Before revision:  Also, we show that there are no constant-factor approximation algorithms that solve the problem in sub-quadratic timein the worst-case and only uses intersection information to solve the problem .
After revision:  We also give a data-structure for computing the length of the curve inside arbitrary query disk, known as the length-query problem. It has O(n^6\log n) construction time, uses O(n^6) space, and has query time O(\log n+k), where k is the number of intersected segments with the query shape .