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% | Cbp Reference Manual:  intro.tex
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% | Nef_polyhedron_S2 reference manual pages
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% | 26.5.2004   Peter Hachenberger
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\RCSdefDate{\Nef_polyhedronRefDate}{$Date$}
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\clearpage
\chapter{Nef Polyhedra embedded on the Sphere}
\label{chapterNefpolyhedronRef}
\ccChapterAuthor{Peter Hachenberger \and Lutz Kettner \and Michael Seel}

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\section*{Summary}

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 \ccc{Nef_polyhedron_S2}. Additionally, there is the
\emph{shalfloop} representing the great circles.

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