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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>
| 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 
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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.
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traits class selected for Nef_polyhedronTraits_3.
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All object (vertices, edges, etc.) are attributed by a Mark.
Mark equals bool.
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size type of Nef_polyhedron_3.
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non-mutable handle to a vertex.
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non-mutable handle to a halfedge.
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non-mutable handle to a halffacet.
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non-mutable handle to a volume.
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non-mutable handle to a svertex.
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non-mutable handle to a shalfedge.
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non-mutable handle to a shalfloop.
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non-mutable handle to a sface.
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non-mutable iterator over all vertices.
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non-mutable iterator over all halfeges.
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non-mutable iterator over all halffacets.
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non-mutable iterator over all volumes.
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non-mutable iterator over all svertices.
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non-mutable iterator over all shalfedges.
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non-mutable iterator over all shalfloops.
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non-mutable iterator over all sfaces.
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non-mutable circulator of shalfedges around a svertex (cw).
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non-mutable circulator of shalfedges around a sface (ccw).
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non-mutable circulator of shalfedges around a halffacet (ccw).
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non-mutable iterator over the cylces of a sface.
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non-mutable iterator over the cylces of a halffacet.
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non-mutable iterator providing an entry to each shell.
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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.
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location of vertices.
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segment represented by a halfedge.
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direction of a halfedge.
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plane of a halffacet lies in.
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affine transformation.
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construction selection.
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construction selection.
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a sphere map.
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a polyhedral surface.
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creates a Nef polyhedron and initializes it to the empty
set if plane == EMPTY
and to the whole space if space == COMPLETE.
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creates a
Nef polyhedron containing the halfspace on the negative side of
p including p if b==INCLUDED, excluding p if
b==EXCLUDED.
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creates a Nef polyhedron, which represents the same point set as
the polyhedral surface P does.
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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.
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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.
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| returns a generic handle to the object (vertex, edge, facet or volume) which contains the point p in its relative interior. |
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| returns the neighborhood of a vertex modeled by a Nef_polyhedron_S2. | ||
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| returns the complement of N . |
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| returns the interior of N . |
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| returns the boundary of N . |
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| returns the closure of N . |
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| returns the regularization, i.e. the closure of the interior, of N . |
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| returns the complement of N . |
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| return the intersection of N and N1. |
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| return the union of N and N1. (Note that ''union'' is a C++ keyword and cannot be used for this operation.) |
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| return the difference between N and N1. |
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| return the symmetric difference of N and N1. |
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| return the intersection of N and N1. |
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| return the union of N and N1. |
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| return the difference between N and N1. |
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| return the symmetric difference of N and N1. |
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| intersects N and N1. |
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| unites N with N1. |
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| subtracts N1 from N . |
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| performs a symmetric intersection of N and N1. |
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| make N the empty set if space == EMPTY and the complete 3D space if space == COMPLETE. | ||
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| applies an affine transformation to N . | ||||
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converts N into a Polyhedron.
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| calls the visit function of V for every item which belongs to the same shell as sf. | ||||
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_3<Traits>
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.cpp
#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;
}