Determining Generic Point Configurations From Unlabeled Path or Loop
Lengths
release_hamflzvdqrdj5fo5fz2xwi5loy
by
Ioannis Gkioulekas, Steven J. Gortler, Louis Theran, Todd Zickler
2017
Abstract
Let p be a configuration of n points in R^d for some
n and some d > 2. Each pair of points defines an edge, which has a
Euclidean length in the configuration. A path is an ordered sequence of the
points, and a loop is a path that has the same endpoints. A path or loop, as a
sequence of edges, also has a Euclidean length. In this paper, we study the
question of when p will be uniquely determined (up to an unknowable
Euclidean transform) from a given set of path or loop lengths. In particular,
we consider the setting where the lengths are given simply as a set of real
numbers, and are not labeled with the combinatorial data describing the paths
or loops that gave rise to the lengths.
Our main result is a condition on the set of paths or loops that is
sufficient to guarantee such a unique determination. We also provide an
algorithm, under a real computational model, for performing a reconstruction of
p from such unlabeled lengths.
To obtain our results, we introduce a new family of algebraic varieties which
we call the unsquared measurement varieties. The family is parameterized by the
number of points n and the dimension d, and our results follow from a
complete characterization of the linear automorphisms of these varieties for
all n and d. The linear automorphisms for the special case of n = 4 and
d = 2 correspond to the so-called Regge symmetries of the tetrahedron.
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