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CGAL 4.7 - 2D Boolean Operations on Nef Polygons Embedded on the Sphere
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CGAL 4.7 - 2D Boolean Operations on Nef Polygons Embedded on the Sphere
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Nef polyhedra are defined as a subset of the d-dimensional space obtained by a finite number of set complement and set intersection operations on halfspaces.
Due to the fact that all other binary set operations like union, difference and symmetric difference can be reduced to intersection and complement calculations, Nef polyhedra are also closed under those operations. Also, Nef polyhedra are closed under topological unary set operations. Given a Nef polyhedron one can determine its interior, its boundary, and its closure.
Additionally, a d-dimensional Nef polyhedron has the property, that its boundary is a (d-1)-dimensional Nef polyhedron. This property can be used as a way to represent 3-dimensional Nef polyhedra by means of planar Nef polyhedra. This is done by intersecting the neighborhood of a vertex in a 3D Nef polyhedron with an \( \epsilon\)-sphere. The result is a planar Nef polyhedron embedded on the sphere.
The intersection of a halfspace going through the center of the \( \epsilon\)-sphere, with the \( \epsilon\)-sphere, results in a halfsphere which is bounded by a great circle. A binary operation of two halfspheres cuts the great circles into great arcs.
The incidence structure of planar Nef polyhedra can be reused. The items are denoted as svertex, shalfedge and sface, analogous to their counterparts in CGAL::Nef_polyhedron_2. Additionally, there is the shalfloop representing the great circles.
Classes | |
| class | CGAL::Qt_widget_Nef_S2< Nef_polyhedron_S2 > |
The class Qt_widget_Nef_S2 uses the OpenGL interface of Qt in order to display a Nef_polyhedron_S2. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits > |
An instance of data type Nef_polyhedron_S2<Traits> is a subset of the sphere \( S_2\) that is the result of forming complements and intersections starting from a finite set H of halfspaces bounded by a plane containing the origin. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::Sphere_circle |
An object c of type Sphere_circle is an oriented great circle on the surface of a unit sphere. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::Sphere_point |
An object p of type Sphere_point<R> is a point on the surface of a unit sphere. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::Sphere_segment |
An object s of type Sphere_segment is a segment in the surface of a unit sphere that is part of a great circle trough the origin. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::SFace_cycle_iterator |
The type SFace_cycle_iterator iterates over a list of Object_handles. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::SFace |
| Figures figureNefS2SVertexIncidences and figureNefS2SHalfloopIncidences illustrate the incidences of an sface. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::SHalfedge |
| A shalfedge is a great arc on a sphere map. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::SHalfloop |
| A sloop is a great circle on a sphere. More... | |
| class | CGAL::Nef_polyhedron_S2< Traits >::SVertex |
| Figure figureNefS2SVertexIncidences illustrates the incidence of a svertex on a sphere map. More... | |
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::Mark |
attributes of objects (vertices, edges, faces).
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::Object_handle |
a generic handle to an object of the underlying plane map.
The kind of object (vertex, halfedge, face) can be determined and the object can be assigned to a corresponding handle by the three functions:
bool assign(Vertex_const_handle& h, Object_handle)bool assign(Halfedge_const_handle& h, Object_handle)bool assign(Face_const_handle& h, Object_handle)where each function returns true iff the assignment to h was done.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SFace_const_handle |
non-mutable handle to sface.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SFace_const_iterator |
non-mutable iterator over all sfaces.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SFace_cycle_const_iterator |
iterating all sface cycles of an sface f.
The iterator has method bool is_svertex(), bool is_shalfedge(), bool is_shalfloop(), and can be converted to the corresponding handles SVertex_const_handle, SHalfedge_const_handle, or SHalfloop_const_handle.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SHalfedge_around_sface_const_circulator |
circulating the sface cycle of an sface f.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SHalfedge_around_svertex_const_circulator |
circulating the adjacency list of an svertex v.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SHalfedge_const_handle |
non-mutable handle to shalfedge.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SHalfedge_const_iterator |
non-mutable iterator over all shalfedges.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SHalfloop_const_handle |
non-mutable handle to shalfloop.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SHalfloop_const_iterator |
non-mutable iterator over all shalfloops.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::size_type |
size type
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SVertex_const_handle |
non-mutable handle to svertex.
| typedef unspecified_type CGAL::Nef_polyhedron_S2< Traits >::SVertex_const_iterator |
non-mutable iterator over all svertices.
| enum CGAL::Nef_polyhedron_S2::Boundary |
| enum CGAL::Nef_polyhedron_S2::Content |
1.8.8