Query-sensitive ray shooting

Ray (segment) shooting is the problem of determining the first intersection between a ray (directed line segment) and a collection of polygonal or polyhedral obstacles. In order to process queries efficiently, the set of obstacle polyhedra is usually preprocessed into a data structure. In this paper we propose a query-sensitive data structure for ray shooting, which means that the performance of our data structure depends on the local geometry of obstacles near the query segment. We measure the complexity of the local geometry near the segment by a parameter called the simple cover complexity, denoted by scc(s) for a segment s. Our data structure consists of a subdivision that partitions the space into a collection of polyhedral cells, each of O(1) complexity. We answer a segment shooting query by walking along the segment through the subdivision. Our first result is that, for any fixed dimension d, there exists a simple hierarchical subdivision in which no query segment s intersecte more than O(scc(s)) cells. Our second result shows that in two dimensions such a subdivision of size O(n) can be constructed in time O(n log n), where n is the total number of vertices in all the obstacles.