Free Functions

Free functions to compute barycentric coordinates.

Functions
template<typename TriangleMesh , typename OutIterator , typename GeomTraits , typename VertexToPointMap >
std::pair< OutIterator, bool >	CGAL::Barycentric_coordinates::boundary_coordinates_3 (const TriangleMesh &triangle_mesh, const typename GeomTraits::Point_3 &query, OutIterator c_begin, const GeomTraits &traits, const VertexToPointMap vertex_to_point_map)
	computes boundary barycentric coordinates. More...
template<typename TriangleMesh , typename Point_3 , typename OutIterator , typename VertexToPointMap = typename property_map_selector<TriangleMesh, CGAL::vertex_point_t>::const_type>
std::pair< OutIterator, bool >	CGAL::Barycentric_coordinates::boundary_coordinates_3 (const TriangleMesh &triangle_mesh, const Point_3 &query, OutIterator c_begin, const VertexToPointMap vertex_to_point_map)
	computes boundary barycentric coordinates. More...
template<typename Point_3 , typename TriangleMesh , typename OutIterator >
OutIterator	CGAL::Barycentric_coordinates::discrete_harmonic_coordinates_3 (const TriangleMesh &triangle_mesh, const Point_3 &query, OutIterator c_begin, const Computation_policy_3 policy=Computation_policy_3::FAST)
	computes 3D discrete harmonic coordinates. More...
template<typename Point_3 , typename TriangleMesh , typename OutIterator >
OutIterator	CGAL::Barycentric_coordinates::mean_value_coordinates_3 (const TriangleMesh &triangle_mesh, const Point_3 &query, OutIterator c_begin, const Computation_policy_3 policy=Computation_policy_3::FAST_WITH_EDGE_CASES)
	computes 3D mean value coordinates. More...
template<typename OutIterator , typename GeomTraits >
OutIterator	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, OutIterator c_begin, const GeomTraits &traits)
	computes tetrahedron coordinates. More...
template<typename GeomTraits >
std::array< typename GeomTraits::FT, 4 >	CGAL::Barycentric_coordinates::tetrahedron_coordinates_in_array (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 tetrahedron coordinates. More...
template<typename Point_3 , typename PolygonMesh , typename OutIterator >
OutIterator	CGAL::Barycentric_coordinates::wachspress_coordinates_3 (const PolygonMesh &polygon_mesh, const Point_3 &query, OutIterator c_begin, const Computation_policy_3 policy=Computation_policy_3::FAST)
	computes 3D Wachspress coordinates. More...

Function Documentation

boundary_coordinates_3() [1/2]

template<typename TriangleMesh , typename Point_3 , typename OutIterator , typename VertexToPointMap = typename property_map_selector<TriangleMesh, CGAL::vertex_point_t>::const_type>

std::pair<OutIterator, bool> CGAL::Barycentric_coordinates::boundary_coordinates_3	(	const TriangleMesh &	triangle_mesh,
		const Point_3 &	query,
		OutIterator	c_begin,
		const VertexToPointMap	vertex_to_point_map
	)

#include <CGAL/Barycentric_coordinates_3/boundary_coordinates_3.h>

computes boundary barycentric coordinates.

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 c_begin.

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
OutIterator	a model of OutputIterator that accepts values of type GeomTraits::FT
VertexToPointMap	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

triangle_mesh	an instance of TriangleMesh, which must be a convex simplicial polyhedron
query	a query point
c_begin	the beginning of the destination range with the computed coordinates
vertex_to_point_map	an instance of VertexToPointMap that maps a vertex from triangle_mesh 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

num_vertices(triangle_mesh) >= 4.

triangle_mesh is simplicial.

boundary_coordinates_3() [2/2]

template<typename TriangleMesh , typename OutIterator , typename GeomTraits , typename VertexToPointMap >

std::pair<OutIterator, bool> CGAL::Barycentric_coordinates::boundary_coordinates_3	(	const TriangleMesh &	triangle_mesh,
		const typename GeomTraits::Point_3 &	query,
		OutIterator	c_begin,
		const GeomTraits &	traits,
		const VertexToPointMap	vertex_to_point_map
	)

#include <CGAL/Barycentric_coordinates_3/boundary_coordinates_3.h>

computes boundary barycentric coordinates.

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 c_begin.

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
OutIterator	a model of OutputIterator that accepts values of type GeomTraits::FT
VertexToPointMap	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

triangle_mesh	an instance of TriangleMesh, which must be a convex simplicial polyhedron
query	a query point
c_begin	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_to_point_map	an instance of VertexToPointMap that maps a vertex from triangle_mesh 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

num_vertices(triangle_mesh) >= 4.

triangle_mesh is simplicial.

discrete_harmonic_coordinates_3()

template<typename Point_3 , typename TriangleMesh , typename OutIterator >

OutIterator CGAL::Barycentric_coordinates::discrete_harmonic_coordinates_3	(	const TriangleMesh &	triangle_mesh,
		const Point_3 &	query,
		OutIterator	c_begin,
		const Computation_policy_3	policy = Computation_policy_3::FAST
	)

#include <CGAL/Barycentric_coordinates_3/Discrete_harmonic_coordinates_3.h>

computes 3D discrete harmonic coordinates.

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 c_begin.

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.
OutIterator	a model of OutputIterator that accepts values of type GeomTraits::FT

Parameters

triangle_mesh	an instance of TriangleMesh, which must be a convex simplicial polyhedron
query	a query point
c_begin	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

num_vertices(triangle_mesh) >= 4.

triangle_mesh is strongly convex.

triangle_mesh is simplicial.

Examples

Barycentric_coordinates_3/discrete_harmonic_coordinates.cpp.

mean_value_coordinates_3()

template<typename Point_3 , typename TriangleMesh , typename OutIterator >

OutIterator CGAL::Barycentric_coordinates::mean_value_coordinates_3	(	const TriangleMesh &	triangle_mesh,
		const Point_3 &	query,
		OutIterator	c_begin,
		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 coordinates.

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 c_begin.

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.
OutIterator	a model of OutputIterator that accepts values of type GeomTraits::FT

Parameters

triangle_mesh	an instance of TriangleMesh, which must be a simplicial polyhedron
query	a query point
c_begin	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

num_vertices(triangle_mesh) >= 4.

triangle_mesh is simplicial.

Examples

Barycentric_coordinates_3/mean_value_coordinates.cpp.

tetrahedron_coordinates()

template<typename OutIterator , typename GeomTraits >

OutIterator 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,
		OutIterator	c_begin,
		const GeomTraits &	traits
	)

#include <CGAL/Barycentric_coordinates_3/tetrahedron_coordinates.h>

computes tetrahedron coordinates.

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 c_begin.

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

OutIterator	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
c_begin	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

Compute_volume_3(p0, p1, p2, p3) != 0

tetrahedron_coordinates_in_array()

template<typename GeomTraits >

std::array<typename GeomTraits::FT, 4> CGAL::Barycentric_coordinates::tetrahedron_coordinates_in_array	(	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 tetrahedron coordinates.

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 a tuple.

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

a array std::array<GeomTraits::FT, 4> with the computed coordinates

Precondition

Compute_volume_3(p0, p1, p2, p3) != 0

Examples

Barycentric_coordinates_3/tetrahedron_coordinates.cpp.

wachspress_coordinates_3()

template<typename Point_3 , typename PolygonMesh , typename OutIterator >

OutIterator CGAL::Barycentric_coordinates::wachspress_coordinates_3	(	const PolygonMesh &	polygon_mesh,
		const Point_3 &	query,
		OutIterator	c_begin,
		const Computation_policy_3	policy = Computation_policy_3::FAST
	)

#include <CGAL/Barycentric_coordinates_3/Wachspress_coordinates_3.h>

computes 3D Wachspress coordinates.

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 c_begin.

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.
PolygonMesh	must be a model of the concept FaceListGraph.
OutIterator	a model of OutputIterator that accepts values of type GeomTraits::FT

Parameters

polygon_mesh	an instance of PolygonMesh, which must be a convex simplicial polyhedron
query	a query point
c_begin	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

num_vertices(polygon_mesh) >= 4.

polygon_mesh is strongly convex.

polygon_mesh is simplicial.