namespace CGAL {

/*!

\mainpage User Manual
\anchor Chapter_Fast_Intersection_and_Distance_Computation
\cgalAutoToc

\authors Pierre Alliez, Stephane Tayeb, and Camille Wormser

\section aabb_tree_introduction Introduction

The AABB tree component offers a static data structure and algorithms
to perform efficient intersection and distance queries against sets of
finite 2D or 3D geometric objects. The set of geometric objects stored in
the data structure can be queried for intersection detection,
intersection computation and distance. The intersection queries can be
of any type, provided that the corresponding intersection predicates
and constructors are implemented in the traits class. The distance
queries are limited to point queries. Examples of intersection queries
include line objects (rays, lines, segments) against sets of
triangles, or plane objects (planes, triangles) against sets of
segments. An example of a distance query consists of finding the
closest point from a point query to a set of triangles.

Note that this component is not suited to the problem of finding all
intersecting pairs of objects. We refer to the component
\ref chapterBoxIntersection "Intersecting Sequences of dD Iso-oriented Boxes"
which can find all intersecting pairs of iso-oriented boxes.

The AABB tree data structure takes as input an iterator range of
geometric data, which is then converted into primitives. From these
primitives a hierarchy of axis-aligned bounding boxes (AABBs) is
constructed and used to speed up intersection and distance queries.
Each primitive gives access to both one input geometric object
(so-called datum) and one reference id to this object. A typical
example primitive wraps a 3D triangle as datum and a face handle of a
polyhedral surface as id. Each intersection query can return the
intersection objects (e.g., 3D points or segments for ray queries) as
well as the id (here the face handle) of the intersected
primitives. Similarly, each distance query can return the closest
point from the point query as well as the id of the closest primitive.

\cgalFigureBegin{aabbtree_meca,anchor.png}
Left: surface triangle mesh of a mechanical part. Right: AABB tree constructed.
\cgalFigureEnd

\section aabb_tree_interface Interface

The main entry point to the component is the class `AABB_tree`
which represents a static AABB tree constructed from an iterator range
of geometric data. Once instantiated an AABB tree can be queried for
intersection and distance queries.

\b Intersections. Assume for example that the tree contains triangle
primitives. The tree can be queried for intersection against line
objects (rays, segments or line) in various ways. We distinguish
intersection *tests* which do not construct any intersection
objects, from *intersections* which construct the intersection
objects.

Tests:

- Function `AABB_tree::do_intersect()` tests if the input primitives are
intersected by the query. This function is fast as it involves only
predicates and stops after the first encountered intersection.
- Function `AABB_tree::number_of_intersected_primitives()` counts all
intersected primitives.
- Function `AABB_tree::all_intersected_primitives()` enumerates all intersected
primitives ids without constructing the corresponding intersection
objects.
- Function `AABB_tree::any_intersected_primitive()` returns the first
encountered intersecting primitive id (if any) without constructing
the corresponding intersection object, and stops after the first
encountered intersection. Note that the traversal order of the tree is
such that first herein does not refer to any particular ordering of
the intersections with respect to the query.
- Function `AABB_tree::first_intersected_primitive()` returns
the intersecting primitive id (if any) of the corresponding
intersection object that is closest to the source of the ray.

Constructions:

- Function  `AABB_tree::all_intersections()` detects and constructs all
intersection objects with the input primitives.
- Function  `AABB_tree::any_intersection()` detects and constructs the first
encountered intersection and constructs the corresponding object. This
function is fast as it stops after the first encountered intersection.
- Function `AABB_tree::first_intersection()` detects and constructs
the intersection object that is closest to the source of the ray.


\b Distance. An AABB tree computes the closest point from a
given point query to the input primitives through the function
`AABB_tree::closest_point()`. In addition, it can compute the id of the
closest primitive from a given point query through the function
`AABB_tree::closest_point_and_primitive()`, i.e., the id of the
primitive which realizes the minimum distance from the point
query. The AABB tree uses a secondary search structure to speed up the
distance queries. The construction of this secondary structure should
be requested by the user by a call to
`AABB_tree::accelerate_distance_queries()` before the first the distance
computation. This data structure is not generated by default because
it is used only for distance computations.

\warning Having degenerate primitives in the AABB-tree is not recommended as the underlying
predicates and constructions of the traits class might not be able to handle them.
For example if one is using `CGAL::AABB_traits` with a Kernel from \cgal,
having degenerate triangles or segments in the AABB-tree will result in an undefined
behavior or a crash.

\section aabb_tree_examples Examples

\subsection aabb_tree_examples_1 Tree of Triangles, for Intersection and Distance Queries

In the following example a set of 3D triangles is stored in a
list. The AABB primitive wraps a triangle as `datum` and an
iterator in the list as `id`. We compute the number of input
triangles intersected by a ray query, as well as the closest point and
the squared distance from a point query.

\cgalExample{AABB_tree/AABB_triangle_3_example.cpp}

\subsection aabb_tree_examples_2 Tree of Polyhedron Triangle Facets for Intersection Queries

In the following example the AABB primitive wraps a facet handle of a
triangle polyhedral surface as `id` and the corresponding 3D
triangle as geometric object. From a segment query we test the
intersections, then compute the number of intersections, compute the
first encountered intersection (generally a point), compute all
intersections (where each intersection is a pair of one \cgal object
and one primitive id - here a face handle) and compute all intersected
primitives. The latter involves only tests and no predicates and is
hence faster than computing all intersections. We also compute the
first encountered intersection with a plane query, which is generally
a segment.  We finally compute the first intersection along a ray.

\cgalExample{AABB_tree/AABB_polyhedron_facet_intersection_example.cpp}


\subsection aabb_ray_shooting Ray Shooting

In the following example we load a closed polyhedral surface and perform a
ray shooting query from the centroid of each face, orthogonally to the face
towards the interior.  As the centroid is computed with floating point
arithmetic the first face hit by the ray may be the face of the centroid.
The skip functor takes care of ignoring the face.

\cgalExample{AABB_tree/AABB_ray_shooting_example.cpp}



\subsection aabb_tree_examples_3 Tree of Polyhedron Triangle Facets for Distance Queries

In the following example the AABB primitive wraps a facet handle of a
triangle polyhedral surface as `id` and the corresponding 3D
triangle as geometric object. From a point query we compute the
squared distance, the closest point as well as the closest point and
primitive id. The latter returns a pair composed of a point and a face
handle.

\cgalExample{AABB_tree/AABB_polyhedron_facet_distance_example.cpp}

\subsection aabb_tree_examples_4 Tree of Segments for Intersection and Distance Queries

In the following example the segments are stored into a list, and the
AABB primitive wraps a segment as `datum` and an iterator in the
list as `id`. We compute the number of intersections with plane
and triangles queries, and the closest point from a point query.
\cgalExample{AABB_tree/AABB_segment_3_example.cpp}

\subsection aabb_tree_examples_8 Tree of Polyline and Polygon Segments for Intersection and Distance Queries

The following example uses segments given by a polyline and a polygon. The difference is that the polygon is closed. The `id` in this case is the iterator pointing to a `Point_2` in the polyline or the polygon. The datum, a `Segment_2`, is created on the fly from the points using the `id` as the source and the following `Point_2` as the target. We count the intersections with a `Segment_2` and the closest point and id from a point query.
\cgalExample{AABB_tree/AABB_polyline_segment_2_example.cpp}

\subsection aabb_tree_examples_5 Tree of Polyhedron Edge Segments for Intersection and Distance Queries

In the following example the AABB primitive wraps a halfedge handle as
`id` and generates a 3D segment on the fly, each time its method
`datum` is called. We compute the number of intersections with a
triangle query and the closest point from a point query.
\cgalExample{AABB_tree/AABB_polyhedron_edge_example.cpp}

\subsection aabb_tree_examples_6 Incremental Insertion of Primitives

The AABB tree is a static data structure, but it allows to insert
primitives, and will internally rebuild triggered by the first query,
or because the user calls the `AABB_tree::build()` method.  The following
example illustrates this for two polyhedral surfaces.

\cgalExample{AABB_tree/AABB_insertion_example.cpp}

\subsection aabb_tree_examples_7 Trees of Custom Primitives

The AABB tree example folder contains three examples of trees
constructed with custom primitives. In \ref AABB_tree/AABB_custom_example.cpp "AABB_custom_example.cpp"
the primitive contains triangles which are defined by three pointers
to custom points. In \ref AABB_tree/AABB_custom_triangle_soup_example.cpp "AABB_custom_triangle_soup_example.cpp" all input
triangles are stored into a single array as to form a triangle
soup. The primitive internally uses a `boost::iterator_adaptor`
as to provide the three functions `AABBPrimitive::id()`, `AABBPrimitive::datum()`,
and `AABBPrimitive::reference_point()`) required by the primitive concept. In
\ref AABB_tree/AABB_custom_indexed_triangle_set_example.cpp "AABB_custom_indexed_triangle_set_example.cpp" the input is an indexed
triangle set stored through two arrays: one array of points and one
array of indices which refer to the point array. Here also the
primitive internally uses a `boost::iterator_adaptor`.

\section aabb_tree_perf Performances

We provide some performance numbers for the case where the AABB tree
contains a set of polyhedron triangle facets. We measure the tree
construction time, the memory occupancy and the number of queries per
second for a variety of intersection and distance queries. The machine
used is a PC running Windows XP64 with an Intel CPU Core2 Extreme
clocked at 3.06 GHz with 4GB of RAM. By default, the kernel used is
`Simple_cartesian<double>` (the fastest in our experiments). The
program has been compiled with Visual C++ 2005 compiler with the O2
option which maximizes speed.

\subsection aabb_tree_perf_cons Construction

The surface triangle mesh chosen for benchmarking the tree
construction is the knot model (14,400 triangles) depicted by
\cgalFigureRef{figAABB-tree-bench}. We measure the tree construction time (both
AABB tree alone and AABB tree with internal KD-tree) for this model as
well as for three denser versions subdivided through the Loop
subdivision scheme which multiplies the number of triangles by four.


|  Number of triangles | Construction (in ms) | Construction with internal KD-tree (in ms) |
| ----: | ----:     | ----:     |
   14,400 |     156 |       157 |
   57,600 |     328 |       328 |
  230,400 |   1,141 |     1,437 |
  921,600 |   4,813 |     5,953 |

\subsection aabb_tree_perf_mem Memory

When using the polyhedron triangle facet primitive (defined in
`AABB_face_graph_triangle_primitive.h`) the AABB tree occupies
approximately 61 bytes per primitive (without constructing the
internal KD-tree). It increases to approximately 150 bytes per
primitive when constructing the internal KD-tree with one reference
point per primitive (the default mode when calling the function
`AABB_tree::accelerate_distance_queries()`). Note that the polyhedron
facet primitive stores only one facet handle as primitive id
and computes on the fly a 3D triangle from the facet handle stored
internally. When explicitly storing a 3D triangle in the primitive the
tree occupies approximately 140 bytes per primitive instead of 60
(without constructing the internal KD-tree).

The following table provides orders of memory occupancy in MBytes for
an increasing number of triangles. As the internal KD-tree used to
accelerate the distance queries dominates the memory occupancy, we
recommend to specify for large models a lower number of reference
point (evenly distributed) to construct the internal KD-tree through
the function `AABB_tree::accelerate_distance_queries()` which
takes an iterator range as input.

|  Number of triangles | AABB tree (in MBytes) | AABB tree with internal KD-tree (in MBytes)|
| ----: | ----: | ----: |
   18,400 |   1.10 |    2.76 |
  102,400 |   6.33 |   14.73 |
1,022,400 |  59.56 |  151.31 |
1,822,400 | 108.34 |  291.84 |


\subsection aabb_tree_perf_inter Intersections

The following table measures the number of intersection queries per
second on the 14,400 triangle version of the knot mesh model for ray,
line, segment and plane queries. Each ray query is generated by
choosing a random source point within the mesh bounding box and a
random vector. A line or segment query is generated by choosing two
random points inside the bounding box. A plane query is generated by
picking a random point inside the bounding box and a random normal
vector. Note that a plane query generally intersects many triangles of
the input surface mesh. This explains the low performance numbers for
the intersection functions which enumerate all intersections.


|  Function                              | Segment |     Ray |    Line |   Plane |
| :---- | ----: | ----: | -: | -: |
|   AABB_tree::do_intersect()                       | 187,868 | 185,649 | 206,096 | 377,969 |
|   AABB_tree::any_intersected_primitive()         | 190,684 | 190,027 | 208,941 | 360,337 |
|   AABB_tree::any_intersection()                   | 147,468 | 143,230 | 148,235 | 229,336 |
|   AABB_tree::number_of_intersected_primitives() |  64,389 |  52,943 |  54,559 |   7,906 |
|   AABB_tree::all_intersected_primitives()        |  65,553 |  54,838 |  53,183 |   5,693 |
|   AABB_tree::all_intersections()                  |  46,507 |  38,471 |  36,374 |   2,644 |


Curve of \cgalFigureRef{figAABB-tree-bench} plots the number of queries per second
(here the `AABB_tree::all_intersections()` function with random segment
queries) against the number of input triangles for the knot triangle
surface mesh.


\cgalFigureBegin{figAABB-tree-bench,bench.png}
Number of queries per second against number of triangles for the knot model with 14K (shown), 57K, 230K and 921K triangles. We call the `all_intersections()` function with segment queries randomly chosen within the bounding box.
\cgalFigureEnd

The following table measures the number of `AABB_tree::all_intersections()`
queries per second against several kernels. We use the 14,400 triangle
version of the knot mesh model for random segment queries. Note how
the `Simple_cartesian` kernel is substantially faster than the
`Cartesian` kernel.

|  Kernel | Queries/s (all_intersections() with segment queries)|
| :---- | ----: |
|`Simple_cartesian<double>`               | 46,507 |
|`Simple_cartesian<float>`                | 43,187 |
|`Cartesian<double>`                      |  5,335 |
|`Cartesian<float>`                       |  5,522 |
|`Exact_predicates_inexact_constructions_kernel` | 18,411 |



\subsection aabb_tree_perf_dist Distances

The surface triangle mesh chosen for benchmarking distances is again
the knot model in four increasing resolutions obtained through Loop
subdivision. In the following table we first measure the tree
construction time (which includes the construction of the internal
KD-tree data structure used to accelerate the distance queries by up
to one order of magnitude in our experiments). We then measure the
number of queries per second for the three types distance queries
(`AABB_tree::closest_point()`, `AABB_tree::squared_distance()` and
`AABB_tree::closest_point_and_primitive()`) from point queries randomly chosen
inside the bounding box.

|  Nb triangles | Construction (ms) | Closest_point() | Squared_distance() | Closest_point_and_primitive() |
| ----:    | ----:     | ----: | ----: | -: |
|   14,400 |   157.000 | 45,132 | 45,626 | 45,770 |
|   57,600 |   328.000 | 21,589 | 21,312 | 21,137 |
|  230,400 |     1.437 | 11,063 | 10,962 | 11,086 |
|  921,600 |     5.953 |  5,636 |  5,722 |  5,703 |

\subsection aabb_tree_perf_summary Summary

The experiments described above are neither exhaustive nor conclusive
as we have chosen one specific case where the input primitives are the
facets of a triangle surface polyhedron. Nevertheless, we now provide
some general observations and advises about how to put the AABB tree
to use with satisfactory performances. While the tree construction
times and memory occupancy do not fluctuate much in our experiments
depending on the input surface triangle mesh, the performance
expressed in number of queries varies greatly depending on a complex
combination of criteria: type of kernel, number of input primitives,
distribution of primitives in space, type of function queried, type of
query and location of query in space.

- %Kernel: The type of \cgal kernel turns out to dominate the final
  execution times, the maximum performances being obtained with the
  `Simple_cartesian` kernel parameterized with the double precision
  number type. In applications where the intersection and distance
  execution times are crucial it is possible to use this kernel for
  the AABB tree in combination with a more robust kernel for the main
  data structure.

- Primitives: Although the number of input primitives plays an obvious
  role in the final performance, their distribution in space is at
  least equally important in order to obtain a well-balanced AABB
  tree. Ideally the primitives must be evenly distributed in space and
  the long primitives spanning the bounding box of the tree root node
  must be avoided as much as possible. It is often beneficial to split
  these long primitives into smaller ones before constructing the
  tree, e.g., through recursive longest edge bisection for triangle
  surface meshes.

- Function: The type of function queried plays another important
  role. Obviously the *exhaustive* functions, which list all
  intersections, are slower than the ones stopping after the first
  intersection. Within each of these functions the ones which call
  only intersection tests (`AABB_tree::do_intersect()`,
  `AABB_tree::number_of_intersected_primitives()`,
  `AABB_tree::any_intersected_primitive()`, `AABB_tree::all_intersected_primitives()`) are
  faster than the ones which explicitly construct the intersections
  (`AABB_tree::any_intersection()` and `AABB_tree::all_intersections()`).

- Query: The type of query (e.g., line, ray, segment or plane used
  above) plays another role, strongly correlated with the type of
  function (exhaustive or not, and whether or not it constructs the
  intersections). When all intersection constructions are needed, the
  final execution times highly depend on the complexity of the general
  intersection object. For example a plane query generally intersects
  a surface triangle mesh into many segments while a segment query
  generally intersects a surface triangle mesh into few
  points. Finally, the location of the query in space also plays an
  obvious role in the performances, especially for the distance
  queries. Assuming the internal KD-tree constructed through the
  function `AABB_tree::accelerate_distance_queries()`, it is preferable
  to specify a query point already close to the surface triangle mesh
  so that the query traverses only few AABBs of the tree. For a large
  number of primitive data (greater than 2M faces in our experiments)
  however we noticed that it is not necessary (and sometimes even
  slower) to use all reference points when constructing the
  KD-tree. In these cases we recommend to specify through the function
  ` AABB_tree::accelerate_distance_queries()` fewer reference
  points (typically not more than 100K) evenly distributed over the
  input primitives.

\section aabb_tree_impl Implementation Details

The AABB tree construction is initialized by computing the AABB of the
whole set of input primitives. All primitives are then sorted along
the longest coordinate axis of this box, and the primitives are
separated into two equal size sets. This procedure is applied
recursively until an AABB contains a single primitive. The tree is
leafless as presented in `OPCODE` \cgalCite{cgal:t-ocdl-05}\. An
intersection query traverses the tree by computing intersection tests
only with respect to the AABBs during traversal, and with respect to
the input primitives at the end of traversal (in the leafs of the
tree).

The reference id is not used internally but simply used by the AABB
tree to refer to the primitive in the results provided to the user. It
follows that, while in most cases each reference id corresponds to a
unique primitive, this is not a requirement of the component. This way
a user may use these reference ids as labels, each of them being
shared by several geometric objects.

A distance query between a query point `q` and the input primitives is
turned into a *ball* query centered at `q`. The ball traverses the
tree while recursively querying intersections with the AABBs, and
computes the closest point `p` from the query point to the input
primitives at the leafs of the tree. The ball radius is then shrunk to
the distance between `p` and `q` for all remaining recursive
traversals of the tree. Efficiency is achieved through setting the
initial ball radius to a small value still guaranteed to intersect the
input primitives. This is achieved by constructing an internal secondary data
structure which provides a good hint to the algorithm at the beginning
of the traversal (done by default).
Calling `AABB_tree::do_not_accelerate_distance_queries()` will disable
the construction and the usage of this internal secondary data structure.

\section aabb_tree_history Design and Implementation History

Camille Wormser and Pierre Alliez started working on a data structure
for efficient collision detection in 2007. The generic design for
implementing both intersection and distance queries, and for generic
queries and primitives was developed by Camille Wormser. In 2009,
Pierre Alliez, Stéphane Tayeb and Camille Wormser made the
implementation CGAL-compliant, with the help of Laurent Rineau for
optimizing the tree construction. Additionally, Andreas
Fabri, Jane Tournois, Mariette Yvinec and Sylvain Lefèbvre are
given thanks for helpful comments and discussions during that period.
Later, Sébastien Loriot contributed several improvements:
thread-safe queries, introduction of shared data stored in the traits
for lighter primitive types, ...
In 2024, the package was made compatible with 2D and 3D primitives by
Andreas Fabri, Sébastien Loriot, and Sven Oesau.

*/

}
