## CGAL::box_self_intersection_d

### Definition

The function box_self_intersection_d computes the pairwise intersecting boxes in a sequence of iso-oriented boxes in arbitrary dimension. The sequence of boxes is given with as a random-access 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 .

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 d-dimensional iso-oriented box is defined as the Cartesian product of d intervals. We call the box half-open if the d intervals { [loi,hii) | 0 i < d} are half-open intervals, and we call the box closed if the d intervals { [loi,hii] | 0 i < d} are closed intervals. Note that closed boxes support zero-width boxes and they can intersect at their boundaries, while non-empty half-open boxes always have a positive volume and they only intersect iff their interiors overlap. The distinction between closed or half-open 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 id-number. 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 self-intersection function creates internally a second copy of the box sequence. The copying has to preserve the id-number of boxes. Note that this implies that the address of the box is not sufficient for the id-number if boxes are copied by value. Boxes of equal id-number 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>

template< class RandomAccessIterator, class Callback >

void

box_self_intersection_d ( | RandomAccessIterator begin, RandomAccessIterator end, Callback callback, std::ptrdiff_t cutoff = 10, Box_intersection_d::Topology topology = Box_intersection_d::CLOSED)

|  | 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.
template< class RandomAccessIterator, class Callback, class BoxTraits >

void

box_self_intersection_d ( | RandomAccessIterator begin, RandomAccessIterator end, Callback callback, BoxTraits box_traits, std::ptrdiff_t cutoff = 10, Box_intersection_d::Topology topology = Box_intersection_d::CLOSED)

|  | Invocation with custom box traits.

### Requirements

- RandomAccessIterator must be a mutable random-access iterator. We call its value type Box_handle in the following.
- Callback must be of the BinaryFunction concept. The Box_handle must be convertible to both argument types. The return type is not used and can be void.
- The Box_handle must be a model of the Assignable concept.
- In addition, if the default box traits is used the Box_handle must be a class type T or a pointer to a class type T, where T must be a model of the BoxIntersectionBox_d concept. In both cases, the default box traits specializes to a suitable implementation.
- BoxTraits must be of the BoxIntersectionTraits_d concept.

### See Also

CGAL::box_intersection_d
CGAL::box_self_intersection_all_pairs_d

CGAL::Box_intersection_d::Box_traits_d<BoxHandle>
BoxIntersectionBox_d
BoxIntersectionTraits_d

### Implementation

See the implementation section of the CGAL::box_intersection_d function on page .

### Example

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 id-number with an explicit counter in the boxes, which is the default choice since it always works. We use the id-number 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 non-intuitive.

    // file: examples/Box_intersection_d/minimal_self.C
    #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;
    }