let p be another point in R. Let C(P, p) be the number of the simplices containing p. Boros and F...

Finding Segments and Triangles Spanned by Points in R3

Visibility Maps of Segments and Triangles in 3D

Let T be a set of n triangles in three-dimensional space, let s be a line segment, and let t be a triangle, both disjoint from T . We consider the visibility map of s with respect to T , i.e., the portions of T that are visible from s. The visibility map of t is defined analogously. We look at two different notions of visibility: strong (complete) visibility, and weak (partial) visibility. The ...