Nef_polyhedron_3<Traits>

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CGAL::Nef_polyhedron_3<Traits>

Definition

A 3D Nef polyhedron is a subset of the 3-dimensional space that is the result of forming complements and intersections starting from a finite set H of 3-dimensional halfspaces. Nef polyhedra are closed under all binary set operations, i.e., intersection, union, difference, complement, and under the topological operations boundary, closure, and interior.

A 3D Nef polyhedron can be represented by the local pyramids of the minimal elements of its incidence structure. Without going into to much detail, a local pyramid essentially reflects the topologic and geometric situation at a certain location in a point set. For finite polyhedra the minimal elements of the incidence structure are vertices only. This means, that it suffices to model the topological and geometric situation of the vertices. For 3D Nef polyhedra, the local pyramid of a vertex is represented by a planar Nef polyhedra embedded on a sphere.

A Nef_polyhedron_3 consists of vertices V, a sphere map for each vertex in V, edges E, facets F, volumes C, a mark for every item, and an incidence relation on them. Each edge and each facet is represented by two halfedges or two halffacets, respectively.

#include <CGAL/Nef_polyhedron_3.h>

Parameters

template <	class Nef_polyhedronTraits_3,
	class Nef_polyhedronItems_3 = CGAL::Default_items<Nef_polyhedronTraits_3>,
	class Nef_polyhedronMarks = bool >
class Nef_polyhedron_3;

The first parameter requires one of the following exact kernels: Homogeneous, Simple_homogeneous, Extended_homogeneous parametrized with Gmpz, leda_integer or any other number type modeling ℤ, or Cartesian, Simple_cartesian, Extended_cartesian parametrized with Gmpq, leda_rational, Quotient<Gmpz> or any other number type modeling ℚ.

The second parameter and the third parameter are for future considerations. Neither Nef_polyhedronItems_3 nor Nef_polyhedronMarks is specifed, yet. Do not use any other than the default types for these two template parameters.

Types

Nef_polyhedron_3<Traits>::Traits
	traits class selected for Nef_polyhedronTraits_3.
Nef_polyhedron_3<Traits>::Mark
	All object (vertices, edges, etc.) are attributed by a Mark. Mark equals bool.

Nef_polyhedron_3<Traits>::size_type
	size type of Nef_polyhedron_3.

Nef_polyhedron_3<Traits>::Vertex_const_handle
	non-mutable handle to a vertex.
Nef_polyhedron_3<Traits>::Halfedge_const_handle
	non-mutable handle to a halfedge.
Nef_polyhedron_3<Traits>::Halffacet_const_handle
	non-mutable handle to a halffacet.
Nef_polyhedron_3<Traits>::Volume_const_handle
	non-mutable handle to a volume.
Nef_polyhedron_3<Traits>::SVertex_const_handle
	non-mutable handle to a svertex.
Nef_polyhedron_3<Traits>::SHalfedge_const_handle
	non-mutable handle to a shalfedge.
Nef_polyhedron_3<Traits>::SHalfloop_const_handle
	non-mutable handle to a shalfloop.
Nef_polyhedron_3<Traits>::SFace_const_handle
	non-mutable handle to a sface.

Nef_polyhedron_3<Traits>::Vertex_const_iterator
	non-mutable iterator over all vertices.
Nef_polyhedron_3<Traits>::Halfedge_const_iterator
	non-mutable iterator over all halfeges.
Nef_polyhedron_3<Traits>::Halffacet_const_iterator
	non-mutable iterator over all halffacets.
Nef_polyhedron_3<Traits>::Volume_const_iterator
	non-mutable iterator over all volumes.
Nef_polyhedron_3<Traits>::SVertex_const_iterator
	non-mutable iterator over all svertices.
Nef_polyhedron_3<Traits>::SHalfedge_const_iterator
	non-mutable iterator over all shalfedges.
Nef_polyhedron_3<Traits>::SHalfloop_const_iterator
	non-mutable iterator over all shalfloops.
Nef_polyhedron_3<Traits>::SFace_const_iterator
	non-mutable iterator over all sfaces.

Nef_polyhedron_3<Traits>::SHalfedge_around_svertex_const_circulator
	non-mutable circulator of shalfedges around a svertex (cw).
Nef_polyhedron_3<Traits>::SHalfedge_around_sface_const_circulator
	non-mutable circulator of shalfedges around a sface (ccw).
Nef_polyhedron_3<Traits>::SHalfedge_around_facet_const_circulator
	non-mutable circulator of shalfedges around a halffacet (ccw).
Nef_polyhedron_3<Traits>::SFace_cycle_const_iterator
	non-mutable iterator over the cylces of a sface.
Nef_polyhedron_3<Traits>::Halffacet_cycle_const_iterator
	non-mutable iterator over the cylces of a halffacet.
Nef_polyhedron_3<Traits>::Shell_entry_const_iterator
	non-mutable iterator providing an entry to each shell.

Nef_polyhedron_3<Traits>::Object_handle
	a generic handle to an object. The kind of object (vertex, halfedge, halffacet, volume, svertex, shalfedge, shalfloop, sface) can be determined and the object can be assigned to a corresponding constant handle by one of the following functions: bool assign(Vertex_const_handle& h, Object_handle) bool assign(Halfedge_const_handle& h, Object_handle) bool assign(Halffacet_const_handle& h, Object_handle) bool assign(Volume_const_handle& h, Object_handle) bool assign(SVertex_const_handle& h, Object_handle) bool assign(SHalfedge_const_handle& h, Object_handle) bool assign(SHalfloop_const_handle& h, Object_handle) bool assign(SFace_const_handle& h, Object_handle) where each function returns true iff the assignment to h could be accomplished.

Nef_polyhedron_3<Traits>::Point_3
	location of vertices.
Nef_polyhedron_3<Traits>::Segment_3
	segment represented by a halfedge.
Nef_polyhedron_3<Traits>::Vector_3
	direction of a halfedge.
Nef_polyhedron_3<Traits>::Plane_3
	plane of a halffacet lies in.
Nef_polyhedron_3<Traits>::Aff_transformation_3
	affine transformation.

Nef_polyhedron_3<Traits>::Polylines_tag
	tag for calling polyline constructor.

enum Boundary { EXCLUDED, INCLUDED};
	construction selection.
enum Content { EMPTY, COMPLETE};
	construction selection.
enum Intersection_mode { CLOSED_HALFSPACE, OPEN_HALFSPACE, PLANE_ONLY};
	intersection selection.

Nef_polyhedron_3<Traits>::Nef_polyhedron_S2
	a sphere map.
Nef_polyhedron_3<Traits>::Polyhedron
	a polyhedral surface.

Creation

Nef_polyhedron_3<Traits> N ( Content space = EMPTY);
	creates a Nef polyhedron and initializes it to the empty set if plane == EMPTY and to the whole space if space == COMPLETE.

Nef_polyhedron_3<Traits> N ( Plane_3 p, Boundary b = INCLUDED);
	creates a Nef polyhedron containing the halfspace on the negative side of p including p if b==INCLUDED, excluding p if b==EXCLUDED.

Nef_polyhedron_3<Traits> N ( Polyhedron& P);
	creates a Nef polyhedron, which represents the same point set as the polyhedral surface P does.

Nef_polyhedron_3<Traits> N ( Input_iterator begin, Input_iterator end);
	creates a Nef polyhedron consisting of a single polygon spanned by the list of points in the iterator range [begin,end). If the points do not on a common supporting plane, the constructor tries to triangulate the polygon into multiple facets.If the construction does not succeed, the empty set is created.

template <class Forward_iterator>
Nef_polyhedron_3<Traits> N ( Forward_iterator it, Forward_iterator end, Polylines_tag);
	The iterator range [it, end) defines a set point ranges, each of which defines a polyline.

Access Member Functions

bool	N.is_simple () const	returns true, if N is a 2-manifold.
bool	N.is_valid () const	checks the integrity of N .

Size_type	N.number_of_vertices () const	returns the number of vertices.
Size_type	N.number_of_halfedges () const	return the number of halfedges.
Size_type	N.number_of_edges () const	returns the number of halfedge pairs.
Size_type	N.number_of_halffacets () const	returns the number of halffacets.
Size_type	N.number_of_facets () const	returns the number of halffacet pairs.
Size_type	N.number_of_volumes () const	returns the number of volumes.

Vertex_const_iterator	N.vertices_begin () const	iterator over all vertices.
Vertex_const_iterator	N.vertices_end () const	past-the-end iterator.

Halfedge_const_iterator	N.halfedges_begin () const	iterator over all halfedges.
Halfedge_const_iterator	N.halfedges_end () const	past-the-end iterator.

Halffacet_const_iterator	N.halffacets_begin () const	iterator over all halffacets.
Halffacet_const_iterator	N.halffacets_end () const	past-the-end iterator.

Volume_const_iterator	N.volumes_begin () const	iterator over all volumes.
Volume_const_iterator	N.volumes_end () const	past-the-end iterator.

The following macros are provided: CGAL_forall_vertices(v,N), CGAL_forall_halfedges(e,N), CGAL_forall_edges(e,N), CGAL_forall_halffacets(f,N), CGAL_forall_facets(f,N), CGAL_forall_volumes(c,N) where N is a Nef_polyhedron_3.

Object_handle	N.locate ( Point_3 p) const	returns a generic handle to the object (vertex, edge, facet or volume) which contains the point p in its relative interior.

Nef_polyhedron_S2	N.get_sphere_map ( Vertex_const_iterator v) const
		returns the neighborhood of a vertex modeled by a Nef_polyhedron_S2.

Point Set Predicates

bool	N.is_empty () const	returns true, if N is the empty point set.
bool	N.is_space () const	returns true, if N is the complete 3D space.

bool	N == N1 const	returns true, if N and N1 comprise the same point sets.
bool	N != N1 const	returns true, if N and N1 comprise different point sets.
bool	N < N1 const	returns true, if N is a proper subset of N1.
bool	N > N1 const	returns true, if N is a proper superset of N1.
bool	N <= N1 const	returns true, if N is a subset of N1.
bool	N >= N1 const	returns true, if N is a superset of N1.

Unary Set Operations

Nef_polyhedron_3<Traits>	N.complement () const	returns the complement of N .
Nef_polyhedron_3<Traits>	N.interior () const	returns the interior of N .
Nef_polyhedron_3<Traits>	N.boundary () const	returns the boundary of N .
Nef_polyhedron_3<Traits>	N.closure () const	returns the closure of N .
Nef_polyhedron_3<Traits>	N.regularization () const	returns the regularization, i.e. the closure of the interior, of N .

Nef_polyhedron_3<Traits>	! N const	returns the complement of N .

Binary Set Operations

Nef_polyhedron_3<Traits>	N.intersection ( N1) const	return the intersection of N and N1.
Nef_polyhedron_3<Traits>	N.join ( N1) const	return the union of N and N1. (Note that ''union'' is a C++ keyword and cannot be used for this operation.)
Nef_polyhedron_3<Traits>	N.difference ( N1) const	return the difference between N and N1.
Nef_polyhedron_3<Traits>	N.symmetric_difference ( N1) const
		return the symmetric difference of N and N1.
Nef_polyhedron_3<Traits>	N.intersection ( Plane_3 p, Intersection_mode im) const
		returns intersection of N with plane (im=PLANE_ONLY), open halfspace (im=OPEN_HALFSPACE), or closed halfspace (im=CLOSED_HALFSPACE). In the latter two cases, the halfspaces are on the negative side of the plane p. The function is written for the use with standard kernels, where halfspaces are not part of the domain. The function does not work in combination with an extended kernels or with an unbounded polyhedron.

Nef_polyhedron_3<Traits>	N N1 const*	return the intersection of N and N1.
Nef_polyhedron_3<Traits>	N + N1 const	return the union of N and N1.
Nef_polyhedron_3<Traits>	N - N1 const	return the difference between N and N1.
Nef_polyhedron_3<Traits>	N ^ N1 const	return the symmetric difference of N and N1.

void	N = N1*	intersects N and N1.
void	N += N1	unites N with N1.
void	N -= N1	subtracts N1 from N .
void	N ^= N1	performs a symmetric intersection of N and N1.

Operations

void

N.clear ( Content space = EMPTY)

make N the empty set if space == EMPTY and the complete 3D space if space == COMPLETE.

void

N.transform ( Aff_transformation_3 aff)

applies an affine transformation to N .

void

N.convert_to_polyhedron ( Polyhedron& P) const

converts N into a Polyhedron.

Precondition:

N is simple.

void

N.visit_shell_objects ( SFace_const_handle f, Visitor& V) const

calls the visit function of V for every item which belongs to the same shell as sf.

CGAL::Nef_polyhedron_3<Traits>

Definition

Parameters

Types

Creation

Access Member Functions

Point Set Predicates

Unary Set Operations

Binary Set Operations

Operations

See Also