  A two-dimensional grid with dots is called a \emph{configuration with
distinct differences} if any two lines which connect two dots are distinct
either in their length or in their slope. These configurations are known to
have many applications such as radar, sonar, physical alignment, and
time-position synchronization. Rather than restricting dots to lie in a square
or rectangle, as previously studied, we restrict the maximum distance between
dots of the configuration; the motivation for this is a new application of such
configurations to key distribution in wireless sensor networks. We consider
configurations in the hexagonal grid as well as in the traditional square grid,
with distances measured both in the Euclidean metric, and in the Manhattan or
hexagonal metrics.
  We note that these configurations are confined inside maximal anticodes in
the corresponding grid. We classify maximal anticodes for each diameter in each
grid. We present upper bounds on the number of dots in a pattern with distinct
differences contained in these maximal anticodes. Our bounds settle (in the
negative) a question of Golomb and Taylor on the existence of honeycomb arrays
of arbitrarily large size. We present constructions and lower bounds on the
number of dots in configurations with distinct differences contained in various
two-dimensional shapes (such as anticodes) by considering periodic
configurations with distinct differences in the square grid.
