CGAL 6.2 - Polygon Mesh Processing

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Typedefs | Functions
Location Functions
Reference Manual

##

Functions to locate points on a mesh, and manipulate such locations.

## Typedefs
template<typename TriangleMesh >
using | CGAL::Polygon_mesh_processing::descriptor_variant = std::variant< typename boost::graph_traits< TriangleMesh >::vertex_descriptor, typename boost::graph_traits< TriangleMesh >::halfedge_descriptor, typename boost::graph_traits< TriangleMesh >::face_descriptor >
| A variant used in the function `get_descriptor_from_location()`.
template<typename FT >
using | CGAL::Polygon_mesh_processing::Barycentric_coordinates = std::array< FT, 3 >
| A triplet of coordinates describing the barycentric coordinates of a point with respect to the vertices of a triangular face.
template<typename TriangleMesh , typename FT >
using | CGAL::Polygon_mesh_processing::Face_location = std::pair< typename boost::graph_traits< TriangleMesh >::face_descriptor, Barycentric_coordinates< FT > >
| If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

## Functions
template<typename GeomTraits , typename Point >
Barycentric_coordinates< typename GeomTraits::FT > | CGAL::Polygon_mesh_processing::barycentric_coordinates (const Point &p, const Point &q, const Point &r, const Point &query, const GeomTraits &gt)
| Given a set of three points and a query point, computes the barycentric coordinates of the query point with respect to the first three points.
template<typename FT , typename TriangleMesh >
descriptor_variant< TriangleMesh > | CGAL::Polygon_mesh_processing::get_descriptor_from_location (const Face_location< TriangleMesh, FT > &loc, const TriangleMesh &tm)
| Given a location, returns a descriptor to the simplex of smallest dimension on which the point corresponding to the location lies.
template<typename FT , typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>
Point | CGAL::Polygon_mesh_processing::construct_point (const Face_location< TriangleMesh, FT > &loc, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| Given a location in a face, returns the geometric position described by these coordinates, as a point.

## Random Location Generation
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::random_location_on_halfedge (typename boost::graph_traits< TriangleMesh >::halfedge_descriptor hd, const TriangleMesh &tm, CGAL::Random &rnd=get_default_random())
| returns a random point over the halfedge `hd`, as a location.
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::random_location_on_face (typename boost::graph_traits< TriangleMesh >::face_descriptor fd, const TriangleMesh &tm, CGAL::Random &rnd=get_default_random())
| returns a random point over the face `fd`, as a location.
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::random_location_on_mesh (const TriangleMesh &tm, CGAL::Random &rnd=get_default_random())
| returns a random point over the mesh `tm`.

## Location Predicates
template<typename FT , typename TriangleMesh >
bool | CGAL::Polygon_mesh_processing::is_on_vertex (const Face_location< TriangleMesh, FT > &loc, const typename boost::graph_traits< TriangleMesh >::vertex_descriptor vd, const TriangleMesh &tm)
| Given a location, returns whether the location is on the vertex `vd` or not.
template<typename FT , typename TriangleMesh >
bool | CGAL::Polygon_mesh_processing::is_on_halfedge (const Face_location< TriangleMesh, FT > &loc, const typename boost::graph_traits< TriangleMesh >::halfedge_descriptor hd, const TriangleMesh &tm)
| Given a location, returns whether this location is on the halfedge `hd` or not.
template<typename FT , typename TriangleMesh >
bool | CGAL::Polygon_mesh_processing::is_in_face (const Barycentric_coordinates< FT > &bar, const TriangleMesh &tm)
| Given a set of barycentric coordinates, returns whether those barycentric coordinates correspond to a point within the face (boundary included), that is, if all the barycentric coordinates are positive.
template<typename FT , typename TriangleMesh >
bool | CGAL::Polygon_mesh_processing::is_in_face (const Face_location< TriangleMesh, FT > &loc, const TriangleMesh &tm)
| Given a location, returns whether the location is in the face (boundary included) or not.
template<typename FT , typename TriangleMesh >
bool | CGAL::Polygon_mesh_processing::is_on_face_border (const Face_location< TriangleMesh, FT > &loc, const TriangleMesh &tm)
| Given a location, returns whether the location is on the boundary of the face or not.
template<typename FT , typename TriangleMesh >
bool | CGAL::Polygon_mesh_processing::is_on_mesh_border (const Face_location< TriangleMesh, FT > &loc, const TriangleMesh &tm)
| Given a location, returns whether the location is on the border of the mesh or not.

## Point Location
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_vertex (typename boost::graph_traits< TriangleMesh >::vertex_descriptor vd, const TriangleMesh &tm)
| returns the location of the given vertex `vd` as a location, that is an ordered pair specifying a face incident to `vd` and the barycentric coordinates of the vertex `vd` in that face.
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_vertex (const typename boost::graph_traits< TriangleMesh >::vertex_descriptor vd, const typename boost::graph_traits< TriangleMesh >::face_descriptor fd, const TriangleMesh &tm)
| returns the location of a given vertex as a location in `fd`, that is an ordered pair composed of `fd` and of the barycentric coordinates of the vertex in `fd`.
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_on_halfedge (const typename boost::graph_traits< TriangleMesh >::halfedge_descriptor hd, const FT t, const TriangleMesh &tm)
| Given a point described by a halfedge `hd` and a scalar `t` as `p = (1 - t) * source(hd, tm) + t * target(hd, tm)`, returns this location along the given edge as a location, that is an ordered pair specifying a face containing the location and the barycentric coordinates of that location in that face.
template<typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_in_face (const Point &query, const typename boost::graph_traits< TriangleMesh >::face_descriptor fd, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| Given a point `query` and a face `fd` of a triangulated surface mesh, returns this location as a location, that is an ordered pair composed of `fd` and of the barycentric coordinates of `query` with respect to the vertices of `fd`.
template<typename FT , typename TriangleMesh >
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_in_adjacent_face (const Face_location< TriangleMesh, FT > &loc, const typename boost::graph_traits< TriangleMesh >::face_descriptor fd, const TriangleMesh &tm)
| Given a location and a second face adjacent to the first, returns the location of the point in the second face.

## Nearest Face Location Queries
The following functions can be used to find the closest point on a triangle mesh, given either a point or a ray. This closest point is computed using a `CGAL::AABB_tree`. Users intending to call location functions on more than a single point (or ray) should first compute an AABB tree to store it (otherwise, it will be recomputed every time). Note that since the AABB tree class is a 3D structure, it might be required to wrap your point property map to convert your point type to the 3D point type (i.e., your traits' `Point_3`) if you are working with a 2D triangle structure.
template<typename TriangleMesh , typename Point3VPM , typename NamedParameters = parameters::Default_named_parameters>
void | CGAL::Polygon_mesh_processing::build_AABB_tree (const TriangleMesh &tm, AABB_tree< AABB_traits_3< Geom_traits, CGAL::AABB_face_graph_triangle_primitive< TriangleMesh, Point3VPM > > > &outTree, const NamedParameters &np=parameters::default_values())
| creates an AABB tree suitable for use with `locate_with_AABB_tree()`.
template<typename TriangleMesh , typename Point3VPM , typename NamedParameters = parameters::Default_named_parameters>
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_with_AABB_tree (const Point &p, const AABB_tree< AABB_traits_3< Geom_traits, AABB_face_graph_triangle_primitive< TriangleMesh, Point3VPM > > > &tree, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| returns the face location nearest to the given point, as a location.
template<typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate (const Point &p, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| returns the nearest face location to the given point.
template<typename TriangleMesh , typename Point3VPM , typename NamedParameters = parameters::Default_named_parameters>
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate_with_AABB_tree (const Ray &ray, const AABB_tree< AABB_traits_3< Geom_traits, AABB_face_graph_triangle_primitive< TriangleMesh, Point3VPM > > > &tree, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| returns the face location along `ray` nearest to its source point.
template<typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>
Face_location< TriangleMesh, FT > | CGAL::Polygon_mesh_processing::locate (const Ray &ray, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| returns the face location along `ray` nearest to its source point.

## Typedef Documentation

## ◆ Barycentric_coordinates

template<typename FT >

using CGAL::Polygon_mesh_processing::Barycentric_coordinates = typedef std::array<FT, 3>

`#include <CGAL/Polygon_mesh_processing/locate.h>`

A triplet of coordinates describing the barycentric coordinates of a point with respect to the vertices of a triangular face.

See also
  `Face_location`

## ◆ Face_location

template<typename TriangleMesh , typename FT >

using CGAL::Polygon_mesh_processing::Face_location = typedef std::pair<typename boost::graph_traits<TriangleMesh>::face_descriptor, Barycentric_coordinates<FT> >

`#include <CGAL/Polygon_mesh_processing/locate.h>`

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

## Function Documentation

## ◆ barycentric_coordinates()

template<typename GeomTraits , typename Point >

Barycentric_coordinates< typename GeomTraits::FT > CGAL::Polygon_mesh_processing::barycentric_coordinates | ( | const Point & | p,
|  | const Point & | q,
|  | const Point & | r,
|  | const Point & | query,
|  | const GeomTraits & | gt
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a set of three points and a query point, computes the barycentric coordinates of the query point with respect to the first three points.

Template Parameters

  GeomTraits | the type of a geometric traits. Must be a model of `Kernel` and be compatible with the template parameter `Point`.
  Point | the type of a geometric 2D or 3D point

Parameters

  p,q,r | three points with respect to whom the barycentric coordinates of `query` will be computed
  query | the query point whose barycentric coordinates will be computed
  gt | an instance of the geometric traits

Precondition
  `p`, `q`, and `r` are not collinear.
  `query` lies on the plane defined by `p`, `q`, and `r`.

## ◆ build_AABB_tree()

template<typename TriangleMesh , typename Point3VPM , typename NamedParameters = parameters::Default_named_parameters>

void CGAL::Polygon_mesh_processing::build_AABB_tree | ( | const TriangleMesh & | tm,
|  | AABB_tree< AABB_traits_3< Geom_traits, CGAL::AABB_face_graph_triangle_primitive< TriangleMesh, Point3VPM > > > & | outTree,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

creates an AABB tree suitable for use with `locate_with_AABB_tree()`.

This function should first be called by users who intend to locate multiple points: in this case, it is better to first build an AABB tree, and use the function `locate_with_AABB_tree()` that takes as parameter an AABB tree, instead of calling `locate()` multiple times, which will build a new AABB tree on every call.

Template Parameters

  TriangleMesh | must be a model of `FaceListGraph`
  Point3VPM | must be a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and the CGAL 3D point type (your traits' `Point_3`) as value type.
  NamedParameters | a sequence of Named Parameters

Parameters

  tm | a triangulated surface mesh
  outTree | output parameter that stores the computed `AABB_tree`
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. Extra: Must be identical to the traits used in the template parameter of the `AABB_traits`. |

Examples
  Polygon_mesh_processing/locate_example.cpp.

## ◆ construct_point()

template<typename FT , typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>

Point CGAL::Polygon_mesh_processing::construct_point | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location in a face, returns the geometric position described by these coordinates, as a point.

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`
  NamedParameters | a sequence of Named Parameters

Parameters

  loc | the location from which a point is constructed
  tm | a triangulated surface mesh
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. Extra: If such traits class is provided, its type `FT` must be identical to the template parameter `FT` of this function. |

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.

Returns
  a point whose type is the same as the value type of the vertex point property map provided by the user or via named parameters, or the internal point map of the mesh `tm`.

Examples
  Polygon_mesh_processing/locate_example.cpp.

## ◆ get_descriptor_from_location()

template<typename FT , typename TriangleMesh >

descriptor_variant< TriangleMesh > CGAL::Polygon_mesh_processing::get_descriptor_from_location | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location, returns a descriptor to the simplex of smallest dimension on which the point corresponding to the location lies.

In other words:

- if the point lies on a vertex, this function returns a `boost::graph_traits<TriangleMesh>::vertex_descriptor` `v`;
- if the point lies on a halfedge, this function returns a `boost::graph_traits<TriangleMesh>::halfedge_descriptor` `hd` (note that in that case, `loc.first == face(hd, tm)` holds).
- otherwise, this function returns a `boost::graph_traits<TriangleMesh>::face_descriptor` `fd` (equal to `loc.first`).

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | a location with `loc.first` a face of `tm`
  tm | a triangulated surface mesh

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.
  `loc` describes the barycentric coordinates of a point that lives within the face (boundary included), meaning the barycentric coordinates are all positive.

## ◆ is_in_face() [1/2]

template<typename FT , typename TriangleMesh >

bool CGAL::Polygon_mesh_processing::is_in_face | ( | const Barycentric_coordinates< FT > & | bar,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a set of barycentric coordinates, returns whether those barycentric coordinates correspond to a point within the face (boundary included), that is, if all the barycentric coordinates are positive.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  bar | an array of barycentric coordinates
  tm | a triangulated surface mesh

## ◆ is_in_face() [2/2]

template<typename FT , typename TriangleMesh >

bool CGAL::Polygon_mesh_processing::is_in_face | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location, returns whether the location is in the face (boundary included) or not.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | a location with `loc.first` a face of `tm`
  tm | a triangulated surface mesh

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.

## ◆ is_on_face_border()

template<typename FT , typename TriangleMesh >

bool CGAL::Polygon_mesh_processing::is_on_face_border | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location, returns whether the location is on the boundary of the face or not.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | a location with `loc.first` a face of `tm`
  tm | a triangulated surface mesh

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.

Examples
  Polygon_mesh_processing/locate_example.cpp.

## ◆ is_on_halfedge()

template<typename FT , typename TriangleMesh >

bool CGAL::Polygon_mesh_processing::is_on_halfedge | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const typename boost::graph_traits< TriangleMesh >::halfedge_descriptor | hd,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location, returns whether this location is on the halfedge `hd` or not.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | a location with `loc.first` a face of `tm`
  hd | a halfedge of `tm`
  tm | a triangulated surface mesh

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.

## ◆ is_on_mesh_border()

template<typename FT , typename TriangleMesh >

bool CGAL::Polygon_mesh_processing::is_on_mesh_border | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location, returns whether the location is on the border of the mesh or not.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | a location with `loc.first` a face of `tm`
  tm | a triangulated surface mesh

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.

Examples
  Polygon_mesh_processing/locate_example.cpp.

## ◆ is_on_vertex()

template<typename FT , typename TriangleMesh >

bool CGAL::Polygon_mesh_processing::is_on_vertex | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const typename boost::graph_traits< TriangleMesh >::vertex_descriptor | vd,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location, returns whether the location is on the vertex `vd` or not.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | a location with `loc.first` a face of `tm`
  vd | a vertex of `tm`
  tm | a triangulated surface mesh

Precondition
  `loc.first` is a face descriptor corresponding to a face of `tm`.

## ◆ locate() [1/2]

template<typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate | ( | const Point & | p,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns the nearest face location to the given point.

Note that this function will build an AABB tree on each call. If you need to call this function more than once, first use `build_AABB_tree()` to create a an AABB tree that you can store and use the function `locate_with_AABB_tree()`.

Template Parameters

  TriangleMesh | must be a model of `FaceListGraph`.
  NamedParameters | a sequence of Named Parameters

Parameters

  p | the point to locate on the input triangulated surface mesh
  tm | a triangulated surface mesh
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. |
  a tolerance value used to snap barycentric coordinates Type: double Default: `0` Extra: Depending on the geometric traits used, the computation of the barycentric coordinates might be an inexact construction, thus leading to sometimes surprising values (e.g. a triplet `[0.5, 0.5, -1-e17]` for a point at the middle of an edge). The coordinates will be snapped towards `0` and `1` if the difference is smaller than the tolerance value, while still ensuring that the total sum of the coordinates is `1`. |

Examples
  Polygon_mesh_processing/locate_example.cpp.

## ◆ locate() [2/2]

template<typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate | ( | const Ray & | ray,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns the face location along `ray` nearest to its source point.

If the ray does not intersect the mesh, a default constructed location is returned.

Note that this function will build an AABB tree on each call. If you need to call this function more than once, use `build_AABB_tree()` to cache a copy of the `AABB_tree`, and use the overloads of this function that accept a reference to an AABB tree as input.

Template Parameters

  TriangleMesh | must be a model of `FaceListGraph`.
  NamedParameters | a sequence of Named Parameters

Parameters

  ray | a ray to intersect with the input triangulated surface mesh
  tm | the input triangulated surface mesh
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. |
  a tolerance value used to snap barycentric coordinates Type: double Default: `0` Extra: Depending on the geometric traits used, the computation of the barycentric coordinates might be an inexact construction, thus leading to sometimes surprising values (e.g. a triplet `[0.5, 0.5, -1-e17]` for a point at the middle of an edge). The coordinates will be snapped towards `0` and `1` if the difference is smaller than the tolerance value, while still ensuring that the total sum of the coordinates is `1`. |

Precondition
  `ray` is an object with the same ambient dimension as the point type (the value type of the vertex point map).

## ◆ locate_in_adjacent_face()

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_in_adjacent_face | ( | const Face_location< TriangleMesh, FT > & | loc,
|  | const typename boost::graph_traits< TriangleMesh >::face_descriptor | fd,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a location and a second face adjacent to the first, returns the location of the point in the second face.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  loc | the first location, with `loc.first` being a face of `tm`
  fd | the second face, adjacent to `loc.first`
  tm | the triangle mesh to which `fd` belongs

Precondition
  `loc` corresponds to a point that lies on a face incident to both `loc.first` and `fd`.

## ◆ locate_in_face()

template<typename TriangleMesh , typename NamedParameters = parameters::Default_named_parameters>

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_in_face | ( | const Point & | query,
|  | const typename boost::graph_traits< TriangleMesh >::face_descriptor | fd,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a point `query` and a face `fd` of a triangulated surface mesh, returns this location as a location, that is an ordered pair composed of `fd` and of the barycentric coordinates of `query` with respect to the vertices of `fd`.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  TriangleMesh | must be a model of `FaceGraph`
  NamedParameters | a sequence of Named Parameters

Parameters

  query | a point, whose type is equal to the value type of the vertex point property map (either user-provided via named parameters or the internal point map of the mesh `tm`)
  fd | a face of `tm`
  tm | a triangulated surface mesh
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. Extra: If such traits class is provided, its type `FT` must be identical to the template parameter `FT` of this function. |
  a tolerance value used to snap barycentric coordinates Type: double Default: `0` Extra: Depending on the geometric traits used, the computation of the barycentric coordinates might be an inexact construction, thus leading to sometimes surprising values (e.g. a triplet `[0.5, 0.5, -1-e17]` for a point at the middle of an edge). The coordinates will be snapped towards `0` and `1` if the difference is smaller than the tolerance value, while still ensuring that the total sum of the coordinates is `1`. |

Precondition
  `fd` is not the null face

Returns
  a face location. The type `FT` is deduced from the geometric traits, either provided by the user via named parameters (with `geom_traits`) or using `CGAL::Kernel_traits` and the point type of the vertex point property map in use.

## ◆ locate_on_halfedge()

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_on_halfedge | ( | const typename boost::graph_traits< TriangleMesh >::halfedge_descriptor | hd,
|  | const FT | t,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

Given a point described by a halfedge `hd` and a scalar `t` as `p = (1 - t) * source(hd, tm) + t * target(hd, tm)`, returns this location along the given edge as a location, that is an ordered pair specifying a face containing the location and the barycentric coordinates of that location in that face.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  hd | a halfedge of `tm`
  t | the parametric distance of the desired point along `hd`
  tm | a triangulated surface mesh

## ◆ locate_vertex() [1/2]

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_vertex | ( | const typename boost::graph_traits< TriangleMesh >::vertex_descriptor | vd,
|  | const typename boost::graph_traits< TriangleMesh >::face_descriptor | fd,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns the location of a given vertex as a location in `fd`, that is an ordered pair composed of `fd` and of the barycentric coordinates of the vertex in `fd`.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  vd | a vertex of `tm` and a vertex of the face `fd`
  fd | a face of `tm`
  tm | a triangulated surface mesh

Precondition
  `fd` is not the null face

## ◆ locate_vertex() [2/2]

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_vertex | ( | typename boost::graph_traits< TriangleMesh >::vertex_descriptor | vd,
|  | const TriangleMesh & | tm
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns the location of the given vertex `vd` as a location, that is an ordered pair specifying a face incident to `vd` and the barycentric coordinates of the vertex `vd` in that face.

If `tm` is the input triangulated surface mesh and given the pair (`f`, `bc`) such that `bc` is the triplet of barycentric coordinates `(w0, w1, w2)`, the correspondence between the coordinates in `bc` and the vertices of the face `f` is the following:

- `w0` corresponds to `source(halfedge(f, tm), tm)`
- `w1` corresponds to `target(halfedge(f, tm), tm)`
- `w2` corresponds to `target(next(halfedge(f, tm), tm), tm)`

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  vd | a vertex of `tm`
  tm | a triangulated surface mesh

Precondition
  `vd` is not an isolated vertex

## ◆ locate_with_AABB_tree() [1/2]

template<typename TriangleMesh , typename Point3VPM , typename NamedParameters = parameters::Default_named_parameters>

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_with_AABB_tree | ( | const Point & | p,
|  | const AABB_tree< AABB_traits_3< Geom_traits, AABB_face_graph_triangle_primitive< TriangleMesh, Point3VPM > > > & | tree,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns the face location nearest to the given point, as a location.

Note that it is possible for the triangle mesh to have ambiant dimension `2` (e.g. the mesh is a 2D triangulation, or a CGAL::Surface_mesh<CGAL::Point_2<Kernel> >), as long as an appropriate vertex point property map is passed in the AABB tree, which will convert from 2D to 3D.

Template Parameters

  TriangleMesh | must be a model of `FaceListGraph`
  Point3VPM | must be a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and the CGAL 3D point type (your traits' `Point_3`) as value type.
  NamedParameters | a sequence of Named Parameters

Parameters

  p | the point to locate on the input triangulated surface mesh
  tree | an AABB tree containing the triangular faces of the input surface mesh to perform the point location with
  tm | a triangulated surface mesh
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. Extra: Must be identical to the traits used in the template parameter of the `AABB_traits`. |
  a tolerance value used to snap barycentric coordinates Type: double Default: `0` Extra: Depending on the geometric traits used, the computation of the barycentric coordinates might be an inexact construction, thus leading to sometimes surprising values (e.g. a triplet `[0.5, 0.5, -1-e17]` for a point at the middle of an edge). The coordinates will be snapped towards `0` and `1` if the difference is smaller than the tolerance value, while still ensuring that the total sum of the coordinates is `1`. |

Returns
  a face location. The type `FT` is deduced from the geometric traits, either provided by the user via named parameters (with `geom_traits`) or using `CGAL::Kernel_traits` and the point type of the vertex point property map in use.

Examples
  Polygon_mesh_processing/locate_example.cpp.

## ◆ locate_with_AABB_tree() [2/2]

template<typename TriangleMesh , typename Point3VPM , typename NamedParameters = parameters::Default_named_parameters>

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::locate_with_AABB_tree | ( | const Ray & | ray,
|  | const AABB_tree< AABB_traits_3< Geom_traits, AABB_face_graph_triangle_primitive< TriangleMesh, Point3VPM > > > & | tree,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns the face location along `ray` nearest to its source point.

If the ray does not intersect the mesh, a default constructed location is returned.

Template Parameters

  TriangleMesh | must be a model of `FaceListGraph`.
  Point3VPM | must be a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and the CGAL 3D point type (your traits' `Point_3`) as value type.
  NamedParameters | a sequence of Named Parameters

Parameters

  ray | a ray to intersect with the input triangulated surface mesh
  tree | an AABB tree containing the triangular faces of the input surface mesh to perform the point location with
  tm | a triangulated surface mesh
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tm)` |
  an instance of a geometric traits class Type: a class model of `Kernel` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. Extra: Must be identical to the traits used in the template parameter of the `AABB_traits`. |
  a tolerance value used to snap barycentric coordinates Type: double Default: `0` Extra: Depending on the geometric traits used, the computation of the barycentric coordinates might be an inexact construction, thus leading to sometimes surprising values (e.g. a triplet `[0.5, 0.5, -1-e17]` for a point at the middle of an edge). The coordinates will be snapped towards `0` and `1` if the difference is smaller than the tolerance value, while still ensuring that the total sum of the coordinates is `1`. |

Precondition
  `ray` is an object with the same ambient dimension as the point type (the value type of the vertex point map).

## ◆ random_location_on_face()

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::random_location_on_face | ( | typename boost::graph_traits< TriangleMesh >::face_descriptor | fd,
|  | const TriangleMesh & | tm,
|  | CGAL::Random & | rnd = `get_default_random()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns a random point over the face `fd`, as a location.

The random point is on the face, meaning that all its barycentric coordinates are positive. It is constructed by uniformly picking a value `u` between `0` and `1`, a value `v` between `1-u`, and setting the barycentric coordinates to `u`, `v`, and `1-u-v` for respectively the source and target of `halfedge(fd, tm)`, and the third point.

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  fd | a face of `tm`
  tm | a triangulated surface mesh
  rnd | optional random number generator

## ◆ random_location_on_halfedge()

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::random_location_on_halfedge | ( | typename boost::graph_traits< TriangleMesh >::halfedge_descriptor | hd,
|  | const TriangleMesh & | tm,
|  | CGAL::Random & | rnd = `get_default_random()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns a random point over the halfedge `hd`, as a location.

The random point is chosen on the halfedge, meaning that all its barycentric coordinates are positive. It is constructed by uniformly generating a value `t` between `0` and `1` and setting the barycentric coordinates to `t`, `1-t`, and `0` for respetively the source and target of `hd`, and the third vertex.

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  hd | a halfedge of `tm`
  tm | a triangulated surface mesh
  rnd | optional random number generator

## ◆ random_location_on_mesh()

template<typename FT , typename TriangleMesh >

Face_location< TriangleMesh, FT > CGAL::Polygon_mesh_processing::random_location_on_mesh | ( | const TriangleMesh & | tm,
|  | CGAL::Random & | rnd = `get_default_random()`
| ) |  |

`#include <CGAL/Polygon_mesh_processing/locate.h>`

returns a random point over the mesh `tm`.

The returned location is obtained by choosing a random face of the mesh and a random point on that face. The barycentric coordinates of the point in the face are thus all positive. Note that all faces have the same probability to be chosen.

Template Parameters

  FT | must be a model of `FieldNumberType`
  TriangleMesh | must be a model of `FaceGraph`

Parameters

  tm | a triangulated surface mesh
  rnd | optional random number generator

See also
  `random_location_on_face()`