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CGAL 5.2.2 - 2D Boolean Operations on Nef Polygons Embedded on the Sphere
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CGAL 5.2.2 - 2D Boolean Operations on Nef Polygons Embedded on the Sphere
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#include <CGAL/Nef_polyhedron_S2.h>
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.
Halfspaces correspond to hemispheres of \( S_2\) and are therefore modeled by oriented great circles of type Sphere_circle. Nef_polyhedron_S2 is closed under all binary set operations intersection, union, difference, complement and under the topological operations boundary, closure, and interior.
Parameters
The first parameter requires one of the following exact kernels: Homogeneous, Simple_homogeneous parametrized with Gmpz, leda_integer or any other number type modeling \(\mathbb{Z}\), or Cartesian, Simple_cartesian parametrized with Gmpq, leda_rational, Quotient<Gmpz> or any other number type modeling \(\mathbb{Q}\).
The second parameter and the third parameter are for future considerations. Neither Nef_polyhedronItems_S2 nor Nef_polyhedronMarks is specifed, yet. Do not use other than the default types for these two template parameters.
Exploration - Point location - Ray shooting
As Nef polyhedra are the result of forming complements and intersections starting from a set H of half-spaces that are defined by oriented lines in the plane, they can be represented by an attributed plane map \( M = (V,E,F)\). For topological queries within M the following types and operations allow exploration access to this structure.
Input and Output
A Nef polyhedron N can be visualized in an open GL window. The output operator is defined in the file CGAL/IO/Nef_polyhedron_2_Window-stream.h.
Implementation
Nef polyhedra are implemented on top of a halfedge data structure and use linear space in the number of vertices, edges and facets. Operations like empty take constant time. The operations clear, complement, interior, closure, boundary, regularization, input and output take linear time. All binary set operations and comparison operations take time \( O(n \log n)\) where \( n\) is the size of the output plus the size of the input.
The point location and ray shooting operations are implemented in the naive way. The operations run in linear query time without any preprocessing.
Classes | |
| class | SFace |
| Figures figureNefS2SVertexIncidences and figureNefS2SHalfloopIncidences illustrate the incidences of an sface. More... | |
| class | SFace_cycle_iterator |
The type SFace_cycle_iterator iterates over a list of Object_handles. More... | |
| class | SHalfedge |
| A shalfedge is a great arc on a sphere map. More... | |
| class | SHalfloop |
| A sloop is a great circle on a sphere. More... | |
| class | Sphere_circle |
An object c of type Sphere_circle is an oriented great circle on the surface of a unit sphere. More... | |
| class | Sphere_point |
An object p of type Sphere_point<R> is a point on the surface of a unit sphere. More... | |
| class | 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 | SVertex |
| Figure figureNefS2SVertexIncidences illustrates the incidence of a svertex on a sphere map. More... | |
Types | |
| enum | Boundary { EXCLUDED, INCLUDED } |
| construction selection. More... | |
| enum | Content { EMPTY, COMPLETE } |
| construction selection. More... | |
| typedef unspecified_type | SVertex_const_handle |
| non-mutable handle to svertex. | |
| typedef unspecified_type | SHalfedge_const_handle |
| non-mutable handle to shalfedge. | |
| typedef unspecified_type | SHalfloop_const_handle |
| non-mutable handle to shalfloop. | |
| typedef unspecified_type | SFace_const_handle |
| non-mutable handle to sface. | |
| typedef unspecified_type | SVertex_const_iterator |
| non-mutable iterator over all svertices. | |
| typedef unspecified_type | SHalfedge_const_iterator |
| non-mutable iterator over all shalfedges. | |
| typedef unspecified_type | SHalfloop_const_iterator |
| non-mutable iterator over all shalfloops. | |
| typedef unspecified_type | SFace_const_iterator |
| non-mutable iterator over all sfaces. | |
| typedef unspecified_type | SHalfedge_around_svertex_const_circulator |
circulating the adjacency list of an svertex v. | |
| typedef unspecified_type | SHalfedge_around_sface_const_circulator |
circulating the sface cycle of an sface f. | |
| typedef unspecified_type | SFace_cycle_const_iterator |
iterating all sface cycles of an sface f. More... | |
| typedef unspecified_type | Mark |
| attributes of objects (vertices, edges, faces). | |
| typedef unspecified_type | size_type |
| size type | |
| typedef unspecified_type | Object_handle |
| a generic handle to an object of the underlying plane map. More... | |
Creation | |
| Nef_polyhedron_S2 (Content sphere=EMPTY) | |
creates an instance N of type Nef_polyhedron_S2<K> and initializes it to the empty set if sphere == EMPTY and to the whole sphere if sphere == COMPLETE. | |
| Nef_polyhedron_S2 (Sphere_circle c, Boundary circle=INCLUDED) | |
creates a Nef polyhedron N containing the half-sphere left of c including c if circle==INCLUDED, excluding c if circle==EXCLUDED. | |
| template<class Forward_iterator > | |
| Nef_polyhedron_S2 (Forward_iterator first, Forward_iterator beyond, Boundary b=INCLUDED) | |
creates a Nef polyhedron N from the set of sphere segments in the iterator range [first,beyond). More... | |
Operations | |
| void | clear (Content plane=EMPTY) |
makes N the empty set if plane == EMPTY and the full plane if plane == COMPLETE. | |
| bool | is_empty () |
returns true if N is empty, false otherwise. | |
| bool | is_sphere () |
returns true if N is the whole sphere, false otherwise. | |
| bool | contains (Object_handle h) |
returns true iff the object h is contained in the set represented by N. | |
| bool | contained_in_boundary (Object_handle h) |
returns true iff the object h is contained in the \( 1\)-skeleton of N. | |
| Object_handle | locate (const Sphere_point &p) |
returns a generic handle h to an object (face, halfedge, vertex) of the underlying plane map that contains the point p in its relative interior. More... | |
| Object_handle | ray_shoot (const Sphere_point &p, const Sphere_direction &d) |
returns a handle h with N.contains(h) that can be converted to a Vertex_/Halfedge_/Face_const_handle as described above. More... | |
| Object_handle | ray_shoot_to_boundary (const Sphere_point &p, const Sphere_direction &d) |
returns a handle h that can be converted to a Vertex_/Halfedge_const_handle as described above. More... | |
Constructive Operations | |
Additionally there are operators There are also the corresponding modification operations There are also comparison operations like | |
| Nef_polyhedron_S2< K > | complement () |
returns the complement of N in the plane. | |
| Nef_polyhedron_S2< K > | interior () |
returns the interior of N. | |
| Nef_polyhedron_S2< K > | closure () |
returns the closure of N. | |
| Nef_polyhedron_S2< K > | boundary () |
returns the boundary of N. | |
| Nef_polyhedron_S2< K > | regularization () |
| returns the regularized polyhedron (closure of interior). | |
| Nef_polyhedron_S2< K > | intersection (const Nef_polyhedron_S2< K > &N1) |
returns N \( \cap\) N1. | |
| Nef_polyhedron_S2< K > | join (const Nef_polyhedron_S2< K > &N1) |
returns N \( \cup\) N1. | |
| Nef_polyhedron_S2< K > | difference (const Nef_polyhedron_S2< K > &N1) |
returns N \( -\) N1. | |
| Nef_polyhedron_S2< K > | symmetric_difference (const Nef_polyhedron_S2< K > &N1) |
returns the symmectric difference N - T \( \cup\) T - N. | |
Statistics and Integrity | |
| Size_type | number_of_svertices () |
| returns the number of svertices. | |
| Size_type | number_of_shalfedges () |
| returns the number of shalfedges. | |
| Size_type | number_of_sedges () |
| returns the number of sedges. | |
| Size_type | number_of_shalfloops () |
| returns the number of shalfloops. | |
| Size_type | number_of_sloops () |
| returns the number of sloops. | |
| Size_type | number_of_sfaces () |
| returns the number of sfaces. | |
| Size_type | number_of_sface_cycles () |
| returns the number of sface cycles. | |
| Size_type | number_of_connected_components () |
calculates the number of connected components of P. | |
| void | print_statistics (std::ostream &os=std::cout) |
print the statistics of P: the number of vertices, edges, and faces. | |
| void | check_integrity_and_topological_planarity (bool faces=true) |
checks the link structure and the genus of P. | |
Iteration | |
| bool | has_shalfloop () const |
| The list of all objects can be accessed via iterator ranges. More... | |
| SHalfloop_const_handle | shalfloop () const |
| returns access to the sloop. | |
| 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_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.
| enum CGAL::Nef_polyhedron_S2::Boundary |
| enum CGAL::Nef_polyhedron_S2::Content |
| CGAL::Nef_polyhedron_S2< Traits >::Nef_polyhedron_S2 | ( | Forward_iterator | first, |
| Forward_iterator | beyond, | ||
| Boundary | b = INCLUDED |
||
| ) |
creates a Nef polyhedron N from the set of sphere segments in the iterator range [first,beyond).
If the set of sphere segments is a simple polygon that separates the sphere surface into two regions, then the polygonal region that is left of the segment *first is selected. The polygonal region includes its boundary if b = INCLUDED and excludes the boundary otherwise. Forward_iterator has to be an iterator with value type Sphere_segment.
| bool CGAL::Nef_polyhedron_S2< Traits >::has_shalfloop | ( | ) | const |
The list of all objects can be accessed via iterator ranges.
For comfortable iteration we also provide iterations macros. The iterator range access operations are of the following kind:
SVertex_iterator svertices_begin()/svertices_end()SHalfedge_iterator shalfedges_begin()/shalfedges_end()SHalfloop_iterator shalfloops_begin()/shalfloops_end()SFace_iterator sfaces_begin()/sfaces_end()The macros are then
CGAL_forall_svertices(v,M),CGAL_forall_shalfedges(e,M),CGAL_forall_sfaces(f,M),CGAL_forall_sface_cycles_of(fc,F)where M is a sphere map and F is a sface.
returns true iff there is a shalfloop.
| Object_handle CGAL::Nef_polyhedron_S2< Traits >::locate | ( | const Sphere_point & | p | ) |
returns a generic handle h to an object (face, halfedge, vertex) of the underlying plane map that contains the point p in its relative interior.
The point p is contained in the set represented by N if N.contains(h) is true. The location mode flag m allows one to choose between different point location strategies.
| Object_handle CGAL::Nef_polyhedron_S2< Traits >::ray_shoot | ( | const Sphere_point & | p, |
| const Sphere_direction & | d | ||
| ) |
returns a handle h with N.contains(h) that can be converted to a Vertex_/Halfedge_/Face_const_handle as described above.
The object returned is intersected by the ray starting in p with direction d and has minimal distance to p. The operation returns an empty Object_handle if the ray shoot along d does not hit any object h of N with N.contains(h).
| Object_handle CGAL::Nef_polyhedron_S2< Traits >::ray_shoot_to_boundary | ( | const Sphere_point & | p, |
| const Sphere_direction & | d | ||
| ) |
returns a handle h that can be converted to a Vertex_/Halfedge_const_handle as described above.
The object returned is part of the \( 1\)-skeleton of N, intersected by the ray starting in p with direction d and has minimal distance to p. The operation returns an empty Object_handle if the ray shoot along d does not hit any \( 1\)-skeleton object h of N. The location mode flag m allows one to choose between different point location strategies.
1.8.13