  A geometrical pattern is a set of points with all pairwise distances (or,
more generally, relative distances) specified. Finding matches to such patterns
has applications to spatial data in seismic, astronomical, and transportation
contexts. For example, a particularly interesting geometric pattern in
astronomy is the Einstein cross, which is an astronomical phenomenon in which a
single quasar is observed as four distinct sky objects (due to gravitational
lensing) when captured by earth telescopes. Finding such crosses, as well as
other geometric patterns, is a challenging problem as the potential number of
sets of elements that compose shapes is exponentially large in the size of the
dataset and the pattern. In this paper, we denote geometric patterns as
constellation queries and propose algorithms to find them in large data
applications. Our methods combine quadtrees, matrix multiplication, and
unindexed join processing to discover sets of points that match a geometric
pattern within some additive factor on the pairwise distances. Our distributed
experiments show that the choice of composition algorithm (matrix
multiplication or nested loops) depends on the freedom introduced in the query
geometry through the distance additive factor. Three clearly identified blocks
of threshold values guide the choice of the best composition algorithm.
Finally, solving the problem for relative distances requires a novel
continuous-to-discrete transformation. To the best of our knowledge this paper
is the first to investigate constellation queries at scale.
