Before revision:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right) , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.
After revision:  Concretely}\left({k\epsilon}}\right) + \mathcal{O}(nm) and has query time in \mathcal{O}}\left(\right). As an indicator of hardness , we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O}(L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}.