Geometric Measure Functions

Functions to compute discrete curvatures, lengths of edges and borders, areas of faces and patches, volumes of closed meshes.

Functions
template<typename PolygonMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::angle_sum (typename boost::graph_traits< PolygonMesh >::vertex_descriptor v, const PolygonMesh &pmesh, const NamedParameters &np=parameters::default_values())
	computes the sum of the angles around a vertex.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::discrete_Gaussian_curvature (typename boost::graph_traits< TriangleMesh >::vertex_descriptor v, const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the discrete Gaussian curvature at a vertex.
template<typename TriangleMesh , typename VertexCurvatureMap , typename NamedParameters = CGAL::parameters::Default_named_parameters>
void	CGAL::Polygon_mesh_processing::discrete_Gaussian_curvatures (const TriangleMesh &tmesh, VertexCurvatureMap vcm, const NamedParameters &np=parameters::default_values())
	computes the discrete Gaussian curvatures at the vertices of a mesh.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::discrete_mean_curvature (typename boost::graph_traits< TriangleMesh >::vertex_descriptor v, const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the discrete mean curvature at a vertex.
template<typename TriangleMesh , typename VertexCurvatureMap , typename NamedParameters = CGAL::parameters::Default_named_parameters>
void	CGAL::Polygon_mesh_processing::discrete_mean_curvatures (const TriangleMesh &tmesh, VertexCurvatureMap vcm, const NamedParameters &np=parameters::default_values())
	computes the discrete mean curvatures at the vertices of a mesh.
template<typename PolygonMesh , typename NamedParameters = parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::edge_length (typename boost::graph_traits< PolygonMesh >::halfedge_descriptor h, const PolygonMesh &pmesh, const NamedParameters &np=parameters::default_values())
	computes the length of an edge of a given polygon mesh.
template<typename PolygonMesh , typename NamedParameters = parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::squared_edge_length (typename boost::graph_traits< PolygonMesh >::halfedge_descriptor h, const PolygonMesh &pmesh, const NamedParameters &np=parameters::default_values())
	computes the squared length of an edge of a given polygon mesh.
template<typename PolygonMesh , typename NamedParameters = parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::face_border_length (typename boost::graph_traits< PolygonMesh >::halfedge_descriptor h, const PolygonMesh &pmesh, const NamedParameters &np=parameters::default_values())
	computes the length of the border polyline that contains a given halfedge.
template<typename PolygonMesh , typename NamedParameters = parameters::Default_named_parameters>
std::pair< halfedge_descriptor, FT >	CGAL::Polygon_mesh_processing::longest_border (const PolygonMesh &pmesh, const NamedParameters &np=parameters::default_values())
	finds the longest border of a given triangulated surface and returns a halfedge that is part of this border as well as the length of this border.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::face_area (typename boost::graph_traits< TriangleMesh >::face_descriptor f, const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the area of a face of a given triangulated surface mesh.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::squared_face_area (typename boost::graph_traits< TriangleMesh >::face_descriptor f, const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the squared area of a face of a given triangulated surface mesh.
template<typename FaceRange , typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::area (FaceRange face_range, const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the area of a range of faces of a given triangulated surface mesh.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::area (const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the surface area of a triangulated surface mesh.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::volume (const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the volume of the domain bounded by a closed triangulated surface mesh.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
FT	CGAL::Polygon_mesh_processing::face_aspect_ratio (typename boost::graph_traits< TriangleMesh >::face_descriptor f, const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the aspect ratio of a face of a given triangulated surface mesh.
template<typename TriangleMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>
Point_3	CGAL::Polygon_mesh_processing::centroid (const TriangleMesh &tmesh, const NamedParameters &np=parameters::default_values())
	computes the centroid of a volume bounded by a closed triangulated surface mesh.
template<typename PolygonMesh1 , typename PolygonMesh2 , typename FacePairOutputIterator , typename FaceOutputIterator1 , typename FaceOutputIterator2 , typename NamedParameters1 = parameters::Default_named_parameters, typename NamedParameters2 = parameters::Default_named_parameters>
void	CGAL::Polygon_mesh_processing::match_faces (const PolygonMesh1 &m1, const PolygonMesh2 &m2, FacePairOutputIterator common, FaceOutputIterator1 m1_only, FaceOutputIterator2 m2_only, const NamedParameters1 &np1=parameters::default_values(), const NamedParameters2 &np2=parameters::default_values())
	identifies faces only present in m1 and m2 as well as the faces present in both polygon meshes.

Function Documentation

angle_sum()

template<typename PolygonMesh , typename NamedParameters = CGAL::parameters::Default_named_parameters>

FT CGAL::Polygon_mesh_processing::angle_sum	(	typename boost::graph_traits< PolygonMesh >::vertex_descriptor	v,
		const PolygonMesh &	pmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/curvature.h>

computes the sum of the angles around a vertex.

The angle sum is given in degrees.

Template Parameters

PolygonMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

v	the vertex whose sum of angles is computed
pmesh	the polygon mesh to which v belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map a property map associating points to the vertices of pmesh Type: a class model of ReadablePropertyMap with boost::graph_traits<PolygonMesh>::vertex_descriptor as key type and Point_3 as value type Default: boost::get(CGAL::vertex_point, pmesh)
geom_traits an instance of a geometric traits class Type: The traits class must provide the nested functor Compute_approximate_angle_3, model of Kernel::ComputeApproximateAngle_3. 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.

Returns

the sum of angles around v. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of pmesh.

Warning

This function involves trigonometry.

area() [1/2]

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

FT CGAL::Polygon_mesh_processing::area	(	const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the surface area of a triangulated surface mesh.

Template Parameters

TriangleMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

tmesh	the triangulated surface mesh
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Returns

the surface area of tmesh. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

Warning

This function involves a square root computation. If Kernel::FT does not support the sqrt() operation, the square root computation will be performed approximately.

See also

face_area()

area() [2/2]

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

FT CGAL::Polygon_mesh_processing::area	(	FaceRange	face_range,
		const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the area of a range of faces of a given triangulated surface mesh.

Template Parameters

FaceRange	range of boost::graph_traits<PolygonMesh>::face_descriptor, model of Range. Its iterator type is InputIterator.
TriangleMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

face_range	the range of faces of whose area is computed
tmesh	the triangulated surface mesh to which the faces of face_range belong
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Returns

sum of face areas of faces. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

Warning

This function involves a square root computation. If Kernel::FT does not support the sqrt() operation, the square root computation will be performed approximately.

See also

face_area()

centroid()

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

Point_3 CGAL::Polygon_mesh_processing::centroid	(	const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the centroid of a volume bounded by a closed triangulated surface mesh.

Template Parameters

TriangleMesh	a model of FaceListGraph
NamedParameters	a sequence of Named Parameters

Parameters

tmesh	the closed triangulated surface mesh bounding the volume
np	an optional sequence of Named Parameters among the ones listed below

Precondition

tmesh is closed

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Returns

the centroid of the domain bounded by tmesh.

discrete_Gaussian_curvature()

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

FT CGAL::Polygon_mesh_processing::discrete_Gaussian_curvature	(	typename boost::graph_traits< TriangleMesh >::vertex_descriptor	v,
		const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/curvature.h>

computes the discrete Gaussian curvature at a vertex.

We refer to Meyer et al. [13] for the definition of discrete Gaussian curvature.

Template Parameters

TriangleMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

v	the vertex whose discrete Gaussian curvature is being computed
tmesh	the triangle mesh to which v belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits an instance of a geometric traits class Type: The traits class must be a 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.

Returns

the discrete Gaussian curvature at v. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

Warning

This function involves trigonometry.

The current formulation is not well defined for border vertices.

Precondition

tmesh is a triangle mesh

discrete_Gaussian_curvatures()

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

void CGAL::Polygon_mesh_processing::discrete_Gaussian_curvatures	(	const TriangleMesh &	tmesh,
		VertexCurvatureMap	vcm,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/curvature.h>

computes the discrete Gaussian curvatures at the vertices of a mesh.

We refer to Meyer et al. [13] for the definition of discrete Gaussian curvature.

Template Parameters

TriangleMesh	a model of FaceGraph
VertexCurvatureMap	must be a model of WritablePropertyMap with key type boost::graph_traits<TriangleMesh>::vertex_descriptor and value type FT, which is either geom_traits::FT if this named parameter is provided, or kernel::FT with the kernel deduced from from the point property map of tmesh.
NamedParameters	a sequence of Named Parameters

Parameters

tmesh	the triangle mesh to which v belongs
vcm	the property map that contains the computed discrete curvatures
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits an instance of a geometric traits class Type: The traits class must be a 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.

Warning

This function involves trigonometry.

The current formulation is not well defined for border vertices.

Precondition

tmesh is a triangle mesh

discrete_mean_curvature()

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

FT CGAL::Polygon_mesh_processing::discrete_mean_curvature	(	typename boost::graph_traits< TriangleMesh >::vertex_descriptor	v,
		const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/curvature.h>

computes the discrete mean curvature at a vertex.

We refer to Meyer et al. [13] for the definition of discrete mean curvature.

Template Parameters

TriangleMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

v	the vertex whose discrete mean curvature is being computed
tmesh	the triangle mesh to which v belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits an instance of a geometric traits class Type: The traits class must be a 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.

Returns

the discrete mean curvature at v. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

Warning

The current formulation is not well defined for border vertices.

Precondition

tmesh is a triangle mesh

discrete_mean_curvatures()

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

void CGAL::Polygon_mesh_processing::discrete_mean_curvatures	(	const TriangleMesh &	tmesh,
		VertexCurvatureMap	vcm,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/curvature.h>

computes the discrete mean curvatures at the vertices of a mesh.

We refer to Meyer et al. [13] for the definition of discrete mean curvature.

Template Parameters

TriangleMesh	a model of FaceGraph
VertexCurvatureMap	must be a model of WritablePropertyMap with key type boost::graph_traits<TriangleMesh>::vertex_descriptor and value type FT, which is either geom_traits::FT if this named parameter is provided, or kernel::FT with the kernel deduced from from the point property map of tmesh.
NamedParameters	a sequence of Named Parameters

Parameters

tmesh	the triangle mesh to which v belongs
vcm	the property map that contains the computed discrete curvatures
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits an instance of a geometric traits class Type: The traits class must be a 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.

Warning

The current formulation is not well defined for border vertices.

Precondition

tmesh is a triangle mesh

edge_length()

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

FT CGAL::Polygon_mesh_processing::edge_length	(	typename boost::graph_traits< PolygonMesh >::halfedge_descriptor	h,
		const PolygonMesh &	pmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the length of an edge of a given polygon mesh.

The edge is given by one of its halfedges, or the edge itself.

Template Parameters

PolygonMesh	a model of HalfedgeGraph
NamedParameters	a sequence of Named Parameters

Parameters

h	one halfedge of the edge whose length is computed
pmesh	the polygon mesh to which h belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map a property map associating points to the vertices of pmesh Type: a class model of ReadablePropertyMap with boost::graph_traits<PolygonMesh>::vertex_descriptor as key type and Point_3 as value type Default: boost::get(CGAL::vertex_point, pmesh)
geom_traits 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.

Returns

the length of h. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of pmesh.

Warning

This function involves a square root computation. If FT does not support the sqrt() operation, the square root computation will be performed approximately.

See also

squared_edge_length()

face_border_length()

face_area()

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

FT CGAL::Polygon_mesh_processing::face_area	(	typename boost::graph_traits< TriangleMesh >::face_descriptor	f,
		const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the area of a face of a given triangulated surface mesh.

Template Parameters

TriangleMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

f	the face whose area is computed
tmesh	the triangulated surface mesh to which f belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Precondition

f != boost::graph_traits<TriangleMesh>::null_face()

Returns

the area of f. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

Warning

This function involves a square root computation. If Kernel::FT does not support the sqrt() operation, the square root computation will be performed approximately.

See also

squared_face_area()

area()

face_aspect_ratio()

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

FT CGAL::Polygon_mesh_processing::face_aspect_ratio	(	typename boost::graph_traits< TriangleMesh >::face_descriptor	f,
		const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the aspect ratio of a face of a given triangulated surface mesh.

Template Parameters

TriangleMesh	a model of HalfedgeGraph
NamedParameters	a sequence of Named Parameters

Parameters

f	the face whose aspect ratio is computed
tmesh	the triangulated surface mesh to which f belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Precondition

f != boost::graph_traits<TriangleMesh>::null_face()

Returns

the aspect ratio of f. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

Warning

This function involves a square root computation. If Kernel::FT does not support the sqrt() operation, the square root computation will be performed approximately.

face_border_length()

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

FT CGAL::Polygon_mesh_processing::face_border_length	(	typename boost::graph_traits< PolygonMesh >::halfedge_descriptor	h,
		const PolygonMesh &	pmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the length of the border polyline that contains a given halfedge.

Template Parameters

PolygonMesh	a model of HalfedgeGraph
NamedParameters	a sequence of Named Parameters

Parameters

h	a halfedge of the border polyline whose length is computed
pmesh	the polygon mesh to which h belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map a property map associating points to the vertices of pmesh Type: a class model of ReadablePropertyMap with boost::graph_traits<PolygonMesh>::vertex_descriptor as key type and Point_3 as value type Default: boost::get(CGAL::vertex_point, pmesh)
geom_traits 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.

Returns

the length of the sequence of border edges of face(h, pmesh). The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of pmesh.

Warning

This function involves a square root computation. If Kernel::FT does not support the sqrt() operation, the square root computation will be performed approximately.

See also

edge_length()

longest_border()

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

std::pair< halfedge_descriptor, FT > CGAL::Polygon_mesh_processing::longest_border	(	const PolygonMesh &	pmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

finds the longest border of a given triangulated surface and returns a halfedge that is part of this border as well as the length of this border.

Template Parameters

PolygonMesh	a model of HalfedgeGraph
NamedParameters	a sequence of Named Parameters

Parameters

pmesh	the polygon mesh
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map a property map associating points to the vertices of pmesh Type: a class model of ReadablePropertyMap with boost::graph_traits<PolygonMesh>::vertex_descriptor as key type and Point_3 as value type Default: boost::get(CGAL::vertex_point, pmesh)
geom_traits 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.

Returns

a pair composed of two members:

• first: a halfedge on the longest border. The return type halfedge_descriptor is a halfedge descriptor. It is deduced from the graph traits corresponding to the type PolygonMesh. first is among the halfedges reported by extract_boundary_cycles().
• second: the length of the longest border The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of pmesh

Warning

This function involves a square root computation. If Kernel::FT does not support the sqrt() operation, the square root computation will be performed approximately.

See also

face_border_length()

extract_boundary_cycles()

match_faces()

template<typename PolygonMesh1 , typename PolygonMesh2 , typename FacePairOutputIterator , typename FaceOutputIterator1 , typename FaceOutputIterator2 , typename NamedParameters1 = parameters::Default_named_parameters, typename NamedParameters2 = parameters::Default_named_parameters>

void CGAL::Polygon_mesh_processing::match_faces	(	const PolygonMesh1 &	m1,
		const PolygonMesh2 &	m2,
		FacePairOutputIterator	common,
		FaceOutputIterator1	m1_only,
		FaceOutputIterator2	m2_only,
		const NamedParameters1 &	np1 = parameters::default_values(),
		const NamedParameters2 &	np2 = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

identifies faces only present in m1 and m2 as well as the faces present in both polygon meshes.

Two faces are matching if they have the same orientation and the same points.

Template Parameters

PolygonMesh1	a model of HalfedgeListGraph and FaceListGraph
PolygonMesh2	a model of HalfedgeListGraph and FaceListGraph
FaceOutputIterator1	model of OutputIterator holding boost::graph_traits<PolygonMesh1>::face_descriptor.
FaceOutputIterator2	model of OutputIterator holding boost::graph_traits<PolygonMesh2>::face_descriptor.
FacePairOutputIterator	model of OutputIterator holding std::pair<boost::graph_traits<PolygonMesh1>::face_descriptor, boost::graph_traits<PolygonMesh2>::face_descriptor.
NamedParameters1	a sequence of Named Parameters
NamedParameters2	a sequence of Named Parameters

Parameters

m1	the first polygon mesh
m2	the second polygon mesh
common	output iterator collecting the faces that are common to both meshes
m1_only	output iterator collecting the faces that are only in m1
m2_only	output iterator collecting the faces that are only in m2
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

vertex_point_map a property map associating points to the vertices of m1 Type: a class model of ReadablePropertyMap with boost::graph_traits<PolygonMesh1>::vertex_descriptor as key type and Point_3 as value type. Point_3 must be LessThanComparable. Default: boost::get(CGAL::vertex_point, m1) Extra: The same holds for m2 and PolygonMesh2 and the point type must be the same for both meshes.

vertex_index_map a property map associating to each vertex of m1 a unique index between 0 and num_vertices(m1) - 1, and similarly for m2. Type: a class model of ReadablePropertyMap with boost::graph_traits<Graph>::vertex_descriptor as key type and std::size_t as value type Default: an automatically indexed internal map Extra: If this parameter is not passed, internal machinery will create and initialize a face index property map, either using the internal property map if it exists or using an external map. The latter might result in - slightly - worsened performance in case of non-constant complexity for index access. The same holds for m2 and PolygonMesh2.

squared_edge_length()

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

FT CGAL::Polygon_mesh_processing::squared_edge_length	(	typename boost::graph_traits< PolygonMesh >::halfedge_descriptor	h,
		const PolygonMesh &	pmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the squared length of an edge of a given polygon mesh.

The edge is given by one of its halfedges, or the edge itself.

Template Parameters

PolygonMesh	a model of HalfedgeGraph
NamedParameters	a sequence of Named Parameters

Parameters

h	one halfedge of the edge whose squared length is computed
pmesh	the polygon mesh to which h belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map a property map associating points to the vertices of pmesh Type: a class model of ReadablePropertyMap with boost::graph_traits<PolygonMesh>::vertex_descriptor as key type and Point_3 as value type Default: boost::get(CGAL::vertex_point, pmesh)
geom_traits 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.

Returns

the squared length of h. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of pmesh.

See also

edge_length()

face_border_length()

squared_face_area()

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

FT CGAL::Polygon_mesh_processing::squared_face_area	(	typename boost::graph_traits< TriangleMesh >::face_descriptor	f,
		const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the squared area of a face of a given triangulated surface mesh.

Template Parameters

TriangleMesh	a model of FaceGraph
NamedParameters	a sequence of Named Parameters

Parameters

f	the face whose squared area is computed
tmesh	the triangulated surface mesh to which f belongs
np	an optional sequence of Named Parameters among the ones listed below

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Precondition

f != boost::graph_traits<TriangleMesh>::null_face()

Returns

the squared area of f. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.

See also

face_area()

volume()

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

FT CGAL::Polygon_mesh_processing::volume	(	const TriangleMesh &	tmesh,
		const NamedParameters &	np = parameters::default_values()
	)

#include <CGAL/Polygon_mesh_processing/measure.h>

computes the volume of the domain bounded by a closed triangulated surface mesh.

Template Parameters

TriangleMesh	a model of HalfedgeGraph
NamedParameters	a sequence of Named Parameters

Parameters

tmesh	the closed triangulated surface mesh bounding the volume
np	an optional sequence of Named Parameters among the ones listed below

Precondition

tmesh is closed

Optional Named Parameters

vertex_point_map 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_3 as value type Default: boost::get(CGAL::vertex_point, tmesh)
geom_traits 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.

Returns

the volume bounded by tmesh. The return type FT is a number type either deduced from the geom_traits Named Parameters if provided, or the geometric traits class deduced from the point property map of tmesh.