  We consider the $(1+\varepsilon)$-Approximate Nearest Neighbour (ANN) Problem
for polygonal curves in $d$-dimensional space under the Fr\'echet distance and
ask to what extent known data structures for doubling spaces can be applied to
this problem. Initially, this approach does not seem viable, since the doubling
dimension of the target space is known to be unbounded -- even for well-behaved
polygonal curves of constant complexity in one dimension. In order to overcome
this, we identify a subspace of curves which has bounded doubling dimension and
small Gromov-Hausdorff distance to the target space. We then apply
state-of-the-art techniques for doubling spaces and show how to obtain a data
structure for the $(1+\varepsilon)$-ANN problem for any set of parametrized
polygonal curves. The expected preprocessing time needed to construct the
data-structure is $F(d,k,S,\varepsilon)n\log n$ and the space used is
$F(d,k,S,\varepsilon)n$, with a query time of $F(d,k,S,\varepsilon)\log n +
F(d,k,S,\varepsilon)^{-\log(\varepsilon)}$, where
$F(d,k,S,\varepsilon)=O\left(2^{O(d)}k\Phi(S)\varepsilon^{-1}\right)^k$ and
$\Phi(S)$ denotes the spread of the set of vertices and edges of the curves in
$S$. We extend these results to the realistic class of $c$-packed curves and
show improved bounds for small values of $c$.
