\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - Intersecting Sequences of dD Iso-oriented Boxes

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Functions
CGAL::box_self_intersection_all_pairs_d()
Intersecting Sequences of dD Iso-oriented Boxes Reference

##

The function `box_self_intersection_all_pairs_d()` computes the pairwise intersecting boxes in a sequence 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_self_intersection_d()` algorithm.

The sequence of boxes is given with a forward iterator range. 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 algorithm is interface compatible with the `box_self_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`.

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.

Requirements

- `ForwardIterator` must be a forward 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_d()`
  `CGAL::box_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 \( O(n^2)\) where \( n\) is the size of the input sequence. This algorithm does not use the id-number of the boxes.

## Functions
template<class ForwardIterator , class Callback >
void | CGAL::box_self_intersection_all_pairs_d (ForwardIterator begin, ForwardIterator end, 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 `ForwardIterator`.
template<class ForwardIterator , class Callback , class BoxTraits >
void | CGAL::box_self_intersection_all_pairs_d (ForwardIterator begin, ForwardIterator end, Callback callback, BoxTraits box_traits, CGAL::Box_intersection_d::Topology topology=CGAL::Box_intersection_d::CLOSED)
| Invocation with custom box traits.