  In this paper we present algorithms for a number of problems in geometric
pattern matching where the input consist of a collections of segments in the
plane. Our work consists of two main parts. In the first, we address problems
and measures that relate to collections of orthogonal line segments in the
plane. Such collections arise naturally from problems in mapping buildings and
robot exploration.
  We propose a new measure of segment similarity called a \emph{coverage
measure}, and present efficient algorithms for maximising this measure between
sets of axis-parallel segments under translations. Our algorithms run in time
$O(n^3\polylog n)$ in the general case, and run in time $O(n^2\polylog n)$ for
the case when all segments are horizontal. In addition, we show that when
restricted to translations that are only vertical, the Hausdorff distance
between two sets of horizontal segments can be computed in time roughly
$O(n^{3/2}{\sl polylog}n)$. These algorithms form significant improvements over
the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the
second part of this paper we address the problem of matching polygonal chains.
We study the well known \Frd, and present the first algorithm for computing the
\Frd under general translations. Our methods also yield algorithms for
computing a generalization of the \Fr distance, and we also present a simple
approximation algorithm for the \Frd that runs in time $O(n^2\polylog n)$.
