mirror of https://github.com/CGAL/cgal
82 lines
3.1 KiB
TeX
82 lines
3.1 KiB
TeX
Subdivision surfaces \cite{cc,ds,loop,sqrt3,qts}
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are the limit surface resulting from the
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application of a subdivision algorithm to a control mesh.
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Subdivision algorithms recursively \emph{refine} (subdivide) the
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control mesh and \emph{modify} (smooth) the geometry according
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to a stencil on the source mesh.
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Further details on subdivisions can be found at \cite{Sub:course:2000}
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and \cite{Warren:subdivision}. The OpenMesh library has
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supports of Loop and $\sqrt{3}$ subdivisions \cite{Abhijit:2004:APISUB}.
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Subdivision algorithms contain two major
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components: \emph{\tr} and \emph{\gm}.
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The \tr\ reparameterizes the source mesh into a refined
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mesh. The \gm\ transforms a submesh on the source mesh
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to a vertex on the refined mesh. The source submesh (with
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the normalized weights) is called the
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\emph{stencil}. A proper combination of a \tr\ and a set of
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rules of \gm\ define a valid subdivision scheme.
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Fig.\ref{fig:RefSchemes} shows the local configurations
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of the major refinements
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employed in subdivision algorithms, which include Catmull-Clark
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subdivision (PQQ) \cite{cc}, Loop subdivision (PTQ) \cite{loop},
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Doo-Sabin subdivision (DQQ) \cite{ds} and $\sqrt{3}$ subdivision
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\cite{sqrt3}. Subdivisions, such as Quad-Triangle subdivision
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\cite{qts,l-pg-03}, may employ a hybrid refinement consisting
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of two different refinements (PQQ and PTQ).
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\begin{figure}
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\centering
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\psfrag{PQQ}[]{\scriptsize PQQ}
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\psfrag{PTQ}[]{\scriptsize PTQ}
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\psfrag{DQQ}[]{\scriptsize DQQ}
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\psfrag{Sqrt3}[]{\scriptsize $\sqrt{3}$}
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\epsfig{file=figs/RefSchemes.eps, width=7cm}
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\caption{Examples of refinement schemes:
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primal quadrilateral quadrisection (PQQ),
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primal triangle quadrisection (PTQ),
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dual quadrilateral quadrisection (DQQ) and
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$\sqrt{3}$ triangulation.}
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\label{fig:RefSchemes}
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\end{figure}
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The \gm\ multiplies the stencils and
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generates the corresponding vertex on the refined mesh.
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%The stencil usually specifies an affine map of the
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%submesh.
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Fig.\ref{fig:RefMap} demonstrates the examples of the
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correspondence between a stencil and its vertex. According
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to the employed refinement scheme, subdivisions
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may have several distinct stencils
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(Fig.\ref{fig:RefMap} (a-c)).
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\begin{figure}
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\centering
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\psfrag{A}[]{(a)}
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\psfrag{B}[]{(b)}
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\psfrag{C}[]{(c)}
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\psfrag{D}[]{(d)}
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\epsfig{file=figs/RefMap.eps, width=7cm}
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\caption{The stencil (weights are not shown) and its
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vertex in the Catmull-Clark subdivision (a-c)
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and Doo-Sabin subdivision (d). Catmull-Clark
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subdivision has three stencils: facet-stencil (a),
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edge-stencil (b) and vertex-stencil (c).
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Doo-Sabin subdivision has only corner-stencil (d).}
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\label{fig:RefMap}
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\end{figure}
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% templated rules: a generic framework for subdivisions
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%\subsection{Generic Subdivisions}
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%\label{sec:subtempl}
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\input subtempl
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\subsection{Sqrt 3}
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% connectivity ops: specific polyhedron algorithms (sqrt3 subdivisions)
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%\subsection{$\sqrt{3}$-Subdivision using Euler Operators}
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\input sqrt3
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% inc builder: specific polyhedron algorithms (qt subdivisions)
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%\subsection{Quad-triangle Subdivision using modifier}
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%\input qt
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