mirror of https://github.com/CGAL/cgal
new manual
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JXM(Wd$]C~>
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JXM(Wd$]C~>
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JXM(Wd$]C~>
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||||
%%EndData
|
||||
showpage
|
||||
%%Trailer
|
||||
end
|
||||
%%EOF
|
||||
|
|
@ -0,0 +1,45 @@
|
|||
\relax
|
||||
\@writefile{toc}{\contentsline {chapter}{\numberline {1}Minkowski Sum of Polyhedra }{1}{chapter.1}}
|
||||
\@writefile{lof}{\addvspace {10\p@ }}
|
||||
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|
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|
||||
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|
||||
\@writefile{lot}{\contentsline {xchapter}{Minkowski Sum of Polyhedra }{1}{chapter.1}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1.1}Introduction}{1}{section.1.1}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1.2}Decomposition Method}{1}{section.1.2}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1.3}Features and Restrictions}{1}{section.1.3}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces The Minkowski sum of a spoon and a star.}}{1}{figure.1.1}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces The Minkowski sum of a spoon and a star.}}{2}{figure.1.2}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces The Minkowski sum of a spoon and a star.}}{2}{figure.1.3}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1.4}Usage}{2}{section.1.4}}
|
||||
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|
||||
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|
||||
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|
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|
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|
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|
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|
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\setcounter{Item}{9}
|
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\setcounter{section@level}{1}
|
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}
|
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|
|
@ -0,0 +1,155 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | 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}
|
||||
|
|
@ -0,0 +1,28 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | 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 %%
|
||||
|
|
@ -0,0 +1,37 @@
|
|||
\relax
|
||||
\@writefile{@@@}{\chapterbegin }
|
||||
\newlabel{chapterMinkowskiSum3Ref}{{1.5}{7}{Minkowski sum of Polyhedra\label {chapterMinkowskiSum3Ref}\\ Reference~Manual\relax }{figure.1.4}{}}
|
||||
\@writefile{toc}{\contentsline {section}{\hbox to\@tempdima {\hfil }{Reference Manual}}{7}{chapter*.2}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1.6}Classified Reference Pages}{7}{section.1.6}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1.7}Alphabetical List of Reference Pages}{7}{section.1.7}}
|
||||
\newlabel{ccRef_CGAL::minkowski_sum_3}{{1.7}{8}{\ifnum \ccNewRefManualStyle =\ccTrue \ifnum \ccCurrentIndexCat =\ccIndexFunctionObjectConceptCat \ccDrawRefTabs {FunctionObjectConcept}{minkowski_sum_3}\\ \else \ifnum \ccCurrentIndexCat =\ccIndexFunctionObjectClassCat \ccDrawRefTabs {FunctionObjectClass}{minkowski_sum_3}\\ \else \ccDrawRefTabs {\ccRefCategory }{minkowski_sum_3}\\ \fi \fi \else \ccRefCategory \fi \expandafter \ccPrintTokens \ccRefPureGlobalScope \ccEnd {\expandafter \ccPrintTokens \ccPureRefScope minkowski_sum_3\ccEnd }\relax }{section*.4}{}}
|
||||
\newlabel{refminkowski_sum_3}{{1.7}{8}{\ifnum \ccNewRefManualStyle =\ccTrue \ifnum \ccCurrentIndexCat =\ccIndexFunctionObjectConceptCat \ccDrawRefTabs {FunctionObjectConcept}{minkowski_sum_3}\\ \else \ifnum \ccCurrentIndexCat =\ccIndexFunctionObjectClassCat \ccDrawRefTabs {FunctionObjectClass}{minkowski_sum_3}\\ \else \ccDrawRefTabs {\ccRefCategory }{minkowski_sum_3}\\ \fi \fi \else \ccRefCategory \fi \expandafter \ccPrintTokens \ccRefPureGlobalScope \ccEnd {\expandafter \ccPrintTokens \ccPureRefScope minkowski_sum_3\ccEnd }\relax }{section*.4}{}}
|
||||
\@setckpt{Minkowski_sum_3_ref/main}{
|
||||
\setcounter{page}{9}
|
||||
\setcounter{equation}{0}
|
||||
\setcounter{enumi}{4}
|
||||
\setcounter{enumii}{0}
|
||||
\setcounter{enumiii}{0}
|
||||
\setcounter{enumiv}{0}
|
||||
\setcounter{footnote}{0}
|
||||
\setcounter{mpfootnote}{0}
|
||||
\setcounter{part}{0}
|
||||
\setcounter{chapter}{1}
|
||||
\setcounter{section}{7}
|
||||
\setcounter{subsection}{0}
|
||||
\setcounter{subsubsection}{0}
|
||||
\setcounter{paragraph}{0}
|
||||
\setcounter{subparagraph}{0}
|
||||
\setcounter{figure}{4}
|
||||
\setcounter{table}{0}
|
||||
\setcounter{r@tfl@t}{0}
|
||||
\setcounter{LT@tables}{0}
|
||||
\setcounter{LT@chunks}{0}
|
||||
\setcounter{Item}{9}
|
||||
\setcounter{Hfootnote}{0}
|
||||
\setcounter{mtc}{1}
|
||||
\setcounter{minitocdepth}{2}
|
||||
\setcounter{ptc}{0}
|
||||
\setcounter{parttocdepth}{2}
|
||||
\setcounter{section@level}{1}
|
||||
}
|
||||
|
|
@ -0,0 +1,10 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | 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}
|
||||
|
|
@ -0,0 +1,27 @@
|
|||
% +------------------------------------------------------------------------+
|
||||
% | 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}
|
||||
Loading…
Reference in New Issue