Free Functions

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 (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 (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>
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 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 OutputIterator , typename GeomTraits >
OutputIterator	CGAL::Barycentric_coordinates::tetrahedron_coordinates (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 (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.
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.

Function Documentation

apply_barycentric_coordinates() [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	(	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() [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	(	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

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 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.
geom_traits 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

computation_policy 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
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 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.
geom_traits 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

computation_policy 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
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 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.
geom_traits 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() [1/2]

template<typename GeomTraits >

std::array< typename GeomTraits::FT, 4 > CGAL::Barycentric_coordinates::tetrahedron_coordinates	(	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.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() [2/2]

template<typename OutputIterator , typename GeomTraits >

OutputIterator CGAL::Barycentric_coordinates::tetrahedron_coordinates	(	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.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.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

computation_policy 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
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 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.
geom_traits 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).