CGAL 6.1.2 - Polygon Mesh Processing: Distance Functions

CGAL 6.1.2 - Polygon Mesh Processing

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Functions
Distance Functions
Reference Manual

##

Functions to compute the distance between meshes, between a mesh and a point set and between a point set and a mesh.

## Functions
template<class PointOutputIterator , class TriangleMesh , class NamedParameters = parameters::Default_named_parameters>
PointOutputIterator | CGAL::Polygon_mesh_processing::sample_triangle_mesh (const TriangleMesh &tm, PointOutputIterator out, const NamedParameters &np=parameters::default_values())
| generates points on `tm` and outputs them to `out`; the sampling method is selected using named parameters.
template<class PointOutputIterator , class TriangleRange , class PointRange , class NamedParameters = parameters::Default_named_parameters>
PointOutputIterator | CGAL::Polygon_mesh_processing::sample_triangle_soup (const PointRange &points, const TriangleRange &triangles, PointOutputIterator out, const NamedParameters &np=parameters::default_values())
| generates points on a triangle soup and puts them to `out`; the sampling method is selected using named parameters.
template<class Concurrency_tag , class TriangleMesh , class PointRange , class NamedParameters = parameters::Default_named_parameters>
double | CGAL::Polygon_mesh_processing::max_distance_to_triangle_mesh (const PointRange &points, const TriangleMesh &tm, const NamedParameters &np=parameters::default_values())
| returns the distance to `tm` of the point from `points` that is the furthest from `tm`.
template<class Concurrency_tag , class TriangleMesh , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>
double | CGAL::Polygon_mesh_processing::approximate_Hausdorff_distance (const TriangleMesh &tm1, const TriangleMesh &tm2, const NamedParameters1 &np1=parameters::default_values(), const NamedParameters2 &np2=parameters::default_values())
| computes the approximate Hausdorff distance from `tm1` to `tm2` by returning the distance of the farthest point from `tm2` amongst a sampling of `tm1` generated with the function `sample_triangle_mesh()` with `tm1` and `np1` as parameter.
template<class Concurrency_tag , class TriangleMesh , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>
double | CGAL::Polygon_mesh_processing::approximate_symmetric_Hausdorff_distance (const TriangleMesh &tm1, const TriangleMesh &tm2, const NamedParameters1 &np1=parameters::default_values(), const NamedParameters2 &np2=parameters::default_values())
| returns the approximate symmetric Hausdorff distance between `tm1` and `tm2`, that is the maximum of `approximate_Hausdorff_distance(tm1, tm2, np1, np2)` and `approximate_Hausdorff_distance(tm2, tm1, np2, np1)`.
template<class TriangleMesh , class PointRange , class NamedParameters = parameters::Default_named_parameters>
double | CGAL::Polygon_mesh_processing::approximate_max_distance_to_point_set (const TriangleMesh &tm, const PointRange &points, const double precision, const NamedParameters &np=parameters::default_values())
| returns an approximation of the distance between `points` and the point lying on `tm` that is the farthest from `points`.
template<class Concurrency_tag , class TriangleMesh1 , class TriangleMesh2 , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>
double | CGAL::Polygon_mesh_processing::bounded_error_Hausdorff_distance (const TriangleMesh1 &tm1, const TriangleMesh2 &tm2, const double error_bound=0.0001, const NamedParameters1 &np1=parameters::default_values(), const NamedParameters2 &np2=parameters::default_values())
| returns an estimate on the Hausdorff distance from `tm1` to `tm2` that is at most `error_bound` away from the actual Hausdorff distance from `tm1` to `tm2`.
template<class Concurrency_tag , class TriangleMesh1 , class TriangleMesh2 , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>
double | CGAL::Polygon_mesh_processing::bounded_error_symmetric_Hausdorff_distance (const TriangleMesh1 &tm1, const TriangleMesh2 &tm2, const double error_bound, const NamedParameters1 &np1=parameters::default_values(), const NamedParameters2 &np2=parameters::default_values())
| returns the the symmetric Hausdorff distance, that is the maximum of `bounded_error_Hausdorff_distance(tm1, tm2, error_bound, np1, np2)` and `bounded_error_Hausdorff_distance(tm2, tm1, error_bound, np2, np1)`.
template<class Concurrency_tag , class TriangleMesh1 , class TriangleMesh2 , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>
bool | CGAL::Polygon_mesh_processing::is_Hausdorff_distance_larger (const TriangleMesh1 &tm1, const TriangleMesh2 &tm2, const double distance_bound, const double error_bound, const NamedParameters1 &np1=parameters::default_values(), const NamedParameters2 &np2=parameters::default_values())
| returns `true` if the Hausdorff distance between two meshes is larger than the user-defined max distance, otherwise it returns `false`.

## Function Documentation

## ◆ approximate_Hausdorff_distance()

template<class Concurrency_tag , class TriangleMesh , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>

double CGAL::Polygon_mesh_processing::approximate_Hausdorff_distance | ( | const TriangleMesh & | tm1,
|  | const TriangleMesh & | tm2,
|  | const NamedParameters1 & | np1 = `parameters::default_values()`,
|  | const NamedParameters2 & | np2 = `parameters::default_values()`
| ) |  |

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

computes the approximate Hausdorff distance from `tm1` to `tm2` by returning the distance of the farthest point from `tm2` amongst a sampling of `tm1` generated with the function `sample_triangle_mesh()` with `tm1` and `np1` as parameter.

A parallel version is provided and requires the executable to be linked against the Intel TBB library. To control the number of threads used, the user may use the `tbb::task_scheduler_init` class. See the TBB documentation for more details.

Template Parameters

  Concurrency_tag | enables sequential versus parallel algorithm. Possible values are `Sequential_tag`, `Parallel_tag`, and `Parallel_if_available_tag`.
  TriangleMesh | a model of the concepts `EdgeListGraph` and `FaceListGraph`
  NamedParameters1 | a sequence of Named Parameters for `tm1`
  NamedParameters2 | a sequence of Named Parameters for `tm2`

Parameters

  tm1 | the triangle mesh that will be sampled
  tm2 | the triangle mesh to compute the distance to
  np1 | an optional sequence of Named Parameters forwarded to `sample_triangle_mesh()`
  np2 | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tm2` 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, tm2)` Extra: If this parameter is omitted, an internal property map for `CGAL::vertex_point_t` must be available in `TriangleMesh`. |

Precondition
  `tm1` and `tm2` are non-empty triangle meshes.

Examples
  Polygon_mesh_processing/hausdorff_distance_remeshing_example.cpp.

## ◆ approximate_max_distance_to_point_set()

template<class TriangleMesh , class PointRange , class NamedParameters = parameters::Default_named_parameters>

double CGAL::Polygon_mesh_processing::approximate_max_distance_to_point_set | ( | const TriangleMesh & | tm,
|  | const PointRange & | points,
|  | const double | precision,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

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

returns an approximation of the distance between `points` and the point lying on `tm` that is the farthest from `points`.

Template Parameters

  PointRange | a range of `Point_3`, model of `Range`
  TriangleMesh | a model of the concept `FaceListGraph`
  NamedParameters | a sequence of Named Parameters

Parameters

  tm | a triangle mesh
  points | a range of points
  precision | for each triangle of `tm`, the distance of its farthest point from `points` is bounded. A triangle is subdivided into sub-triangles so that the difference of its distance bounds is smaller than `precision`. `precision` must be strictly positive to avoid infinite loops.
  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)` 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 `PMPDistanceTraits` 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. |

Precondition
  `tm` is a non-empty triangle mesh and `points` is not empty.

Examples
  Poisson_surface_reconstruction_3/poisson_reconstruction_example.cpp.

## ◆ approximate_symmetric_Hausdorff_distance()

template<class Concurrency_tag , class TriangleMesh , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>

double CGAL::Polygon_mesh_processing::approximate_symmetric_Hausdorff_distance | ( | const TriangleMesh & | tm1,
|  | const TriangleMesh & | tm2,
|  | const NamedParameters1 & | np1 = `parameters::default_values()`,
|  | const NamedParameters2 & | np2 = `parameters::default_values()`
| ) |  |

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

returns the approximate symmetric Hausdorff distance between `tm1` and `tm2`, that is the maximum of `approximate_Hausdorff_distance(tm1, tm2, np1, np2)` and `approximate_Hausdorff_distance(tm2, tm1, np2, np1)`.

See the function `approximate_Hausdorff_distance()` for a complete description of the parameters and requirements.

## ◆ bounded_error_Hausdorff_distance()

template<class Concurrency_tag , class TriangleMesh1 , class TriangleMesh2 , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>

double CGAL::Polygon_mesh_processing::bounded_error_Hausdorff_distance | ( | const TriangleMesh1 & | tm1,
|  | const TriangleMesh2 & | tm2,
|  | const double | error_bound = `0.0001`,
|  | const NamedParameters1 & | np1 = `parameters::default_values()`,
|  | const NamedParameters2 & | np2 = `parameters::default_values()`
| ) |  |

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

returns an estimate on the Hausdorff distance from `tm1` to `tm2` that is at most `error_bound` away from the actual Hausdorff distance from `tm1` to `tm2`.

Template Parameters

  Concurrency_tag | enables sequential versus parallel algorithm. Possible values are `Sequential_tag` and `Parallel_tag`. Currently, the parallel version is not implemented and the sequential version is always used whatever tag is chosen!
  TriangleMesh1 | a model of the concept `FaceListGraph`
  TriangleMesh2 | a model of the concept `FaceListGraph`
  NamedParameters | a sequence of Named Parameters

Parameters

  tm1 | a triangle mesh
  tm2 | another triangle mesh
  error_bound | a maximum bound by which the Hausdorff distance estimate is allowed to deviate from the actual Hausdorff distance.
  np1 | an optional sequence of Named Parameters among the ones listed below
  np2 | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  a property map associating points to the vertices of `tmX` Type: a class model of `ReadablePropertyMap` with `boost::graph_traits<TriangleMeshX>::vertex_descriptor` as key type and `Point_3` as value type Default: `boost::get(CGAL::vertex_point, tmX)` Extra: If this parameter is omitted, an internal property map for `CGAL::vertex_point_t` must be available in `TriangleMeshX`. |
  a boolean tag that turns on the preprocessing step that filters out all faces which belong to both meshes and hence do not contribute to the final distance Type: Boolean Default: true Extra: Both `np1` and `np2` must have this tag set to `true` in order to activate this preprocessing. |

Precondition
  `tm1` and `tm2` are non-empty triangle meshes.

Returns
  the one-sided Hausdorff distance

## ◆ bounded_error_symmetric_Hausdorff_distance()

template<class Concurrency_tag , class TriangleMesh1 , class TriangleMesh2 , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>

double CGAL::Polygon_mesh_processing::bounded_error_symmetric_Hausdorff_distance | ( | const TriangleMesh1 & | tm1,
|  | const TriangleMesh2 & | tm2,
|  | const double | error_bound,
|  | const NamedParameters1 & | np1 = `parameters::default_values()`,
|  | const NamedParameters2 & | np2 = `parameters::default_values()`
| ) |  |

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

returns the the symmetric Hausdorff distance, that is the maximum of `bounded_error_Hausdorff_distance(tm1, tm2, error_bound, np1, np2)` and `bounded_error_Hausdorff_distance(tm2, tm1, error_bound, np2, np1)`.

This function optimizes all internal calls to shared data structures in order to speed up the computation.

See the function `CGAL::Polygon_mesh_processing::bounded_error_Hausdorff_distance()` for a complete description of the parameters and requirements.

## ◆ is_Hausdorff_distance_larger()

template<class Concurrency_tag , class TriangleMesh1 , class TriangleMesh2 , class NamedParameters1 = parameters::Default_named_parameters, class NamedParameters2 = parameters::Default_named_parameters>

bool CGAL::Polygon_mesh_processing::is_Hausdorff_distance_larger | ( | const TriangleMesh1 & | tm1,
|  | const TriangleMesh2 & | tm2,
|  | const double | distance_bound,
|  | const double | error_bound,
|  | const NamedParameters1 & | np1 = `parameters::default_values()`,
|  | const NamedParameters2 & | np2 = `parameters::default_values()`
| ) |  |

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

returns `true` if the Hausdorff distance between two meshes is larger than the user-defined max distance, otherwise it returns `false`.

The distance used to compute the proximity of the meshes is the bounded-error Hausdorff distance. Instead of computing the full distance and checking it against the user-provided value, this function returns early if certain criteria show that the meshes do not satisfy the provided `distance_bound`.

See the function `CGAL::Polygon_mesh_processing::bounded_error_Hausdorff_distance()` for a complete description of the parameters and requirements. The following extra named parameter is available for `np1`:

Optional Named Parameters

  a boolean tag indicating if the one-sided Hausdorff distance should be used. Type: Boolean Default: `true` Extra: If this tag is set to `false`, the symmetric Hausdorff distance is used. |

## ◆ max_distance_to_triangle_mesh()

template<class Concurrency_tag , class TriangleMesh , class PointRange , class NamedParameters = parameters::Default_named_parameters>

double CGAL::Polygon_mesh_processing::max_distance_to_triangle_mesh | ( | const PointRange & | points,
|  | const TriangleMesh & | tm,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

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

returns the distance to `tm` of the point from `points` that is the furthest from `tm`.

Template Parameters

  PointRange | a range of `Point_3`, model of `Range`. Its iterator type is `RandomAccessIterator`.
  TriangleMesh | a model of the concepts `EdgeListGraph` and `FaceListGraph`
  NamedParameters | a sequence of Named Parameters

Parameters

  points | the range of points of interest
  tm | the triangle mesh to compute the distance to
  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)` 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 `PMPDistanceTraits` 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. |

Precondition
  `tm` is a non-empty triangle mesh and `points` is not empty.

## ◆ sample_triangle_mesh()

template<class PointOutputIterator , class TriangleMesh , class NamedParameters = parameters::Default_named_parameters>

PointOutputIterator CGAL::Polygon_mesh_processing::sample_triangle_mesh | ( | const TriangleMesh & | tm,
|  | PointOutputIterator | out,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

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

generates points on `tm` and outputs them to `out`; the sampling method is selected using named parameters.

Template Parameters

  TriangleMesh | a model of the concepts `EdgeListGraph` and `FaceListGraph`
  PointOutputIterator | a model of `OutputIterator` holding objects of the same point type as the value type of the point type associated to the mesh `tm`, i.e., the value type of the vertex point map property map, if provided, or the value type of the internal point property map otherwise
  NamedParameters | a sequence of Named Parameters

Parameters

  tm | the triangle mesh to be sampled
  out | output iterator to be filled with sample points
  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)` 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 `PMPDistanceTraits` 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 value to seed the random number generator Type: unsigned int Default: a value generated with `std::time()` |
  If `true` is passed, points are generated uniformly at random on faces and/or edges of `tm`. If `do_sample_faces` is `true`, random points will be iteratively generated uniformly at random in the triangle of a face selected with probability proportional to its area. If `do_sample_edges` is `true`, random points will be iteratively generated uniformly at random in the segment of an edge selected with probability proportional to its length. Type: Boolean Type: `true` Extra: For faces, the number of sample points is the value passed to the named parameter `number_of_points_on_faces`. If not set, the value passed to the named parameter `number_of_points_per_area_unit` is multiplied by the area of `tm` to get the number of sample points. If none of these parameters is set, the number of points sampled is `num_vertices(tm)`. For edges, the number of the number of sample points is the value passed to the named parameter `number_of_points_on_edges`. If not set, the value passed to the named parameter `number_of_points_per_distance_unit` is multiplied by the sum of the length of edges of `tm` to get the number of sample points. If none of these parameters is set, the number of points sampled is `num_vertices(tm)`. |
  If `true` is passed, points are generated on a grid in each triangle, with a minimum of one point per triangle. Type: Boolean Default: `false` Extra: The distance between two consecutive points in the grid is that of the length of the smallest non-null edge of `tm` or the value passed to the named parameter `grid_spacing`. Edges are also split using the same distance, if requested. |
  if `true` is passed, points are generated randomly in each triangle and/or on each edge. Type: Boolean Default: `false` Extra: For faces, the number of points per triangle is the value passed to the named parameter `number_of_points_per_face`. If not set, the value passed to the named parameter `number_of_points_per_area_unit` is used to pick a number of points per face proportional to the triangle area with a minimum of one point per face. If none of these parameters is set, 2 divided by the square of the length of the smallest non-null edge of `tm` is used as if it was passed to `number_of_points_per_area_unit`. For edges, the number of points per edge is the value passed to the named parameter `number_of_points_per_edge`. If not set, the value passed to the named parameter `number_of_points_per_distance_unit` is used to pick a number of points per edge proportional to the length of the edge with a minimum of one point per face. If none of these parameters is set, 1 divided by the length of the smallest non-null edge of `tm` is used as if it was passed to `number_of_points_per_distance_unit`. |
  If `true` is passed, the vertices of `tm` are part of the sample. Type: Boolean Default: `true` |
  If `true` is passed, edges of `tm` are sampled. Type: Boolean Default: `true` |
  If `true` is passed, faces of `tm` are sampled. Type: Boolean Default: `true` |
  a value used as the grid spacing for the grid sampling method Type: double Default: the length of the shortest, non-degenerate edge of `tm` |
  a value used for the random sampling method as the number of points to pick exclusively on edges Type: unsigned int Default: `num_vertices(tm)` or a value based on `nb_points_per_distance_unit`, if it is defined |
  a value used for the random sampling method as the number of points to pick on the surface Type: unsigned int Default: `num_vertices(tm)` or a value based on `nb_points_per_area_unit`, if it is defined |
  a value used for the random sampling and the Monte Carlo sampling methods to respectively determine the total number of points on edges and the number of points per edge Type: double Default: `1` divided by the length of the shortest, non-degenerate edge of `tm` |
  a value used by the Monte-Carlo sampling method as the number of points per edge to pick Type: unsigned int Default: `0` |
  a value used for the random sampling and the Monte Carlo sampling methods to respectively determine the total number of points inside faces and the number of points per face Type: double Default: `2` divided by the squared length of the shortest, non-degenerate edge of `tm` |
  a value used by the Monte-Carlo sampling method as the number of points per face to pick Type: unsigned int Default: `0` |

See also
  `CGAL::Polygon_mesh_processing::sample_triangle_soup()`

## ◆ sample_triangle_soup()

template<class PointOutputIterator , class TriangleRange , class PointRange , class NamedParameters = parameters::Default_named_parameters>

PointOutputIterator CGAL::Polygon_mesh_processing::sample_triangle_soup | ( | const PointRange & | points,
|  | const TriangleRange & | triangles,
|  | PointOutputIterator | out,
|  | const NamedParameters & | np = `parameters::default_values()`
| ) |  |

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

generates points on a triangle soup and puts them to `out`; the sampling method is selected using named parameters.

Template Parameters

  PointRange | a model of the concept `RandomAccessContainer` whose value type is the point type.
  TriangleRange | a model of the concept `RandomAccessContainer` whose `value_type` is itself a model of the concept `RandomAccessContainer` whose `value_type` is an unsigned integral value.
  PointOutputIterator | a model of `OutputIterator` holding objects of the same type as `PointRange`'s value type
  NamedParameters | a sequence of Named Parameters

Parameters

  points | the points of the soup
  triangles | a `TriangleRange` containing the triangles of the soup to be sampled
  out | output iterator to be filled with sample points
  np | an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

  an instance of a geometric traits class Type: a class model of `PMPDistanceTraits` Default: a CGAL Kernel deduced from the point type, using `CGAL::Kernel_traits` Extra: The geometric traits class must be compatible with the point range's point type. |
  a value to seed the random number generator Type: unsigned int Default: a value generated with `std::time()` |
  If `true` is passed, points are generated in a random and uniform way over the triangles of the soup. Type: Boolean Type: `true` Extra: The number of sample points is the value passed to the named parameter `number_of_points_on_faces`. If not set, the value passed to the named parameter `number_of_points_per_area_unit` is multiplied by the area of the soup to get the number of sample points. If none of these parameters is set, the number of points sampled is `points.size()`. |
  If `true` is passed, points are generated on a grid in each triangle, with a minimum of one point per triangle. Type: Boolean Default: `false` Extra: The distance between two consecutive points in the grid is that of the length of the smallest non-null edge of the soup or the value passed to the named parameter `grid_spacing`. |
  if `true` is passed, points are generated randomly in each triangle. Type: Boolean Default: `false` Extra: The number of points per triangle is the value passed to the named parameter `number_of_points_per_face`. If not set, the value passed to the named parameter `number_of_points_per_area_unit` is used to pick a number of points per face proportional to the triangle area with a minimum of one point per face. If none of these parameters is set, the number of points per area unit is set to 2 divided by the square of the length of the smallest non-null edge of the soup. |
  If `true` is passed, the points of `points` are part of the sample. Type: Boolean Default: `true` |
  If `true` is passed, faces of the soup are sampled. Type: Boolean Default: `true` |
  a value used as the grid spacing for the grid sampling method Type: double Default: the length of the shortest, non-degenerate edge of the soup |
  a value used for the random sampling method as the number of points to pick on the surface Type: unsigned int Default: `points.size()` or a value based on `nb_points_per_area_unit`, if it is defined |
  a value used by the Monte-Carlo sampling method as the number of points per face to pick Type: unsigned int Default: `0` |
  a value used for the random sampling and the Monte Carlo sampling methods to respectively determine the total number of points inside faces and the number of points per face Type: double Default: `2` divided by the squared length of the shortest, non-degenerate edge of the soup |

Attention
  Contrary to `sample_triangle_mesh()`, this method does not allow to sample edges.

See also
  `CGAL::Polygon_mesh_processing::sample_triangle_mesh()`