mirror of https://github.com/CGAL/cgal
81 lines
3.0 KiB
TeX
81 lines
3.0 KiB
TeX
% subdivision.tex
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% ------------------------------------------------------------------------
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\begin{figure}[tb]
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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 ({\itshape top blue}) and its
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vertex ({\itshape bottom red}) in
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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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The stencil weights are not shown.}
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\label{fig:RefMap}\vspace*{-4mm}
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\end{figure}
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%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 \cite{Warren:subdivision, Sub:course:2000}
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contain two major steps: \emph{\tr} and \emph{\gm}.
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The \tr\ reparameterizes the control mesh into a refined
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mesh. The \gm\ transforms a submesh on the control mesh
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to a vertex on the refined mesh. The submesh (with
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the normalized weights) is called the
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\emph{stencil}. A subdivision algorithm recursively
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applies these two steps and approximates the limit surface.
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%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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The topological refinement is illustrated
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in Fig.~\ref{fig:RefSchemes} for 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.
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The \gm\ is illustrated with examples of the correspondence between
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stencil and its vertex in Fig.~\ref{fig:RefMap}, where Catmull-Clark
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subdivision has three distinct stencils and Doo-Sabin subdivision has
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only one stencil.
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\begin{figure}
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\centering{\includegraphics[width=7.0cm]{figs/sqrt3}}
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\caption{$\sqrt{3}$-Subdivision of the mannequin mesh.}
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\label{fig:sqrt3}\vspace*{-3mm}
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\end{figure}
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%\subsubsection{Sqrt 3}
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% connectivity ops: specific polyhedron algorithms (sqrt3 subdivisions)
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\newcommand{\mySqrt}{$\sqrt{3}$}
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\subsubsection*{Sqrt-3 Subdivision using Euler Operators}
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\input sqrt3
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% templated rules: a generic framework for subdivisions
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\subsubsection*{Generic Subdivision Solution}
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%\label{sec:subtempl}
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\input subtempl
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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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