Circle Shooting in a Simple Polygon

Abstract

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 log 3 n), into a data structure of size O( n log 3 n), so that, for a query circle π, Φ( P , π, s) can be computed in O(log 4 n) 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)log 4( m + n)).