  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++.
