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CGAL 4.8 - 3D Fast Intersection and Distance Computation (AABB Tree)

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CGAL::AABB_tree< AABBTraits > Class Template Reference
AABB Tree Reference

`#include <CGAL/AABB_tree.h>`

## Definition

Class AABB_tree is a static data structure for efficient intersection and distance computations in 3D.

It builds a hierarchy of axis-aligned bounding boxes (an AABB tree) from a set of 3D geometric objects, and can receive intersection and distance queries, provided that the corresponding predicates are implemented in the traits class AABBTraits. An instance of the class `AABBTraits` is internally stored.

See Also
  `AABBTraits`
  `AABBPrimitive`

Examples:
  AABB_tree/AABB_custom_example.cpp, AABB_tree/AABB_custom_indexed_triangle_set_array_example.cpp, AABB_tree/AABB_custom_indexed_triangle_set_example.cpp, AABB_tree/AABB_custom_triangle_soup_example.cpp, AABB_tree/AABB_face_graph_triangle_example.cpp, AABB_tree/AABB_halfedge_graph_edge_example.cpp, AABB_tree/AABB_insertion_example.cpp, AABB_tree/AABB_polyhedron_edge_example.cpp, AABB_tree/AABB_polyhedron_facet_distance_example.cpp, AABB_tree/AABB_polyhedron_facet_intersection_example.cpp, AABB_tree/AABB_segment_3_example.cpp, and AABB_tree/AABB_triangle_3_example.cpp.

## Types
typedef AABBTraits::FT | FT
| Number type returned by the distance queries.
typedef AABBTraits::Point_3 | Point
| Type of 3D point.
typedef AABBTraits::Primitive | Primitive
| Type of input primitive.
typedef Primitive::Id | Primitive_id
| Identifier for a primitive in the tree.
typedef Primitives::size_type | size_type
| Unsigned integral size type.
typedef AABBTraits::Bounding_box | Bounding_box
| Type of bounding box.
typedef AABBTraits::Point_and_primitive_id | Point_and_primitive_id
typedef AABBTraits::Object_and_primitive_id | Object_and_primitive_id
template<typename Query >
using | Intersection_and_primitive_id = AABBTraits::Intersection_and_primitive_id< Query >
| An alias to `AABBTraits::Intersection_and_primitive_id<Query>`

## Creation
| AABB_tree (const AABBTraits &traits=AABBTraits())
| Constructs an empty tree, and initializes the internally stored traits class using `traits`. More...
template<typename InputIterator , typename... T>
| AABB_tree (InputIterator first, InputIterator beyond, T &&...)
| Builds the datastructure from a sequence of primitives. More...

## Operations
template<typename ConstPrimitiveIterator , typename... T>
void | rebuild (ConstPrimitiveIterator first, ConstPrimitiveIterator beyond, T &&...)
| Equivalent to calling `clear()` and then `insert(first,last,t...)`. More...
template<typename InputIterator , typename... T>
void | insert (InputIterator first, InputIterator beyond, T &&...)
| Add a sequence of primitives to the set of primitives of the AABB tree. More...
void | insert (const Primitive &p)
| Adds a primitive to the set of primitives of the tree.
| ~AABB_tree ()
| Clears and destroys the tree.
const AABBTraits & | traits () const
| Returns a const reference to the internally stored traits class.
void | clear ()
| Clears the tree.
const Bounding_box | bbox () const
| Returns the axis-aligned bounding box of the whole tree. More...
size_type | size () const
| Returns the number of primitives in the tree.
bool | empty () const
| Returns `true`, iff the tree contains no primitive.

## Advanced
void | build ()
| After one or more calls to `AABB_tree::insert()` the internal data structure of the tree must be reconstructed. More...

## Intersection Tests
template<typename Query >
bool | do_intersect (const Query &query) const
| Returns `true`, iff the query intersects at least one of the input primitives. More...
template<typename Query >
size_type | number_of_intersected_primitives (const Query &query) const
| Returns the number of primitives intersected by the query. More...
template<typename Query , typename OutputIterator >
OutputIterator | all_intersected_primitives (const Query &query, OutputIterator out) const
| Outputs to the iterator the list of all intersected primitives ids. More...
template<typename Query >
boost::optional< Primitive_id > | any_intersected_primitive (const Query &query) const
| Returns the first encountered intersected primitive id, iff the query intersects at least one of the input primitives. More...

## Intersections
template<typename Query , typename OutputIterator >
OutputIterator | all_intersections (const Query &query, OutputIterator out) const
| Outputs the list of all intersections, as objects of `Intersection_and_primitive_id<Query>::Type`, between the query and the input data to the iterator. More...
template<typename Query >
boost::optional< typename Intersection_and_primitive_id < Query >::Type > | any_intersection (const Query &query) const
| Returns the first encountered intersection. More...

## Distance Queries
FT | squared_distance (const Point &query) const
| Returns the minimum squared distance between the query point and all input primitives. More...
Point | closest_point (const Point &query) const
| Returns the point in the union of all input primitives which is closest to the query. More...
Point_and_primitive_id | closest_point_and_primitive (const Point &query) const
| Returns a `Point_and_primitive_id` which realizes the smallest distance between the query point and all input primitives. More...

## Accelerating the Distance Queries
In the following paragraphs, we discuss details of the implementation of the distance queries. We explain the internal use of hints, how the user can pass his own hints to the tree, and how the user can influence the construction of the secondary data structure used for accelerating distance queries. Internally, the distance queries algorithms are initialized with some hint, which has the same type as the return type of the query, and this value is refined along a traversal of the tree, until it is optimal, that is to say until it realizes the shortest distance to the primitives. In particular, the exact specification of these internal algorithms is that they minimize the distance to the object composed of the union of the primitives and the hint. It follows that in order to return the exact distance to the set of primitives, the algorithms need the hint to be exactly on the primitives; if this is not the case, and if the hint happens to be closer to the query point than any of the primitives, then the hint is returned. This second observation is reasonable, in the sense that providing a hint to the algorithm means claiming that this hint belongs to the union of the primitives. These considerations about the hints being exactly on the primitives or not are important: in the case where the set of primitives is a triangle soup, and if some of the primitives are large, one may want to provide a much better hint than a vertex of the triangle soup could be. It could be, for example, the barycenter of one of the triangles. But, except with the use of an exact constructions kernel, one cannot easily construct points other than the vertices, that lie exactly on a triangle soup. Hence, providing a good hint sometimes means not being able to provide it exactly on the primitives. In rare occasions, this hint can be returned as the closest point. In order to accelerate distance queries significantly, the AABB tree builds an internal KD-tree containing a set of potential hints, when the method `accelerate_distance_queries()` is called. This KD-tree provides very good hints that allow the algorithms to run much faster than with a default hint (such as the `reference_point` of the first primitive). The set of potential hints is a sampling of the union of the primitives, which is obtained, by default, by calling the method `reference_point` of each of the primitives. However, such a sampling with one point per primitive may not be the most relevant one: if some primitives are very large, it helps inserting more than one sample on them. Conversely, a sparser sampling with less than one point per input primitive is relevant in some cases.
bool | accelerate_distance_queries () const
| Constructs internal search tree from a point set taken on the internal primitives returns `true` iff successful memory allocation.
template<typename ConstPointIterator >
bool | accelerate_distance_queries (ConstPointIterator first, ConstPointIterator beyond) const
| Constructs an internal KD-tree containing the specified point set, to be used as the set of potential hints for accelerating the distance queries. More...
FT | squared_distance (const Point &query, const Point &hint) const
| Returns the minimum squared distance between the query point and all input primitives. More...
Point | closest_point (const Point &query, const Point &hint) const
| Returns the point in the union of all input primitives which is closest to the query. More...
Point_and_primitive_id | closest_point_and_primitive (const Point &query, const Point_and_primitive_id &hint) const
| Returns a `Point_and_primitive_id` which realizes the smallest distance between the query point and all input primitives. More...

## Member Typedef Documentation

template<typename AABBTraits >

typedef AABBTraits::Object_and_primitive_id CGAL::AABB_tree< AABBTraits >::Object_and_primitive_id

Deprecated:

## Constructor & Destructor Documentation

template<typename AABBTraits >

CGAL::AABB_tree< AABBTraits >::AABB_tree | ( | const AABBTraits & | traits = `AABBTraits()`) |  |

Constructs an empty tree, and initializes the internally stored traits class using `traits`.

template<typename AABBTraits >
template<typename InputIterator , typename... T>

CGAL::AABB_tree< AABBTraits >::AABB_tree | ( | InputIterator | first,
|  | InputIterator | beyond,
|  | T && | ...
| ) |  |

Builds the datastructure from a sequence of primitives.

Parameters

  first | iterator over first primitive to insert
  beyond | past-the-end iterator

It is equivalent to constructing an empty tree and calling `insert(first,last,t...)`. For compilers that do not support variadic templates, overloads up to 5 template arguments are provided. The tree stays empty if the memory allocation is not successful.

## Member Function Documentation

template<typename AABBTraits >
template<typename ConstPointIterator >

bool CGAL::AABB_tree< AABBTraits >::accelerate_distance_queries | ( | ConstPointIterator | first,
|  | ConstPointIterator | beyond
| ) |  | const

Constructs an internal KD-tree containing the specified point set, to be used as the set of potential hints for accelerating the distance queries.

Template Parameters

  ConstPointIterator | is an iterator with value type `Point_and_primitive_id`.

template<typename Tr >
template<typename Query , typename OutputIterator >

OutputIterator CGAL::AABB_tree< Tr >::all_intersected_primitives | ( | const Query & | query,
|  | OutputIterator | out
| ) |  | const

Outputs to the iterator the list of all intersected primitives ids.

This function does not compute the intersection points and is hence faster than the function `all_intersections()` function below.

Template Parameters

  Query | must be a type for which `do_intersect` predicates are defined in the traits class `AABBTraits`.

template<typename Tr >
template<typename Query , typename OutputIterator >

OutputIterator CGAL::AABB_tree< Tr >::all_intersections | ( | const Query & | query,
|  | OutputIterator | out
| ) |  | const

Outputs the list of all intersections, as objects of `Intersection_and_primitive_id<Query>::Type`, between the query and the input data to the iterator.

`do_intersect()` predicates and intersections must be defined for `Query` in the `AABBTraits` class.

template<typename AABBTraits >
template<typename Query >

boost::optional<Primitive_id> CGAL::AABB_tree< AABBTraits >::any_intersected_primitive | ( | const Query & | query) |  | const

Returns the first encountered intersected primitive id, iff the query intersects at least one of the input primitives.

No particular order is guaranteed over the tree traversal, such that, e.g, the primitive returned is not necessarily the closest from the source point of a ray query.

Template Parameters

  Query | must be a type for which `do_intersect` predicates are defined in the traits class `AABBTraits`.

template<typename AABBTraits >
template<typename Query >

boost::optional< typename Intersection_and_primitive_id<Query>::Type > CGAL::AABB_tree< AABBTraits >::any_intersection | ( | const Query & | query) |  | const

Returns the first encountered intersection.

No particular order is guaranteed over the tree traversal, e.g, the primitive returned is not necessarily the closest from the source point of a ray query. Type `Query` must be a type for which `do_intersect` predicates and intersections are defined in the traits class AABBTraits.

template<typename AABBTraits >

const Bounding_box CGAL::AABB_tree< AABBTraits >::bbox | ( | ) |  | const

Returns the axis-aligned bounding box of the whole tree.

Precondition
  `!empty()`

template<typename Tr >

void CGAL::AABB_tree< Tr >::build | ( | ) |  |

After one or more calls to `AABB_tree::insert()` the internal data structure of the tree must be reconstructed.

This procedure has a complexity of \(O(n log(n))\), where \(n\) is the number of primitives of the tree. This procedure is called implicitly at the first call to a query member function. You can call AABB_tree::build() explicitly to ensure that the next call to query functions will not trigger the reconstruction of the data structure.

template<typename Tr >

AABB_tree< Tr >::Point CGAL::AABB_tree< Tr >::closest_point | ( | const Point & | query) |  | const

Returns the point in the union of all input primitives which is closest to the query.

In case there are several closest points, one arbitrarily chosen closest point is returned. Method `accelerate_distance_queries()` should be called before the first distance query, so that an internal secondary search structure is build, for improving performance.

Precondition
  `!empty()`

template<typename Tr >

AABB_tree< Tr >::Point CGAL::AABB_tree< Tr >::closest_point | ( | const Point & | query,
|  | const Point & | hint
| ) |  | const

Returns the point in the union of all input primitives which is closest to the query.

In case there are several closest points, one arbitrarily chosen closest point is returned. The internal KD-tree is not used.

Precondition
  `!empty()`

template<typename Tr >

AABB_tree< Tr >::Point_and_primitive_id CGAL::AABB_tree< Tr >::closest_point_and_primitive | ( | const Point & | query) |  | const

Returns a `Point_and_primitive_id` which realizes the smallest distance between the query point and all input primitives.

Method `accelerate_distance_queries()` should be called before the first distance query, so that an internal secondary search structure is build, for improving performance.

Precondition
  `!empty()`

template<typename Tr >

AABB_tree< Tr >::Point_and_primitive_id CGAL::AABB_tree< Tr >::closest_point_and_primitive | ( | const Point & | query,
|  | const Point_and_primitive_id & | hint
| ) |  | const

Returns a `Point_and_primitive_id` which realizes the smallest distance between the query point and all input primitives.

The internal KD-tree is not used.

Precondition
  `!empty()`

template<typename Tr >
template<typename Query >

bool CGAL::AABB_tree< Tr >::do_intersect | ( | const Query & | query) |  | const

Returns `true`, iff the query intersects at least one of the input primitives.

Template Parameters

  Query | must be a type for which `do_intersect` predicates are defined in the traits class `AABBTraits`.

template<typename AABBTraits >
template<typename InputIterator , typename... T>

void CGAL::AABB_tree< AABBTraits >::insert | ( | InputIterator | first,
|  | InputIterator | beyond,
|  | T && | ...
| ) |  |

Add a sequence of primitives to the set of primitives of the AABB tree.

`InputIterator` is any iterator and the parameter pack `T` are any types such that `Primitive` has a constructor with the following signature: `Primitive(InputIterator, T...)`. If `Primitive` is a model of the concept `AABBPrimitiveWithSharedData`, a call to `AABBTraits::set_shared_data(t...)` is made using the internally stored traits. For compilers that do not support variadic templates, overloads up to 5 template arguments are provided.

template<typename AABBTraits >
template<typename Query >

size_type CGAL::AABB_tree< AABBTraits >::number_of_intersected_primitives | ( | const Query & | query) |  | const

Returns the number of primitives intersected by the query.

Template Parameters

  Query | must be a type for which `do_intersect` predicates are defined in the traits class `AABBTraits`.

template<typename AABBTraits >
template<typename ConstPrimitiveIterator , typename... T>

void CGAL::AABB_tree< AABBTraits >::rebuild | ( | ConstPrimitiveIterator | first,
|  | ConstPrimitiveIterator | beyond,
|  | T && | ...
| ) |  |

Equivalent to calling `clear()` and then `insert(first,last,t...)`.

For compilers that do not support variadic templates, overloads up to 5 template arguments are provided.

template<typename Tr >

AABB_tree< Tr >::FT CGAL::AABB_tree< Tr >::squared_distance | ( | const Point & | query) |  | const

Returns the minimum squared distance between the query point and all input primitives.

Method `accelerate_distance_queries()` should be called before the first distance query, so that an internal secondary search structure is build, for improving performance.

Precondition
  `!empty()`

template<typename Tr >

AABB_tree< Tr >::FT CGAL::AABB_tree< Tr >::squared_distance | ( | const Point & | query,
|  | const Point & | hint
| ) |  | const

Returns the minimum squared distance between the query point and all input primitives.

The internal KD-tree is not used.

Precondition
  `!empty()`