\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D Boolean Operations on Nef Polygons Embedded on the Sphere

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2D Boolean Operations on Nef Polygons Embedded on the Sphere Reference

##

Peter Hachenberger and Lutz Kettner
This package offers the equivalent to 2D Nef Polygons in the plane. Here halfplanes correspond to half spheres delimited by great circles.

Introduced in: CGAL 3.1
Depends on: 2D Boolean Operations on Nef Polygons
BibTeX: cgal:hk-bonpes2-16a
License: GPL

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.

## Classified Reference Pages

## Classes
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...