On reconstructing n-point configurations from the distribution of distances or areas

One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations. In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples.Comment: 21 pages, late

Lossless Representation of Graphs using Distributions

We consider complete graphs with edge weights and/or node weights taking values in some set. In the first part of this paper, we show that a large number of graphs are completely determined, up to isomorphism, by the distribution of their sub-triangles. In the second part, we propose graph representations in terms of one-dimensional distributions (e.g., distribution of the node weights, sum of adjacent weights, etc.). For the case when the weights of the graph are real-valued vectors, we show that all graphs, except for a set of measure zero, are uniquely determined, up to isomorphism, from these distributions. The motivating application for this paper is the problem of browsing through large sets of graphs.Comment: 19 page

Path Tracking using Echoes in an Unknown Environment: the Issue of Symmetries and How to Break Them

This paper deals with the problem of reconstructing the path of a vehicle in an unknown environment consisting of planar structures using sound. Many systems in the literature do this by using a loudspeaker and microphones mounted on a vehicle. Symmetries in the environment lead to solution ambiguities for such systems. We propose to resolve this issue by placing the loudspeaker at a fixed location in the environment rather than on the vehicle. The question of whether this will remove ambiguities regardless of the environment geometry leads to a question about breaking symmetries that can be phrased in purely mathematical terms. We solve this question in the affirmative if the geometry is in dimension three or bigger, and give counterexamples in dimension two. Excluding the rare situations where the counterexamples arise, we also give an affirmative answer in dimension two. Our results lead to a simple path reconstruction algorithm for a vehicle carrying four microphones navigating within an environment in which a loudspeaker at a fixed position emits short bursts of sounds. This algorithm could be combined with other methods from the literature to construct a path tracking system for vehicles navigating within a potentially symmetric environment