Before revision:  Our algorithm runs in O(n^5) time . We also give a data-structure for computing the length of the curve inside arbitrary query disk, known as the length-query problem .
After revision:  Our algorithm runs in O(n^5) time , which is the first exact algorithm for disks. We show using \alpha-fat shapes instead of disks adds a factor \alpha^2 to the approximation. We also give a data-structure for computing the length of the curve inside arbitrary query disk, known as the length-query problem .