## 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::SNC_items,
| class Nef_polyhedronMarks = bool >
class Nef_polyhedron_3;

The first parameter requires one of the following exact kernels: Homogeneous, Simple_homogeneous, Extended_homogeneous_3 parametrized with Gmpz, leda_integer or any other number type modeling , or Cartesian, Simple_cartesian, Extended_cartesian_3 parametrized with Gmpq, leda_rational, Quotient<Gmpz> or any other number type modeling .

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_aroud_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.
enum Boundary { EXCLUDED , INCLUDED };
| construction selection.
enum Content { EMPTY , COMPLETE };
| construction 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 left 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.

### Access Member Functions

bool | N.is_simple () | returns true, if N is a 2-manifold.
bool | N.is_valid () | checks the integrity of N .
Size_type | N.number_of_vertices ()
|  | returns the number of vertices.
Size_type | N.number_of_halfedges ()
|  | return the number of halfedges.
Size_type | N.number_of_edges ()
|  | returns the number of halfedge pairs.
Size_type | N.number_of_halffacets ()
|  | returns the number of halffacets.
Size_type | N.number_of_facets ()
|  | returns the number of halffacet pairs.
Size_type | N.number_of_volumes ()
|  | returns the number of volumes.
Vertex_const_iterator
| N.vertices_begin ()
|  | iterator over all vertices.
Vertex_const_iterator
| N.vertices_end () | past-the-end iterator.
Halfedge_const_iterator
| N.halfedges_begin ()
|  | iterator over all halfedges.
Halfedge_const_iterator
| N.halfedges_end () | past-the-end iterator.
Halffacet_const_iterator
| N.halffacets_begin ()
|  | iterator over all halffacets.
Halffacet_const_iterator
| N.halffacets_end ()
|  | past-the-end iterator.
Volume_const_iterator
| N.volumes_begin () | iterator over all volumes.
Volume_const_iterator
| N.volumes_end () | 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)
|  | 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)
|  | returns the neighborhood of a vertex modeled by a Nef_polyhedron_S2.

### Point Set Predicates

bool | N.is_empty () | returns true, if N is the empty point set.
bool | N.is_space () | returns true, if N is the complete 3D space.
bool | N == N1 | returns true, if N and N1 comprise the same point sets.
bool | N != N1 | returns true, if N and N1 comprise different point sets.
bool | N < N1 | returns true, if N is a proper subset of N1.
bool | N > N1 | returns true, if N is a proper superset of N1.
bool | N <= N1 | returns true, if N is a subset of N1.
bool | N >= N1 | returns true, if N is a superset of N1.

### Unary Set Operations

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

### Binary Set Operations

Nef_polyhedron_3<Traits>
| N.intersection ( N1)
|  | return the intersection of N and N1.
Nef_polyhedron_3<Traits>
| N.join ( N1) | 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)
|  | return the difference between N and N1.
Nef_polyhedron_3<Traits>
| N.symmetric_difference ( N1)
|  | return the symmetric difference of N and N1.
Nef_polyhedron_3<Traits>
| N * N1 | return the intersection of N and N1.
Nef_polyhedron_3<Traits>
| N + N1 | return the union of N and N1.
Nef_polyhedron_3<Traits>
| N - N1 | return the difference between N and N1.
Nef_polyhedron_3<Traits>
| N ^ N1 | 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)
|  | converts N into a Polyhedron. Precondition: N is simple.

void

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

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

### See Also

CGAL::Nef_polyhedron_3<Traits>::Vertex
CGAL::Nef_polyhedron_3<Traits>::Halfedge
CGAL::Nef_polyhedron_3<Traits>::Halffacet
CGAL::Nef_polyhedron_3<Traits>::Volume
CGAL::Nef_polyhedron_3<Traits>::SHalfedge
CGAL::Nef_polyhedron_3<Traits>::SHalfloop
CGAL::Nef_polyhedron_3<Traits>::SFace
CGAL::Nef_polyhedron_S2<Traits>
CGAL::Polyhedron<Traits>

### Example

This example program creates two Nef polyhedra - one representing the empty point set, one representing the whole 3D space. The complement of the latter one is computeted and compared with the first one.

    // file: examples/Nef_3/simple.C

    #include <CGAL/Gmpz.h>
    #include <CGAL/Homogeneous.h>
    #include <CGAL/Nef_polyhedron_3.h>

    typedef CGAL::Homogeneous<CGAL::Gmpz>  Kernel;
    typedef CGAL::Nef_polyhedron_3<Kernel> Nef_polyhedron;

    int main() {
      Nef_polyhedron N0(Nef_polyhedron::EMPTY);
      Nef_polyhedron N1(Nef_polyhedron::COMPLETE);

      CGAL_assertion (N0 == N1.complement());
      return 0;
    }