## CGAL::box_intersection_all_pairs_d

### Definition

The function box_intersection_all_pairs_d computes the pairwise intersecting boxes between two sequences of iso-oriented boxes in arbitrary dimension. It does so by comparing all possible pairs of boxes and is thus inferior to the fast CGAL::box_intersection_d algorithm on page .

The sequences of boxes are given with two forward iterator ranges. The sequences are not modified. 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 first sequence, the second argument a box from the second sequence.

The algorithm is interface compatible with the CGAL::box_intersection_d function. Similarly, we call the value_type of the iterators the box handle, which is either our box type or a pointer type to our 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. Boxes with equal id-number are not reported since they obviously intersect 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.

An important special application of this algorithm is the test for self-intersections where the second box sequence is an identical copy of the first sequence including the preserved id-number. We offer a specialized implementation CGAL::box_self_intersection_all_pairs for this application.

#include <CGAL/box_intersection_d.h>

template< class ForwardIterator1, class ForwardIterator2, class Callback >

void

box_intersection_all_pairs_d ( | ForwardIterator1 begin1, ForwardIterator1 end1, ForwardIterator2 begin2, ForwardIterator2 end2, Callback callback, 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 ForwardIterator1.
template< class ForwardIterator1, class ForwardIterator2, class Callback, class BoxTraits >

void

box_intersection_all_pairs_d ( | ForwardIterator1 begin1, ForwardIterator1 end1, ForwardIterator2 begin2, ForwardIterator2 end2, Callback callback, BoxTraits box_traits, Box_intersection_d::Topology topology = Box_intersection_d::CLOSED)

|  | Invocation with custom box traits.

### Requirements

- ForwardIterator1, and ...2, must be forward iterators and both value types must be the same. We call this 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_d
CGAL::box_self_intersection_all_pairs_d

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

### Implementation

The algorithm is trivially testing all pairs and runs therefore in time O(nm) where n is the size of the first sequence and m is the size of the second sequence.