Before revision:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any two points on P, other than the intersection points, a 2 (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons  , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.
After revision:  If the ratio \frac L \delta is fixed, where L is the total length of P and \delta is the minimum distance between any pair of points on two non-intersecting edges of P, (1+\epsilon)-approximation algorithm exists for disks, and a (1+\epsilon)-approximation algorithm exists for convex polygons , with time complexities O(\log (L/\delta){\epsilon}n^3) and O( \frac{\log (L/\delta){\epsilon}} {\epsilon^{5/2}}} n^3\log n), respectively.