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

template<typename PointRange , typename CoordinateRange , typename GeomTraits = typename boost::range_value<PointRange>::type>
boost::range_value< PointRange >::type	CGAL::Barycentric_coordinates::apply_barycentric_coordinates (const PointRange &points, const CoordinateRange &coordinates)
	computes a point location from barycentric coordinates with respect to a set of points.

template<typename TriangleMesh , typename OutputIterator , typename GeomTraits , typename VertexPointMap >
std::pair< OutputIterator, bool >	CGAL::Barycentric_coordinates::boundary_coordinates_3 (const TriangleMesh &tmesh, const typename GeomTraits::Point_3 &query, OutputIterator oi, const GeomTraits &traits, const VertexPointMap vertex_point_map)
	computes boundary barycentric coordinates with respect to a closed convex triangle mesh.

template<typename TriangleMesh , typename Point_3 , typename OutputIterator , typename VertexPointMap = typename boost::property_map<TriangleMesh, CGAL::vertex_point_t>::const_type>
std::pair< OutputIterator, bool >	CGAL::Barycentric_coordinates::boundary_coordinates_3 (const TriangleMesh &tmesh, const Point_3 &query, OutputIterator oi, const VertexPointMap vertex_point_map)
	computes boundary barycentric coordinates with respect to a closed convex triangle mesh.

template<typename Point_3 , typename TriangleMesh , typename OutputIterator >
OutputIterator	CGAL::Barycentric_coordinates::discrete_harmonic_coordinates_3 (const TriangleMesh &tmesh, const Point_3 &query, OutputIterator oi, const Computation_policy_3 policy=Computation_policy_3::FAST)
	computes 3D discrete harmonic coordinates with respect to a closed convex triangle mesh.

template<typename Point_3 , typename TriangleMesh , typename OutputIterator >
OutputIterator	CGAL::Barycentric_coordinates::mean_value_coordinates_3 (const TriangleMesh &tmesh, const Point_3 &query, OutputIterator oi, const Computation_policy_3 policy=Computation_policy_3::FAST_WITH_EDGE_CASES)
	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 Point_3 , typename TriangleMesh , typename OutputIterator >
OutputIterator	CGAL::Barycentric_coordinates::wachspress_coordinates_3 (const TriangleMesh &tmesh, const Point_3 &query, OutputIterator oi, const Computation_policy_3 policy=Computation_policy_3::FAST)
	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 boost::range_value<PointRange>::type>

boost::range_value< PointRange >::type CGAL::Barycentric_coordinates::apply_barycentric_coordinates	(	const PointRange &	points,
		const CoordinateRange &	coordinates
	)

#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`

Parameters

points	a range of input points
coordinates	barycentric coordinates of the point

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
	)

#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`

Parameters

tmesh	an instance of `TriangleMesh`
coordinates	barycentric coordinates of the point
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() [1/2]

template<typename TriangleMesh , typename Point_3 , typename OutputIterator , typename VertexPointMap = typename boost::property_map<TriangleMesh, CGAL::vertex_point_t>::const_type>

std::pair< OutputIterator, bool > CGAL::Barycentric_coordinates::boundary_coordinates_3	(	const TriangleMesh &	tmesh,
		const Point_3 &	query,
		OutputIterator	oi,
		const VertexPointMap	vertex_point_map
	)

#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 one, while all other coordinates are zero. If query is on the face, the three corresponding coordinates are triangle coordinates, while all other coordinates are set to zero. If query is not on the boundary, all the coordinates are set to zero.

Template Parameters

TriangleMesh	must be a model of the concept `FaceListGraph`.
Point_3	a model of `Kernel::Point_3`
OutputIterator	a model of `OutputIterator` that accepts values of type `GeomTraits::FT`
VertexPointMap	a property map with boost::graph_traits<TriangleMesh>::vertex_descriptor as key type and Point_3 as value type. The default is `property_map_selector<TriangleMesh, CGAL::vertex_point_t>`.

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
vertex_point_map	an instance of `VertexPointMap` that maps a vertex from `tmesh` to `Point_3`; the default initialization is provided

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

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

◆ boundary_coordinates_3() [2/2]

template<typename TriangleMesh , typename OutputIterator , typename GeomTraits , typename VertexPointMap >

std::pair< OutputIterator, bool > CGAL::Barycentric_coordinates::boundary_coordinates_3	(	const TriangleMesh &	tmesh,
		const typename GeomTraits::Point_3 &	query,
		OutputIterator	oi,
		const GeomTraits &	traits,
		const VertexPointMap	vertex_point_map
	)

#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 convex simplicial 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 one, while all other coordinates are zero. If query is on the face, the three corresponding coordinates are triangle coordinates, while all other coordinates are set to zero. If query is not on the boundary, all the coordinates are set to zero.

Template Parameters

TriangleMesh	must be a model of the concept `FaceListGraph`.
GeomTraits	a model of `BarycentricTraits_3`
OutputIterator	a model of `OutputIterator` that accepts values of type `GeomTraits::FT`
VertexPointMap	a property map with boost::graph_traits<TriangleMesh>::vertex_descriptor as key type and `GeomTraits::Point_3` as value type

Parameters

tmesh	an instance of `TriangleMesh`
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; the default initialization is provided
vertex_point_map	an instance of `VertexPointMap` that maps a vertex from `tmesh` to `Point_3`; the default initialization is provided

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

Precondition: boost::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 Point_3 , typename TriangleMesh , typename OutputIterator >

OutputIterator CGAL::Barycentric_coordinates::discrete_harmonic_coordinates_3	(	const TriangleMesh &	tmesh,
		const Point_3 &	query,
		OutputIterator	oi,
		const Computation_policy_3	policy = `Computation_policy_3::FAST`
	)

#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

Point_3	A model of `Kernel::Point_3`
TriangleMesh	must be a model of the concept `FaceListGraph`
OutputIterator	a model of `OutputIterator` that accepts values of type `GeomTraits::FT`

Parameters

tmesh	an instance of `TriangleMesh`
query	a query point
oi	the beginning of the destination range with the computed coordinates
policy	one of the `CGAL::Barycentric_coordinates::Computation_policy_3`; the default is `Computation_policy_3::FAST_WITH_EDGE_CASES`

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

Precondition: boost::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 Point_3 , typename TriangleMesh , typename OutputIterator >

OutputIterator CGAL::Barycentric_coordinates::mean_value_coordinates_3	(	const TriangleMesh &	tmesh,
		const Point_3 &	query,
		OutputIterator	oi,
		const Computation_policy_3	policy = `Computation_policy_3::FAST_WITH_EDGE_CASES`
	)

#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 a polyhedron with triangular faces, 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

Point_3	A model of `Kernel::Point_3`
TriangleMesh	must be a model of the concept `FaceListGraph`
OutputIterator	a model of `OutputIterator` that accepts values of type `Kernel_traits<Point_3>::Kernel::FT`

Parameters

tmesh	an instance of `TriangleMesh`
query	a query point
oi	the beginning of the destination range with the computed coordinates
policy	one of the `CGAL::Barycentric_coordinates::Computation_policy_3`; the default is `Computation_policy_3::FAST_WITH_EDGE_CASES`

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

Precondition: boost::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 Point_3 , typename TriangleMesh , typename OutputIterator >

OutputIterator CGAL::Barycentric_coordinates::wachspress_coordinates_3	(	const TriangleMesh &	tmesh,
		const Point_3 &	query,
		OutputIterator	oi,
		const Computation_policy_3	policy = `Computation_policy_3::FAST`
	)

#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 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 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

Point_3	A model of `Kernel::Point_3`
TriangleMesh	must be a model of the concept `FaceListGraph`
OutputIterator	a model of `OutputIterator` that accepts values of type `GeomTraits::FT`

Parameters

tmesh	an instance of `TriangleMesh`
query	a query point
oi	the beginning of the destination range with the computed coordinates
policy	one of the `CGAL::Barycentric_coordinates::Computation_policy_3`; the default is `Computation_policy_3::FAST_WITH_EDGE_CASES`

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

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

Functions

Function Documentation

◆ apply_barycentric_coordinates() [1/2]

◆ apply_barycentric_coordinates() [2/2]

◆ boundary_coordinates_3() [1/2]

◆ boundary_coordinates_3() [2/2]

◆ discrete_harmonic_coordinates_3()

◆ mean_value_coordinates_3()

◆ tetrahedron_coordinates() [1/2]

◆ tetrahedron_coordinates() [2/2]

◆ wachspress_coordinates_3()