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new manual

This commit is contained in:
Peter Hachenberger 2008-06-15 20:22:17 +00:00
parent 08d1f279b0
commit eb1ea04d8c
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@ -2736,6 +2736,13 @@ Minkowski_sum_2/test/Minkowski_sum_2/data/rooms_part1.dat -text
Minkowski_sum_2/test/Minkowski_sum_2/data/rooms_part2.dat -text Minkowski_sum_2/test/Minkowski_sum_2/data/rooms_part2.dat -text
Minkowski_sum_2/test/Minkowski_sum_2/data/wheels_part1.dat -text Minkowski_sum_2/test/Minkowski_sum_2/data/wheels_part1.dat -text
Minkowski_sum_2/test/Minkowski_sum_2/data/wheels_part2.dat -text Minkowski_sum_2/test/Minkowski_sum_2/data/wheels_part2.dat -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3/fig/decomposition_method.eps -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3/fig/decomposition_method.pdf -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3/fig/glide.eps -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3/fig/spoon_star.ps -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3/fig/tight_passage.ps -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3/main.aux -text
Minkowski_sum_3/doc_tex/Minkowski_sum_3_ref/main.aux -text
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% +------------------------------------------------------------------------+
% | CGAL User Manual:
% +------------------------------------------------------------------------+
% |
% | 10.07.2008 Peter Hachenberger
% |
\RCSdef{\MinkowskiSum3Rev}{$Id$}
\RCSdefDate{\MinkowskiSum3Date}{$Date$}
% +------------------------------------------------------------------------+
\ccParDims
\ccUserChapter{Minkowski Sum of Polyhedra \label{chapterMinkowskiSum3}}
\ccChapterRelease{\MinkowskiSum3Rev. \ \MinkowskiSum3Date}
\ccChapterAuthor{Peter Hachenberger}
%\input{Minkowski_sum_3/PkgDescription.tex}
% +------------------------------------------------------------------------+
\section{Introduction}
\begin{figure}
\begin{ccTexOnly}
\begin{center}
\includegraphics[width=0.8\textwidth]{Minkowski_sum_3/fig/spoon_star}
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<p><center>
<img src="./fig/spoon_star.gif" border=0 alt="Minkowski sum example">
</center>
\end{ccHtmlOnly}
\caption{The Minkowski sum of a spoon and a star.}
\end{figure}
The Minkowski sum of two point sets $P$ and $Q$ in $R^d$, denoted by
$P \oplus Q$, is defined as the set $\{p+q:p \in P, q \in Q
\}$. Minkowski sums are used in a wide range of applications such as
robot motion planning and computer-aided design. This
package provides a function that computes the Minkowski sum of two Nef
polyhedra.
% +------------------------------------------------------------------------+
\section{Decomposition Method}
The decomposition method for computing the Minkowski sum of non-convex
polyhedra is based on the fact that the Minkowski sum of convex
polyhedra is rather easy to compute. The method decomposes both
polyhedra into convex pieces computes all pairwise Minkowski sums of
the convex pieces and merges the pairwise sums.
\begin{figure}
\begin{ccTexOnly}
\begin{center}
\includegraphics[width=0.8\textwidth]{Minkowski_sum_3/fig/decomposition_method}
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<p><center>
<img src="./fig/decomposition_method.gif" border=0 alt="Minkowski sum example">
</center>
\end{ccHtmlOnly}
\caption{The Minkowski sum of a spoon and a star.}
\end{figure}
The Minkowski sum is an iherent complex method. Using the
decomposition method, each polyhedron might be divided into a
quadratic number of pieces, which is worst-case optimal. Then up to
$n^2m^2$ pairwise sums have to be computed and merged, where $n$ and
$m$ is the complexity of the two input polyhedra.
% +------------------------------------------------------------------------+
\section{Features and Restrictions}
This package was written to allow the computation of Minkowski sums of
full-dimensional polyhedra even in so-called tight-passage scenarios,
i.e., solve motion planing problems with passages that are exactly as
wide as the robot. In these scenarios at least one polyhedron---the
obstacles or the robot---must be modeled as an open set. This is
possible with the current implementation.
\begin{figure}
\begin{ccTexOnly}
\begin{center}
\includegraphics[width=0.8\textwidth]{Minkowski_sum_3/fig/tight_passage}
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<p><center>
<img src="./fig/tigh_passage.gif" border=0 alt="Minkowski sum example">
</center>
\end{ccHtmlOnly}
\caption{The Minkowski sum of a spoon and a star.}
\end{figure}
We strife for extending the package to work for arbitrary
polyhedra. Yet we added several features, but are not complete. At
the moment we allow an input polyhedron to consist of:
\begin{enumerate}
\item singular vertices
\item singular edges
\item singular convex facets without holes
\item surface with convex facets that have no holes.
\item open or closed solids
\end{enumerate}
Taking a different viewpoint, the implementation is restricted as
follows:
\begin{enumerate}
\item The input polyhedra must be finite point sets.
\item Every convex Minkowski sum must be full-dimensional, i.e., one
of the two input polyhedra must not include lower-dimensional
featueres. Note that lower-dimensional holes are still possible.
\item All sets of coplanar facets that form a side of a full-dimensional
featuer, must have the same selection mark.
\item All facets of lower-dimensional features need to be convex and
must not have holes.
\end{enumerate}
% +------------------------------------------------------------------------+
\section{Usage}
The following example code illustrates the usage of the function
\ccc{minkowski_sum_3}. Note that the two input polyhedra will be
destroyed by the function. So, if they are further on needed, they
need to be copied, first. The copying is not done by the function
itself to keep the memory usage as small as possible.
\ccIncludeExampleCode{Minkowski_sum_3/minkowski_sum.cpp}
% +------------------------------------------------------------------------+
\section{Glide}
\begin{figure}
\begin{ccTexOnly}
\begin{center}
\includegraphics[width=0.8\textwidth]{Minkowski_sum_3/fig/glide}
\end{center}
\end{ccTexOnly}
\begin{ccHtmlOnly}
<p><center>
<img src="./fig/glide.gif" border=0 alt="Minkowski sum example">
</center>
\end{ccHtmlOnly}
\caption{The Minkowski sum of a spoon and a star.}
\end{figure}
With the function \ccc{minkowski_sum_3} it is also possible to realize
other interesting geometric operations like glide operation, which
computes the point set swept by polyhedron that moves along a
polygonal path. The following example shows how to construct a
polygonal path and then compute the glide operation by calling the
function \ccc{minkowski_sum_3}.
\ccIncludeExampleCode{Minkowski_sum_3/glide.cpp}

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% +------------------------------------------------------------------------+
% | CGAL Reference Manual: intro.tex
% +------------------------------------------------------------------------+
% | Minkowski sum 3 reference manual pages
% |
\RCSdef{\MinkowskiSum3RefRev}{$Id$}
\RCSdefDate{\MinkowskiSum3RefDate}{$Date$}
% +------------------------------------------------------------------------+
\ccRefChapter{Minkowski sum of Polyhedra\label{chapterMinkowskiSum3Ref}}
\ccChapterAuthor{Peter Hachenberger}
% +------------------------------------------------------------------------+
The function \ccc{convex_decomposition_3} takes a
\ccc{Nef_polyhedron_3} $N$ as input parameter and inserts additional facets,
such that each marked volume (except for the outer volume) is
subdivided into convex pieces.
\section{Classified Reference Pages}
\subsection*{Functions}
\ccFunction{Nef_polyhedron_3 minkowski_sum_3(Nef_polyhedron_3& N0, Nef_polyhedron_3& N1);}
{\lcRawHtml{<A HREF="convex_decomposition_3.html">(go there)</A>}
\lcTex{\hfill page~\pageref{refConvex_decomposition_3}}}
%% EOF %%

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% +------------------------------------------------------------------------+
% | CBP Reference Manual: main.tex
% +------------------------------------------------------------------------+
% | Automatically generated driver file for the reference manual chapter
% | of this package. Do not edit manually, you may loose your changes.
% +------------------------------------------------------------------------+
\input{Minkowski_sum_3_ref/intro.tex}
\input{Minkowski_sum_3_ref/minkowski_sum_3.tex}

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% +------------------------------------------------------------------------+
% | Reference manual page: minkowski_sum_3.tex
% +------------------------------------------------------------------------+
% | 11.06.2008 Peter Hachenberger
% | Package: Minkowski_sum_3
% |
\RCSdef{\RCSminkowski_sum_3Rev}{$Id$}
\RCSdefDate{\RCminkowski_sum_3Date}{$Date$}
% |
%%RefPage: end of header, begin of main body
% +------------------------------------------------------------------------+
\ccHtmlNoClassLinks
\begin{ccRefFunction}{minkowski_sum_3}
\label{refminkowski_sum_3}
\ccDefinition
This function takes two \ccc{Nef_polyhedron_3} and returns their
Minkowski sum.
\ccGlobalFunction{Nef_polyhedron_3 convex_decomposition_3(Nef_polyhedron_3& N0, Nef_polyhedron_3 N1);}
\ccSeeAlso
\ccRefIdfierPage{CGAL::Nef_polyhedron_3<Traits>}\\
\end{ccRefFunction}