The function box_self_intersection_d computes the pairwise intersecting boxes in a sequence of isooriented boxes in arbitrary dimension. The sequence of boxes is given with as a randomaccess iterator range and will be reordered in the course of the algorithm. For each intersecting pair of boxes a callback function object is called with the two intersecting boxes as argument; the first argument is a box from the sequence, the second argument is a copy of a box from the sequence. The performance of the algorithm can be tuned with a cutoff parameter, see the implementation section of the CGAL::box_intersection_d function on page \icon.
The algorithm creates a second copy of the boxes and reorders the boxes in the course of the algorithm. Now, depending on the size of a box it can be faster to copy the boxes, or to work with pointers to boxes and copy only pointers. We offer automatic support for both options. To simplify the description, let us call the value_type of the iterators box handle. The box handle can either be our box type itself or a pointer (or const pointer) to the box type.
A ddimensional isooriented box is defined as the Cartesian product of d intervals. We call the box halfopen if the d intervals { [lo_{i},hi_{i})  0 ≤ i < d} are halfopen intervals, and we call the box closed if the d intervals { [lo_{i},hi_{i}]  0 ≤ i < d} are closed intervals. Note that closed boxes support zerowidth boxes and they can intersect at their boundaries, while nonempty halfopen boxes always have a positive volume and they only intersect iff their interiors overlap. The distinction between closed or halfopen boxes does not require a different representation of boxes, just a different interpretation when comparing boxes, which is selected with the topology parameter and its two values, CGAL::Box_intersection_d::HALF_OPEN and CGAL::Box_intersection_d::CLOSED.
In addition, a box has an unique idnumber. It is used to order boxes consistently in each dimension even if boxes have identical coordinates. In consequence, the algorithm guarantees that a pair of intersecting boxes is reported only once. This selfintersection function creates internally a second copy of the box sequence. The copying has to preserve the idnumber of boxes. Note that this implies that the address of the box is not sufficient for the idnumber if boxes are copied by value. Boxes of equal idnumber are not reported as intersecting pairs since they are always intersecting trivially.
The algorithm uses a traits class of the BoxIntersectionTraits_d concept to access the boxes. A default traits class is provided that assumes that the box type is a model of the BoxIntersectionBox_d concept and that the box handle, i.e., the iterators value type, is identical to the box type or a pointer to the box type.
#include <CGAL/box_intersection_d.h>
 

 
Invocation of box intersection with default box traits CGAL::Box_intersection_d::Box_traits_d<Box_handle>, where Box_handle corresponds to the iterator value type of RandomAccessIterator.  
 

 
Invocation with custom box traits. 
CGAL::box_intersection_d
CGAL::box_self_intersection_all_pairs_d
CGAL::Box_intersection_d::Box_traits_d<BoxHandle>
BoxIntersectionBox_d
BoxIntersectionTraits_d
See the implementation section of the CGAL::box_intersection_d function on page \icon.
The box implementation provided with CGAL::Box_intersection_d::Box_d<double,2> has a special constructor for the Cgal bounding box type CGAL::Bbox_2 (and similar for dimension 3). We use this in the example to create 3 × 3 boxes in a grid layout.
The default policy of the box type implements the idnumber with an explicit counter in the boxes, which is the default choice since it always works. We use the idnumber in our callback function to report the result of the intersection algorithm call. The result will be 20 pairwise intersections, but the order in which they are reported is nonintuitive.
File: examples/Box_intersection_d/minimal_self.cpp
#include <CGAL/box_intersection_d.h> #include <CGAL/Bbox_2.h> #include <iostream> typedef CGAL::Box_intersection_d::Box_d<double,2> Box; typedef CGAL::Bbox_2 Bbox; // 9 boxes of a grid Box boxes[9] = { Bbox( 0,0,1,1), Bbox( 1,0,2,1), Bbox( 2,0,3,1), // low Bbox( 0,1,1,2), Bbox( 1,1,2,2), Bbox( 2,1,3,2), // middle Bbox( 0,2,1,3), Bbox( 1,2,2,3), Bbox( 2,2,3,3)};// upper void callback( const Box& a, const Box& b ) { std::cout << "box " << a.id() << " intersects box " << b.id() << std::endl; } int main() { CGAL::box_self_intersection_d( boxes, boxes+9, callback); return 0; }