CGAL::Barycentric_coordinates::Wachspress_coordinates_3< PolygonMesh, GeomTraits, VertexToPointMap > Class Template Reference

#include <CGAL/Barycentric_coordinates_3/Wachspress_coordinates_3.h>

Definition

template<typename PolygonMesh, typename GeomTraits, typename VertexToPointMap = typename property_map_selector<PolygonMesh, CGAL::vertex_point_t>::const_type>
class CGAL::Barycentric_coordinates::Wachspress_coordinates_3< PolygonMesh, GeomTraits, VertexToPointMap >

3D Wachspress coordinates.

This class implements 3D Wachspress coordinates ( [1], [3] ), which can be computed at any point inside a convex polyhedron with triangular faces.

Wachspress coordinates are well-defined and non-negative in the closure of a convex polyhedron with triangular faces. The coordinates are computed analytically.

Template Parameters

PolygonMesh	must be a model of the concept FaceListGraph.
GeomTraits	a model of BarycentricTraits_3
VertexToPointMap	a property map with boost::graph_traits<PolygonMesh>::vertex_descriptor as key type and Point_3 as value type. The default is property_map_selector<PolygonMesh, CGAL::vertex_point_t>.

Examples

Barycentric_coordinates_3/wachspress_coordinates.cpp.

Types
typedef GeomTraits::FT	FT
	Number type.
typedef GeomTraits::Point_3	Point_3
	Point type.
typedef GeomTraits::Vector_3	Vector_3
	Vector type.

Initialization
	Wachspress_coordinates_3 (const PolygonMesh &polygon_mesh, const Computation_policy_3 policy, const VertexToPointMap vertex_to_point_map, const GeomTraits traits=GeomTraits())
	initializes all internal data structures. More...

Access
template<typename OutIterator >
OutIterator	operator() (const Point_3 &query, OutIterator c_begin)
	computes 3D Wachspress coordinates. More...

Constructor & Destructor Documentation

Wachspress_coordinates_3()

template<typename PolygonMesh , typename GeomTraits , typename VertexToPointMap = typename property_map_selector<PolygonMesh, CGAL::vertex_point_t>::const_type>

CGAL::Barycentric_coordinates::Wachspress_coordinates_3< PolygonMesh, GeomTraits, VertexToPointMap >::Wachspress_coordinates_3	(	const PolygonMesh &	polygon_mesh,
		const Computation_policy_3	policy,
		const VertexToPointMap	vertex_to_point_map,
		const GeomTraits	traits = GeomTraits()
	)

initializes all internal data structures.

This class implements the behavior of Wachspress coordinates for 3D query points.

Parameters

polygon_mesh	an instance of PolygonMesh, which must be a convex simplicial polyhedron
policy	one of the CGAL::Barycentric_coordinates::Computation_policy_3; the default is Computation_policy_3::FAST_WITH_EDGE_CASES
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 polygon_mesh to Point_3; the default initialization is provided

Precondition

num_vertices(polygon_mesh) >= 4.

polygon_mesh is strongly convex.

polygon_mesh is simplicial.

Member Function Documentation

operator()()

template<typename PolygonMesh , typename GeomTraits , typename VertexToPointMap = typename property_map_selector<PolygonMesh, CGAL::vertex_point_t>::const_type>

template<typename OutIterator >

OutIterator CGAL::Barycentric_coordinates::Wachspress_coordinates_3< PolygonMesh, GeomTraits, VertexToPointMap >::operator()	(	const Point_3 &	query,
		OutIterator	c_begin
	)

computes 3D Wachspress coordinates.

This function fills c_begin with 3D Wachspress coordinates computed at the query point with respect to the vertices of the input polyhedron.

The number of returned coordinates equals to the number of vertices.

After the coordinates \(b_i\) with \(i = 1\dots n\) are computed, where \(n\) is the number of vertices, the query point \(q\) can be obtained as \(q = \sum_{i = 1}^{n}b_ip_i\), where \(p_i\) are the polyhedron vertices.

Template Parameters

OutIterator

a model of OutputIterator that accepts values of type FT

Parameters

query	a query point
c_begin	the beginning of the destination range with the computed coordinates

Returns

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