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

Abstract

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.

Which Point Configurations Are Determined by the Distribution of Their Pairwise Distances?

“…For n = 1, 2 or 3, it is easy to see that the distribution of distances completely characterize the n-point configuration up to a rigid motion. For n ≥ m + 2, we proved that, most of the time, this distribution completely characterizes the shape of the point configuration (see [4,Theorem 2.6]). …”

“…For close enough point configurations, we have proved in [4] that one does not need to keep track of the labeling of the points. The proof is very short and we reproduce it here for completeness.…”

“…The answer is yes. Some examples can be found in Boutin and Kemper [4]. Fortunately, non-reconstructible configurations are rare.…”

“…In fact, our guess is that this problem lies in the complexity class NP; it might even be NP-complete. Theorem 2.6 of Boutin and Kemper [4] actually implies that there exists an open and dense subset Ω ⊂ (R m ) n of reconstructible point configurations. In Section 1, we concentrate on the planar case m = 2 and give an algorithm in O(n 11 ) steps to determine whether a point lies in Ω.…”

Lossless Shape Representation using Invariant Statistics: the Case of Point-sets

“…Kemper and Boutin [4] have proposed to use the bag of distances of a point-set to represent its shape, i.e. to represent the point-set up to a global rotation, reflection and translation.…”