Weak visibility queries of line segments in simple polygons

Given a simple polygon P in the plane, we present new data structures for computing the weak visibility polygon from any query line segment in P  . We build a data structure in O(n)O(n) time and space that can compute the visibility polygon for any query line segment s   in O(klog⁡n)O(klog⁡n) time, where k is the size of the visibility polygon of s and n is the number of vertices of P  . Alternatively, we build a data structure in O(n3)O(n3) time and space that can compute the visibility polygon for any query line segment in O(k+log⁡n)O(k+log⁡n) time. In order to develop these data structures, we obtain many other results that may be interesting in their own right.