CGAL 6.2 - 3D Generalized Barycentric Coordinates

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Functions
Free Functions
3D Generalized Barycentric Coordinates Reference

##

Free functions to compute barycentric coordinates.

## Functions
template<typename TriangleMesh , typename CoordinateRange , typename VertexPointMap , typename GeomTraits = typename CGAL::Kernel_traits<typename boost::property_traits<VertexPointMap>::value_type>::type>
boost::property_traits< VertexPointMap >::value_type | CGAL::Barycentric_coordinates::apply_barycentric_coordinates_3 (const TriangleMesh &tmesh, const CoordinateRange &coordinates, VertexPointMap vpm, GeomTraits geom_traits=GeomTraits())
| computes a point location from barycentric coordinates with respect to a triangle mesh.
template<typename PointRange , typename CoordinateRange , typename GeomTraits = typename Kernel_traits<typename boost::range_value<PointRange>::type>::Kernel>
boost::range_value< PointRange >::type | CGAL::Barycentric_coordinates::apply_barycentric_coordinates_3 (const PointRange &points, const CoordinateRange &coordinates, GeomTraits geom_traits=GeomTraits())
| computes a point location from barycentric coordinates with respect to a set of points.
template<typename TriangleMesh , typename Point , typename OutputIterator , typename NamedParameters = parameters::Default_named_parameters>
OutputIterator | CGAL::Barycentric_coordinates::discrete_harmonic_coordinates_3 (const TriangleMesh &tmesh, const Point &query, OutputIterator oi, const NamedParameters &np=parameters::default_values())
| computes 3D discrete harmonic coordinates with respect to a closed convex triangle mesh.
template<typename TriangleMesh , typename Point , typename OutputIterator , typename NamedParameters = parameters::Default_named_parameters>
OutputIterator | CGAL::Barycentric_coordinates::mean_value_coordinates_3 (const TriangleMesh &tmesh, const Point &query, OutputIterator oi, const NamedParameters &np=parameters::default_values())
| computes 3D mean value barycentric coordinates with respect to a closed triangle mesh.
template<typename TriangleMesh , typename Point , typename OutputIterator , typename NamedParameters = parameters::Default_named_parameters>
OutputIterator | CGAL::Barycentric_coordinates::wachspress_coordinates_3 (const TriangleMesh &tmesh, const Point query, OutputIterator oi, const NamedParameters &np=parameters::default_values())
| computes 3D Wachspress coordinates with respect to a closed convex triangle mesh.
template<typename TriangleMesh , typename Point , typename OutputIterator , typename NamedParameters = parameters::Default_named_parameters>
std::pair< OutputIterator, bool > | CGAL::Barycentric_coordinates::boundary_coordinates_3 (const TriangleMesh &tmesh, const Point &query, OutputIterator oi, const NamedParameters &np=parameters::default_values())
| computes boundary barycentric coordinates with respect to a closed convex triangle mesh.
template<typename OutputIterator , typename GeomTraits >
OutputIterator | CGAL::Barycentric_coordinates::tetrahedron_coordinates_3 (const typename GeomTraits::Point_3 &p0, const typename GeomTraits::Point_3 &p1, const typename GeomTraits::Point_3 &p2, const typename GeomTraits::Point_3 &p3, const typename GeomTraits::Point_3 &query, OutputIterator oi, const GeomTraits &traits)
| computes barycentric coordinates with respect to a tetrahedron.
template<typename GeomTraits >
std::array< typename GeomTraits::FT, 4 > | CGAL::Barycentric_coordinates::tetrahedron_coordinates_3 (const typename GeomTraits::Point_3 &p0, const typename GeomTraits::Point_3 &p1, const typename GeomTraits::Point_3 &p2, const typename GeomTraits::Point_3 &p3, const typename GeomTraits::Point_3 &query, const GeomTraits &traits)
| computes barycentric coordinates with respect to a tetrahedron.

## Function Documentation

## ◆ apply_barycentric_coordinates_3() [1/2]

template<typename PointRange , typename CoordinateRange , typename GeomTraits = typename Kernel_traits<typename boost::range_value<PointRange>::type>::Kernel>

boost::range_value< PointRange >::type CGAL::Barycentric_coordinates::apply_barycentric_coordinates_3 | ( | const PointRange & | points,
|  | const CoordinateRange & | coordinates,
|  | GeomTraits | geom_traits = `GeomTraits()`
| ) |  |

`#include <CGAL/Barycentric_coordinates_3.h>`

computes a point location from barycentric coordinates with respect to a set of points.

This function a computes point location from barycentric coordinates with respect to a set of points.

Template Parameters

  PointRange | is a model of `ConstRange` and `RandomAccessRange` with value type `GeomTraits::Point_3`
  CoordinateRange | a range whose iterator is a model of `ForwardIterator` with value type `GeomTraits::FT`
  GeomTraits | a model of `BarycentricTraits_3`, deduced from the value type of `PointRange` by default.

Parameters

  points | a range of input points
  coordinates | barycentric coordinates of the point
  geom_traits | an instance of `GeomTraits`

Returns
  point with type `GeomTraits::Point_3`

Precondition
  `pts.size()` == `coords.size()`

## ◆ apply_barycentric_coordinates_3() [2/2]

template<typename TriangleMesh , typename CoordinateRange , typename VertexPointMap , typename GeomTraits = typename CGAL::Kernel_traits<typename boost::property_traits<VertexPointMap>::value_type>::type>

boost::property_traits< VertexPointMap >::value_type CGAL::Barycentric_coordinates::apply_barycentric_coordinates_3 | ( | const TriangleMesh & | tmesh,
|  | const CoordinateRange & | coordinates,
|  | VertexPointMap | vpm,
|  | GeomTraits | geom_traits = `GeomTraits()`
| ) |  |

`#include <CGAL/Barycentric_coordinates_3.h>`

computes a point location from barycentric coordinates with respect to a triangle mesh.

This function computes a point location from barycentric coordinates with respect to the vertices of `tmesh`

Template Parameters

  TriangleMesh | must be a model of the concept `FaceListGraph`
  CoordinateRange | a range whose iterator is a model of `ForwardIterator` with value type `GeomTraits::FT`
  VertexPointMap | a property map with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `GeomTraits::Point_3` as value type.
  GeomTraits | a model of `BarycentricTraits_3`, deduced from the value type of `VertexPointMap`, using `CGAL::Kernel_traits`.

Parameters

  tmesh | an instance of `TriangleMesh`
  coordinates | barycentric coordinates of the point
  geom_traits | an instance of `GeomTraits`
  vpm | an instance of `VertexPointMap` that maps a vertex from `tmesh` to `GeomTraits::Point_3`

Returns
  point with type `boost::property_traits<VertexPointMap>::value_type`

Precondition
  `vertices(tmesh).size()` == `coords.size()`

Examples
  Barycentric_coordinates_3/shape_deformation_3.cpp.

## ◆ boundary_coordinates_3()

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

std::pair< OutputIterator, bool > CGAL::Barycentric_coordinates::boundary_coordinates_3 | ( | const TriangleMesh & | tmesh,
|  | const Point & | query,
|  | OutputIterator | oi,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Barycentric_coordinates_3/boundary_coordinates_3.h>`

computes boundary barycentric coordinates with respect to a closed convex triangle mesh.

This function computes boundary barycentric coordinates at a given `query` point with respect to the vertices of a simple `polyhedron`, that is one coordinate per vertex. The coordinates are stored in a destination range beginning at `oi`.

If `query` is at the vertex, the corresponding coordinate is set to 1, while all other coordinates are 0. If `query` is on the face, the three corresponding coordinates are triangle coordinates, while all other coordinates are set to 0. If `query` is not on the boundary, all the coordinates are set to 0.

Template Parameters

  TriangleMesh | must be a model of the concept `FaceListGraph`.
  Point | A model of `GeomTraits::Point_3` with `GeomTraits` being the type of the named parameter `geom_traits`.
  OutputIterator | must be an output iterator accepting `GeomTraits::FT` with `GeomTraits` being the type of the named parameter `geom_traits`.
  NamedParameters | a sequence of Named Parameters

Parameters

  tmesh | an instance of `TriangleMesh`, which must be a convex simplicial polyhedron
  query | a query point
  oi | the beginning of the destination range with the computed coordinates
  np | an optional sequence of Named Parameters among the ones listed below

Returns
  an output iterator to the element in the destination range, one past the last coordinate stored + the flag indicating whether the query point belongs to the polyhedron boundary

Optional Named Parameters

  a property map associating points to the vertices of `tmesh` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point` as value type Default: `boost::get(CGAL::vertex_point, tmesh)` Extra: If this parameter is omitted, an internal property map for `CGAL::vertex_point_t` must be available in `TriangleMesh`. |
  an instance of a geometric traits class Type: a class model of `BarycentricTraits_3` Default: a CGAL Kernel deduced from the value type of the vertex-point map, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. |

Precondition
  num_vertices(`tmesh`) >= 4.
  CGAL::is_triangle_mesh(`tmesh`)
  CGAL::is_closed(`tmesh`).
  CGAL::is_strongly_convex_3(`tmesh`).

## ◆ discrete_harmonic_coordinates_3()

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

OutputIterator CGAL::Barycentric_coordinates::discrete_harmonic_coordinates_3 | ( | const TriangleMesh & | tmesh,
|  | const Point & | query,
|  | OutputIterator | oi,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Barycentric_coordinates_3/Discrete_harmonic_coordinates_3.h>`

computes 3D discrete harmonic coordinates with respect to a closed convex triangle mesh.

This function computes 3D discrete harmonic coordinates at a given `query` point with respect to the vertices of a convex `polyhedron` with triangular faces, that is one coordinate per vertex. The coordinates are stored in a destination range beginning at `oi`.

Internally, the class `Discrete_harmonic_coordinates_3` is used. If one wants to process multiple query points, it is better to use that class. When using the free function, internal memory is allocated for each query point, while when using the class, it is allocated only once, which is much more efficient. However, for a few query points, it is easier to use this function. It can also be used when the processing time is not a concern.

Template Parameters

  TriangleMesh | must be a model of the concept `FaceListGraph`
  Point | A model of `GeomTraits::Point_3` with `GeomTraits` being the type of the named parameter `geom_traits`.
  OutputIterator | must be an output iterator accepting `GeomTraits::FT` with `GeomTraits` being the type of the named parameter `geom_traits`.
  NamedParameters | a sequence of Named Parameters

Parameters

  tmesh | an instance of `TriangleMesh`
  query | a query point
  oi | the beginning of the destination range with the computed coordinates
  np | an optional sequence of Named Parameters among the ones listed below

Returns
  an output iterator to the element in the destination range, one past the last coordinate stored

Optional Named Parameters

  The computation policy for handling points near the boundary of the `tmesh`. Type: `CGAL::Barycentric_coordinates::Computation_policy_3` Default: `Computation_policy_3::FAST_WITH_EDGE_CASES` |
  a property map associating points to the vertices of `tmesh` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point` as value type Default: `boost::get(CGAL::vertex_point, tmesh)` Extra: If this parameter is omitted, an internal property map for `CGAL::vertex_point_t` must be available in `TriangleMesh`. |
  an instance of a geometric traits class Type: a class model of `BarycentricTraits_3` Default: a CGAL Kernel deduced from the value type of the vertex-point map, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. |

Precondition
  num_vertices(`tmesh`) >= 4.
  CGAL::is_triangle_mesh(`tmesh`).
  CGAL::is_closed(`tmesh`).
  CGAL::is_strongly_convex_3(`tmesh`).

Examples
  Barycentric_coordinates_3/discrete_harmonic_coordinates_3.cpp.

## ◆ mean_value_coordinates_3()

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

OutputIterator CGAL::Barycentric_coordinates::mean_value_coordinates_3 | ( | const TriangleMesh & | tmesh,
|  | const Point & | query,
|  | OutputIterator | oi,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Barycentric_coordinates_3/Mean_value_coordinates_3.h>`

computes 3D mean value barycentric coordinates with respect to a closed triangle mesh.

This function computes 3D mean value coordinates at a given `query` point with respect to the vertices of `tmesh`, that is one weight per vertex. The coordinates are stored in a destination range beginning at `oi`.

Internally, the class `Mean_value_coordinates_3` is used. If one wants to process multiple query points, it is better to use that class. When using the free function, internal memory is allocated for each query point, while when using the class, it is allocated only once, which is much more efficient. However, for a few query points, it is easier to use this function. It can also be used when the processing time is not a concern.

Template Parameters

  TriangleMesh | must be a model of the concept `FaceListGraph`
  Point | A model of `GeomTraits::Point_3` with `GeomTraits` being the type of the named parameter `geom_traits`.
  OutputIterator | must be an output iterator accepting `GeomTraits::FT` with `GeomTraits` being the type of the named parameter `geom_traits`.
  NamedParameters | a sequence of Named Parameters

Parameters

  tmesh | an instance of `TriangleMesh`
  query | a query point
  oi | the beginning of the destination range with the computed coordinates
  np | an optional sequence of Named Parameters among the ones listed below

Returns
  an output iterator to the element in the destination range, one past the last coordinates stored

Optional Named Parameters

  The computation policy for handling points near the boundary of the `tmesh`. Type: `CGAL::Barycentric_coordinates::Computation_policy_3` Default: `Computation_policy_3::FAST_WITH_EDGE_CASES` |
  a property map associating points to the vertices of `tmesh` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point` as value type Default: `boost::get(CGAL::vertex_point, tmesh)` Extra: If this parameter is omitted, an internal property map for `CGAL::vertex_point_t` must be available in `TriangleMesh`. |
  an instance of a geometric traits class Type: a class model of `BarycentricTraits_3` Default: a CGAL Kernel deduced from the value type of the vertex-point map, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. |

Precondition
  num_vertices(`tmesh`) >= 4.
  CGAL::is_triangle_mesh(`tmesh`)
  CGAL::is_closed(`tmesh`).

Examples
  Barycentric_coordinates_3/mean_value_coordinates_3.cpp.

## ◆ tetrahedron_coordinates_3() [1/2]

template<typename GeomTraits >

std::array< typename GeomTraits::FT, 4 > CGAL::Barycentric_coordinates::tetrahedron_coordinates_3 | ( | const typename GeomTraits::Point_3 & | p0,
|  | const typename GeomTraits::Point_3 & | p1,
|  | const typename GeomTraits::Point_3 & | p2,
|  | const typename GeomTraits::Point_3 & | p3,
|  | const typename GeomTraits::Point_3 & | query,
|  | const GeomTraits & | traits
| ) |  |

`#include <CGAL/Barycentric_coordinates_3/tetrahedron_coordinates_3.h>`

computes barycentric coordinates with respect to a tetrahedron.

This function computes barycentric coordinates at a given `query` point with respect to the points `p0`, `p1`, `p2`, and `p3`, which form a tetrahedron, that is one coordinate per point. The coordinates are returned in an array.

After the coordinates \(b_0\), \(b_1\), \(b_2\), and \(b_3\) are computed, the query point \(q\) can be obtained as \(q = b_0p_0 + b_1p_1 + b_2p_2 + b_3p_3\).

Template Parameters

  GeomTraits | a model of `BarycentricTraits_3`

Parameters

  p0 | the first vertex of a tetrahedron
  p1 | the second vertex of a tetrahedron
  p2 | the third vertex of a tetrahedron
  p3 | the fourth vertex of a tetrahedron
  query | a query point
  traits | a traits class with geometric objects, predicates, and constructions; this parameter can be omitted if the traits class can be deduced from the point type

Returns
  an array `std::array<GeomTraits::FT, 4>` with the computed coordinates

Precondition
  `traits.compute_volume_3_object()(p0, p1, p2, p3) != 0`

## ◆ tetrahedron_coordinates_3() [2/2]

template<typename OutputIterator , typename GeomTraits >

OutputIterator CGAL::Barycentric_coordinates::tetrahedron_coordinates_3 | ( | const typename GeomTraits::Point_3 & | p0,
|  | const typename GeomTraits::Point_3 & | p1,
|  | const typename GeomTraits::Point_3 & | p2,
|  | const typename GeomTraits::Point_3 & | p3,
|  | const typename GeomTraits::Point_3 & | query,
|  | OutputIterator | oi,
|  | const GeomTraits & | traits
| ) |  |

`#include <CGAL/Barycentric_coordinates_3/tetrahedron_coordinates_3.h>`

computes barycentric coordinates with respect to a tetrahedron.

This function computes barycentric coordinates at a given `query` point with respect to the points `p0`, `p1`, `p2`, and `p3`, which form a tetrahedron, that is one coordinate per point. The coordinates are stored in a destination range beginning at `oi`.

After the coordinates \(b_0\), \(b_1\), \(b_2\), and \(b_2\) are computed, the query point \(q\) can be obtained as \(q = b_0p_0 + b_1p_1 + b_2p_2 + b_3p_3\).

Template Parameters

  OutputIterator | a model of `OutputIterator` that accepts values of type `GeomTraits::FT`
  GeomTraits | a model of `BarycentricTraits_3`

Parameters

  p0 | the first vertex of a tetrahedron
  p1 | the second vertex of a tetrahedron
  p2 | the third vertex of a tetrahedron
  p3 | the fourth vertex of a tetrahedron
  query | a query point
  oi | the beginning of the destination range with the computed coordinates
  traits | a traits class with geometric objects, predicates, and constructions; this parameter can be omitted if the traits class can be deduced from the point type

Returns
  an output iterator to the element in the destination range, one past the last coordinate stored

Precondition
  `traits.compute_volume_3_object()(p0, p1, p2, p3) != 0`

Examples
  Barycentric_coordinates_3/tetrahedron_coordinates_3.cpp.

## ◆ wachspress_coordinates_3()

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

OutputIterator CGAL::Barycentric_coordinates::wachspress_coordinates_3 | ( | const TriangleMesh & | tmesh,
|  | const Point | query,
|  | OutputIterator | oi,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

`#include <CGAL/Barycentric_coordinates_3/Wachspress_coordinates_3.h>`

computes 3D Wachspress coordinates with respect to a closed convex triangle mesh.

This function computes 3D Wachspress coordinates at a given `query` point with respect to the vertices of `tmesh`, that is one coordinate per vertex. The coordinates are stored in a destination range beginning at `oi`.

Internally, the class `Wachspress_coordinates_3` is used. If one wants to process multiple query points, it is better to use that class. When using the free function, internal memory is allocated for each query point, while when using the class, it is allocated only once, which is much more efficient. However, for a few query points, it is easier to use this function. It can also be used when the processing time is not a concern.

Template Parameters

  TriangleMesh | must be a model of the concept `FaceListGraph`
  Point | A model of `GeomTraits::Point_3` with `GeomTraits` being the type of the named parameter `geom_traits`.
  OutputIterator | must be an output iterator accepting `GeomTraits::FT` with `GeomTraits` being the type of the named parameter `geom_traits`.
  NamedParameters | a sequence of Named Parameters

Parameters

  tmesh | an instance of `TriangleMesh`
  query | a query point
  oi | the beginning of the destination range with the computed coordinates
  np | an optional sequence of Named Parameters among the ones listed below

Returns
  an output iterator to the element in the destination range, one past the last coordinate stored

Optional Named Parameters

  The computation policy for handling points near the boundary of the `tmesh`. Type: `CGAL::Barycentric_coordinates::Computation_policy_3` Default: `Computation_policy_3::FAST_WITH_EDGE_CASES` |
  a property map associating points to the vertices of `tmesh` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMesh>::vertex_descriptor` as key type and `Point` as value type Default: `boost::get(CGAL::vertex_point, tmesh)` Extra: If this parameter is omitted, an internal property map for `CGAL::vertex_point_t` must be available in `TriangleMesh`. |
  an instance of a geometric traits class Type: a class model of `BarycentricTraits_3` Default: a CGAL Kernel deduced from the value type of the vertex-point map, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the vertex point type. |

Precondition
  num_vertices(`tmesh`) >= 4.
  CGAL::is_triangle_mesh(`tmesh`).
  CGAL::is_closed(`tmesh`).
  CGAL::is_strongly_convex_3(`tmesh`).