Canonical polygon Queries on the plane: a New Approach
  The polygon retrieval problem on points is the problem of preprocessing a set of $n$ points on the plane, so that given a polygon query, the subset of points lying inside it can be reported efficiently.   It is of great interest in areas such as Computer Graphics, CAD applications, Spatial Databases and GIS developing tasks. In this paper we study the problem of canonical $k$-vertex polygon queries on the plane. A canonical $k$-vertex polygon query always meets the following specific property: a point retrieval query can be transformed into a linear number (with respect to the number of vertices) of point retrievals for orthogonal objects such as rectangles and triangles (throughout this work we call a triangle orthogonal iff two of its edges are axis-parallel).   We present two new algorithms for this problem. The first one requires $O(n\log^2{n})$ space and $O(k\frac{log^3n}{loglogn}+A)$ query time. A simple modification scheme on first algorithm lead us to a second solution, which consumes $O(n^2)$ space and $O(k \frac{logn}{loglogn}+A)$ query time, where $A$ denotes the size of the answer and $k$ is the number of vertices.   The best previous solution for the general polygon retrieval problem uses $O(n^2)$ space and answers a query in $O(k\log{n}+A)$ time, where $k$ is the number of vertices. It is also very complicated and difficult to be implemented in a standard imperative programming language such as C or C++. 
Spyros Sioutas, Dimitrios Sofotassios, Kostas Tsichlas, Dimitrios   Sotiropoulos, Panayiotis Vlamos