CGAL 6.2 - Fast Intersection and Distance Computation (AABB Tree): CGAL::AABB_tree< AABBTraits > Class Template Reference

CGAL 6.2 - Fast Intersection and Distance Computation (AABB Tree)

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CGAL::AABB_tree< AABBTraits > Class Template Reference
Reference Manual

`#include <CGAL/AABB_tree.h>`

## Definition

template<typename AABBTraits>
class CGAL::AABB_tree< AABBTraits >

Static data structure for efficient intersection and distance computations in 2D and 3D.

It builds a hierarchy of axis-aligned bounding boxes (an AABB tree) from a set of 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_indexed_triangle_2_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_polyline_segment_2_example.cpp, AABB_tree/AABB_ray_shooting_example.cpp, AABB_tree/AABB_segment_2_example.cpp, AABB_tree/AABB_segment_3_example.cpp, AABB_tree/AABB_triangle_2_example.cpp, and AABB_tree/AABB_triangle_3_example.cpp.

## Public Member Functions
AABBTraits::Primitive::Datum_reference | datum (Primitive &p) const
| Returns the datum (geometric object) represented `p`.

## Types
typedef AABBTraits::FT | FT
| Number type returned by the distance queries.
typedef AABBTraits::Point | Point
| Type of 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
| Point and Primitive Id type.
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`.
| AABB_tree (Self &&) noexcept
| move constructor
Self & | operator= (Self &&) noexcept
| assignment operator
| AABB_tree (const Self &)=delete
Self & | operator= (const Self &)=delete
template<typename InputIterator , typename ... T>
| AABB_tree (InputIterator first, InputIterator beyond, T &&...)
| builds the data structure from a sequence of primitives.
template<typename ... T>
void | build (T &&...)
| triggers the (re)construction of the internal tree structure.

## Operations
template<typename ConstPrimitiveIterator , typename ... T>
void | rebuild (ConstPrimitiveIterator first, ConstPrimitiveIterator beyond, T &&...)
| is equivalent to calling `clear()`, `insert(first,last,t...)`,
template<typename InputIterator , typename ... T>
void | insert (InputIterator first, InputIterator beyond, T &&...)
| adds a sequence of primitives to the set of primitives of the AABB tree.
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 and the search tree if it was constructed, and switches on the usage of the search tree to find the hint for the distance queries
const Bounding_box | bbox () const
| returns the axis-aligned bounding box of the whole tree.
size_type | size () const
| returns the number of primitives in the tree.
bool | empty () const
| returns `true`, iff the tree contains no primitive.

## 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.
template<typename Query >
size_type | number_of_intersected_primitives (const Query &query) const
| returns the number of primitives intersected by the query.
template<typename Query , typename OutputIterator >
OutputIterator | all_intersected_primitives (const Query &query, OutputIterator out) const
| puts in `out` the ids of all intersected primitives.
template<typename Query >
std::optional< Primitive_id > | any_intersected_primitive (const Query &query) const
| returns the id of the intersected primitive that is encountered first in the tree traversal, iff the query intersects at least one of the input primitives.

## Intersections
template<typename Query , typename OutputIterator >
OutputIterator | all_intersections (const Query &query, OutputIterator out) const
| puts in `out` all intersections, as objects of `Intersection_and_primitive_id<Query>::Type`, between the query and the input data to the iterator.
template<typename Query >
std::optional< typename Intersection_and_primitive_id< Query >::Type > | any_intersection (const Query &query) const
| returns if any the intersection that is encountered first in the tree traversal.
template<typename Ray , typename SkipFunctor >
std::optional< typename Intersection_and_primitive_id< Ray >::Type > | first_intersection (const Ray &query, const SkipFunctor &skip) const
| returns the intersection and primitive id closest to the source point of the ray query.
template<typename Ray , typename SkipFunctor >
std::optional< Primitive_id > | first_intersected_primitive (const Ray &query, const SkipFunctor &skip) const
| returns the primitive id closest to the source point of the ray query.

## Distance Queries
FT | squared_distance (const Point &query) const
| returns the minimum squared distance between the query point and all input primitives.
Point | closest_point (const Point &query) const
| returns the point in the union of all input primitives which is closest to the query.
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.

## 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 a kernel with exact constructions, 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. This KD-tree provides very good hints that allow the algorithms to run much faster than when `do_not_accelerate_distance_queries()` that makes the hint to always be 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. The internal KD-tree is always used if no call to `do_not_accelerate_distance_queries()` was made since object creation or the last call to `clear()`. It will be built by the first distance query or by a call to `accelerate_distance_queries()`.
bool | accelerate_distance_queries ()
| constructs the internal search tree from a point set taken on the internal primitives returns `true` iff successful memory allocation
void | do_not_accelerate_distance_queries ()
| turns off the usage of the internal search tree and clears it if it was already constructed.
template<typename ConstPointIterator >
bool | accelerate_distance_queries (ConstPointIterator first, ConstPointIterator beyond)
| constructs an internal KD-tree containing the specified point set, to be used as the set of potential hints for accelerating the distance queries.
FT | squared_distance (const Point &query, const Point &hint) const
| returns the minimum squared distance between the query point and all input primitives.
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.
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.

## Member Typedef Documentation

## ◆ Object_and_primitive_id

template<typename AABBTraits >

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

Deprecated:

## Constructor & Destructor Documentation

## ◆ AABB_tree()

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

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

builds the data structure from a sequence of primitives.

Parameters

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

constructs an empty tree followed by a call to `insert(first,last,t...)`. The tree stays empty if the memory allocation is not successful.

## Member Function Documentation

## ◆ accelerate_distance_queries()

template<typename AABBTraits >
template<typename ConstPointIterator >

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

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

Note that the search tree built in this function will not be invalidated by the insertion of a new primitive, and an explicit call to `accelerate_distance_queries()` is needed to update the search tree.

Template Parameters

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

## ◆ all_intersected_primitives()

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

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

puts in `out` the ids of all intersected primitives.

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` operators are defined in the traits class `AABBTraits`.

## ◆ all_intersections()

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

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

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

Template Parameters

  Query | must be a type for which `Do_intersect` and `Intersection` operators are defined in the traits class `AABBTraits`.

## ◆ any_intersected_primitive()

template<typename AABBTraits >
template<typename Query >

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

returns the id of the intersected primitive that is encountered first in the tree traversal, 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` operators are defined in the traits class `AABBTraits`.

## ◆ any_intersection()

template<typename AABBTraits >
template<typename Query >

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

returns if any the intersection that is encountered first in the tree traversal.

No particular order is guaranteed over the tree traversal, e.g, the primitive returned is not necessarily the closest from the query.

Template Parameters

  Query | must be a type for which `Do_intersect` and `Intersection` operators are defined in the traits class `AABBTraits`.

## ◆ build()

template<typename AABBTraits >
template<typename ... T>

void CGAL::AABB_tree< AABBTraits >::build | ( | T && | ... | ) |

triggers the (re)construction of the internal tree structure.

The internal tree structure is automatically invalidated by the insertion of any primitives after one or more calls to `insert()`. This procedure is called implicitly at the first call to a query member function. An explicit call to `build()` must be made to ensure that the next call to a query function will not trigger the construction of the data structure. A call to `AABBTraits::set_shared_data(t...)` This procedure has a complexity of \(O(n log(n))\), where \(n\) is the number of primitives of the tree.

Examples
  AABB_tree/AABB_polyline_segment_2_example.cpp.

## ◆ closest_point() [1/2]

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.

Precondition
  `!empty()`

## ◆ closest_point() [2/2]

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()`

## ◆ closest_point_and_primitive() [1/2]

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.

Precondition
  `!empty()`

## ◆ closest_point_and_primitive() [2/2]

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()`

## ◆ do_intersect()

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` operators are defined in the traits class `AABBTraits`.

## ◆ first_intersected_primitive()

template<typename AABBTraits >
template<typename Ray , typename SkipFunctor >

std::optional< Primitive_id > CGAL::AABB_tree< AABBTraits >::first_intersected_primitive | ( | const Ray & | query,
|  | const SkipFunctor & | skip
| ) |  | const

returns the primitive id closest to the source point of the ray query.

Template Parameters

  Ray | must be the same as `AABBTraits::Ray` and `do_intersect` predicates and intersections for it must be defined.
  Skip | a functor with an operator `bool operator()(const Primitive_id& id) const` that returns `true` in order to skip the primitive. Defaults to a functor that always returns `false`.

`AABBTraits` must be a model of `AABBRayIntersectionTraits` to call this member function.

## ◆ first_intersection()

template<typename AABBTraits >
template<typename Ray , typename SkipFunctor >

std::optional< typename Intersection_and_primitive_id< Ray >::Type > CGAL::AABB_tree< AABBTraits >::first_intersection | ( | const Ray & | query,
|  | const SkipFunctor & | skip
| ) |  | const

returns the intersection and primitive id closest to the source point of the ray query.

Template Parameters

  Ray | must be the same as `AABBTraits::Ray` and `do_intersect` predicates and intersections for it must be defined.
  Skip | a functor with an operator `bool operator()(const Primitive_id& id) const` that returns `true` in order to skip the primitive. Defaults to a functor that always returns `false`.

Note
  `skip` might be given some primitives that are not intersected by `query` because the intersection test is done after the skip test. Also note that the order the primitives are given to `skip` is not necessarily the intersection order with `query`.

`AABBTraits` must be a model of `AABBRayIntersectionTraits` to call this member function.

## ◆ insert()

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

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

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

`InputIterator` is any iterator and the parameter pack `T` contains 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.

## ◆ number_of_intersected_primitives()

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` operators are defined in the traits class `AABBTraits`.

## ◆ squared_distance() [1/2]

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.

Precondition
  `!empty()`

## ◆ squared_distance() [2/2]

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()`