mirror of https://github.com/CGAL/cgal
49 lines
2.1 KiB
TeX
49 lines
2.1 KiB
TeX
% +------------------------------------------------------------------------+
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% | Cbp Reference Manual: intro.tex
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% +------------------------------------------------------------------------+
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% | Nef_polyhedron_S2 reference manual pages
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% |
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% | 26.5.2004 Peter Hachenberger
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\RCSdef{\Nef_polyhedronRefRev}{$Id$}
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\RCSdefDate{\Nef_polyhedronRefDate}{$Date$}
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% +------------------------------------------------------------------------+
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\clearpage
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\ccRefChapter{Nef Polyhedra embedded on the Sphere}
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\label{chapterNefpolyhedronRef}
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\ccChapterAuthor{Peter Hachenberger \and Lutz Kettner \and Michael Seel}
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% +------------------------------------------------------------------------+
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Nef polyhedra are defined as a subset of the d-dimensional space obtained by
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a finite number of set complement and set intersection operations on
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halfspaces.
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Due to the fact that all other binary set operations like union,
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difference and symmetric difference can be reduced to intersection and
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complement calculations, Nef polyhedra are also closed under those
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operations. Also, Nef polyhedra are closed under topological unary
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set operations. Given a Nef polyhedron one can determine its interior, its
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boundary, and its closure.
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Additionally, a d-dimensional Nef polyhedron has the property, that its boundary
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is a (d-1)-dimensional Nef polyhedron. This property can be used as a way to
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represent 3-dimensional Nef polyhedra by means of planar Nef polyhedra.
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This is done by intersecting the neighborhood of a vertex in a 3D Nef polyhedron
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with an $\epsilon$-sphere. The result is a planar Nef polyhedron embedded
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on the sphere.
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The intersection of a halfspace going through the center of the $\epsilon$-sphere,
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with the $\epsilon$-sphere, results in a halfsphere which is bounded by
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a great circle. A binary operation of two halfspheres cuts the great circles
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into great arcs.
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The incidence structure of planar Nef polyhedra can be reused. The items
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are denoted as $svertex$, $shalfedge$ and $sface$, analogous
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to their counterparts in \ccc{Nef_polyhedron_S2}. Additionally, there is the
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\emph{shalfloop} representing the great circles.
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%\input{Nef_S2_ref/Spherical_geometry.tex}
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