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CGAL 4.14.2 - Intersecting Sequences of dD Iso-oriented Boxes

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 box_intersection_d() algorithm.

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 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 \( \{ [lo_i,hi_i) \,|\, 0 \leq i < d\}\) are half-open intervals, and we call the box closed if the \( d\) intervals \( \{ [lo_i,hi_i] \,|\, 0 \leq i < d\}\) 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, Box_intersection_d::HALF_OPEN and Box_intersection_d::CLOSED.

In addition, a box has a 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 box_self_intersection_all_pairs_d() for this application.


See also


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.


template<class ForwardIterator1 , class ForwardIterator2 , class Callback >
void CGAL::box_intersection_all_pairs_d (ForwardIterator1 begin1, ForwardIterator1 end1, ForwardIterator2 begin2, ForwardIterator2 end2, Callback callback, CGAL::Box_intersection_d::Topology topology=CGAL::Box_intersection_d::CLOSED)
 Invocation of box intersection with default box traits 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 CGAL::box_intersection_all_pairs_d (ForwardIterator1 begin1, ForwardIterator1 end1, ForwardIterator2 begin2, ForwardIterator2 end2, Callback callback, BoxTraits box_traits, CGAL::Box_intersection_d::Topology topology=CGAL::Box_intersection_d::CLOSED)
 Invocation with custom box traits.