On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any $n$ simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least $(1-o(1))n^2$. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let $S$ be a family of the graphs of $n$ continuous real functions defined on $\mathbb{R}$, no three of which pass through the same point. If there are $nt$ pairs of touching curves in $S$, then the number of crossing points is $\Omega(nt\sqrt{\log t/\log\log t})$.Comment: To appear in SODA 201

Tangencies between families of disjoint regions in the plane

AbstractLet C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O(kn), and that this bound cannot be improved. If we only assume that our sets are connected and vertically convex, that is, their intersection with any vertical line is either a segment or the empty set, then the number of tangencies can be superlinear in n, but it cannot exceed O(nlog2n). Our results imply a new upper bound on the number of regular intersection points on the boundary of ⋃C

Homotopy Measures for Representative Trajectories

An important task in trajectory analysis is defining a meaningful representative for a cluster of similar trajectories. Formally defining and computing such a representative r is a challenging problem. We propose and discuss two new definitions, both of which use only the geometry of the input trajectories. The definitions are based on the homotopy area as a measure of similarity between two curves, which is a minimum area swept by all possible deformations of one curve into the other. In the first definition we wish to minimize the maximum homotopy area between r and any input trajectory, whereas in the second definition we wish to minimize the sum of the homotopy areas between r and the input trajectories. For both definitions computing an optimal representative is NP-hard. However, for the case of minimizing the sum of the homotopy areas, an optimal representative can be found efficiently in a natural class of restricted inputs, namely, when the arrangement of trajectories forms a directed acyclic graph

Barycentric systems and stretchability

AbstractUsing a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible

Beyond the Richter-Thomassen conjecture

If two closed Jordan curves in the plane have precisely one point in common, then it is called a touching point-All other intersection points are called crossing points. The main result of this paper is a Crossing Lemma for closed curves: In any family of n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, the number of crossing points exceeds the number of touching points by a factor of fK(loglogn)1/8). As a corollary, we prove the following long-standing conjecture of Richter and Thomassen: The total number of intersection points between any n pairwise intersecting simple closed curves in the plane, no three of which pass through the same point, is at least (1 - o(l))n2