Circle shooting in a simple polygon

Consider the following problem: Given a simple n-gon P, preprocess it so that for a query circle π and a point s on π, one can quickly compute Φ(P, π, s), the first intersection point between P and π as we follow π from s in clockwise direction. We show that P can be preprocessed, in time O(n log³n), into a data structure of size O(n log³n), so that, for a query circle π, Φ(P, π, s) can be computed in O(log⁴n) time. We apply the circle shooting algorithm to report all K intersections between a set of m circular arcs and another set of n circular arcs in time O((m√n + n√m)log^2.5(m + n) + (K + m + n)log⁴(m + n)).