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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346% generated by GAPDoc2LaTeX from XML source (Frank Luebeck)1\documentclass[a4paper,11pt]{report}23\usepackage{a4wide}4\sloppy5\pagestyle{myheadings}6\usepackage{amssymb}7\usepackage[utf8]{inputenc}8\usepackage{makeidx}9\makeindex10\usepackage{color}11\definecolor{FireBrick}{rgb}{0.5812,0.0074,0.0083}12\definecolor{RoyalBlue}{rgb}{0.0236,0.0894,0.6179}13\definecolor{RoyalGreen}{rgb}{0.0236,0.6179,0.0894}14\definecolor{RoyalRed}{rgb}{0.6179,0.0236,0.0894}15\definecolor{LightBlue}{rgb}{0.8544,0.9511,1.0000}16\definecolor{Black}{rgb}{0.0,0.0,0.0}1718\definecolor{linkColor}{rgb}{0.0,0.0,0.554}19\definecolor{citeColor}{rgb}{0.0,0.0,0.554}20\definecolor{fileColor}{rgb}{0.0,0.0,0.554}21\definecolor{urlColor}{rgb}{0.0,0.0,0.554}22\definecolor{promptColor}{rgb}{0.0,0.0,0.589}23\definecolor{brkpromptColor}{rgb}{0.589,0.0,0.0}24\definecolor{gapinputColor}{rgb}{0.589,0.0,0.0}25\definecolor{gapoutputColor}{rgb}{0.0,0.0,0.0}2627%% for a long time these were red and blue by default,28%% now black, but keep variables to overwrite29\definecolor{FuncColor}{rgb}{0.0,0.0,0.0}30%% strange name because of pdflatex bug:31\definecolor{Chapter }{rgb}{0.0,0.0,0.0}32\definecolor{DarkOlive}{rgb}{0.1047,0.2412,0.0064}333435\usepackage{fancyvrb}3637\usepackage{mathptmx,helvet}38\usepackage[T1]{fontenc}39\usepackage{textcomp}404142\usepackage[43pdftex=true,44bookmarks=true,45a4paper=true,46pdftitle={Written with GAPDoc},47pdfcreator={LaTeX with hyperref package / GAPDoc},48colorlinks=true,49backref=page,50breaklinks=true,51linkcolor=linkColor,52citecolor=citeColor,53filecolor=fileColor,54urlcolor=urlColor,55pdfpagemode={UseNone},56]{hyperref}5758\newcommand{\maintitlesize}{\fontsize{50}{55}\selectfont}5960% write page numbers to a .pnr log file for online help61\newwrite\pagenrlog62\immediate\openout\pagenrlog =\jobname.pnr63\immediate\write\pagenrlog{PAGENRS := [}64\newcommand{\logpage}[1]{\protect\write\pagenrlog{#1, \thepage,}}65%% were never documented, give conflicts with some additional packages6667\newcommand{\GAP}{\textsf{GAP}}6869%% nicer description environments, allows long labels70\usepackage{enumitem}71\setdescription{style=nextline}7273%% depth of toc74\setcounter{tocdepth}{1}757677787980%% command for ColorPrompt style examples81\newcommand{\gapprompt}[1]{\color{promptColor}{\bfseries #1}}82\newcommand{\gapbrkprompt}[1]{\color{brkpromptColor}{\bfseries #1}}83\newcommand{\gapinput}[1]{\color{gapinputColor}{#1}}848586\begin{document}8788\logpage{[ 0, 0, 0 ]}89\begin{titlepage}90\mbox{}\vfill9192\begin{center}{\maintitlesize \textbf{\textsf{Convex}\mbox{}}}\\93\vfill9495\hypersetup{pdftitle=\textsf{Convex}}96\markright{\scriptsize \mbox{}\hfill \textsf{Convex} \hfill\mbox{}}97{\Huge \textbf{A \textsf{GAP} package for handling convex objects.\mbox{}}}\\98\vfill99100{\Huge Version 2013.12.05\mbox{}}\\[1cm]101{August 2012\mbox{}}\\[1cm]102\mbox{}\\[2cm]103{\Large \textbf{Sebastian Gutsche\\104\mbox{}}}\\105\hypersetup{pdfauthor=Sebastian Gutsche\\106}107\mbox{}\\[2cm]108\begin{minipage}{12cm}\noindent109\\110\\111This manual is best viewed as an \textsc{HTML} document. An \textsc{offline} version should be included in the documentation subfolder of the package. \\112\\113\end{minipage}114115\end{center}\vfill116117\mbox{}\\118{\mbox{}\\119\small \noindent \textbf{Sebastian Gutsche\\120} Email: \href{mailto://sebastian.gutsche@rwth-aachen.de} {\texttt{sebastian.gutsche@rwth-aachen.de}}\\121Homepage: \href{http://wwwb.math.rwth-aachen.de/~gutsche} {\texttt{http://wwwb.math.rwth-aachen.de/\texttt{\symbol{126}}gutsche}}\\122Address: \begin{minipage}[t]{8cm}\noindent123Lehrstuhl B f{\"u}r Mathematik, RWTH Aachen, Templergraben 64, 52056 Aachen,124Germany \end{minipage}125}\\126\end{titlepage}127128\newpage\setcounter{page}{2}129{\small130\section*{Copyright}131\logpage{[ 0, 0, 1 ]}132{\copyright} 2011-2012 by Sebastian Gutsche133134This package may be distributed under the terms and conditions of the GNU135Public License Version 2. \mbox{}}\\[1cm]136{\small137\section*{Acknowledgements}138\logpage{[ 0, 0, 2 ]}139\mbox{}}\\[1cm]140\newpage141142\def\contentsname{Contents\logpage{[ 0, 0, 3 ]}}143144\tableofcontents145\newpage146147\index{\textsf{Convex}}148\chapter{\textcolor{Chapter }{Introduction}}\label{intro}149\logpage{[ 1, 0, 0 ]}150\hyperdef{L}{X7DFB63A97E67C0A1}{}151{152153\section{\textcolor{Chapter }{What is the goal of the \textsf{Convex} package?}}\label{WhyToricVarieties}154\logpage{[ 1, 1, 0 ]}155\hyperdef{L}{X7B061C0C87A36AD1}{}156{157\textsf{Convex} provides structures and algorithms for convex geometry. It can handle convex,158fans and polytopes. Not only the structures are provided, but also a159collection of algorithms to handle those objects. Basically, it provides160convex geometry to \textsf{GAP}. It is capable of communicating with the CAS polymake via the package \textsf{PolymakeInterface} and also provides several methods by itself. }161162}163164165\chapter{\textcolor{Chapter }{Installation of the \textsf{Convex} Package}}\label{install}166\logpage{[ 2, 0, 0 ]}167\hyperdef{L}{X781CA2768080E873}{}168{169To install this package just extract the package's archive file to the \textsf{GAP} \texttt{pkg} directory.170171By default the \textsf{Convex} package is not automatically loaded by \textsf{GAP} when it is installed. You must load the package with \\172\\173\texttt{LoadPackage}( "Convex" ); \\174\\175before its functions become available.176177Please, send me an e-mail if you have any questions, remarks, suggestions,178etc. concerning this package. Also, I would be pleased to hear about179applications of this package and about any suggestions for new methods to add180to the package. \\181\\182\\183Sebastian Gutsche }184185186\chapter{\textcolor{Chapter }{Convex Objects}}\label{ConvexObject}187\logpage{[ 3, 0, 0 ]}188\hyperdef{L}{X8359268B7FDA6AEC}{}189{190Convex objects are the main structure of \textsf{Convex}. All other structures, namely fans, cones, and polytopes are derived from191this structure. So all methods of this structure also apply to the other data192types.193\section{\textcolor{Chapter }{Convex Objects: Category and Representations}}\label{ConvexObject:Category}194\logpage{[ 3, 1, 0 ]}195\hyperdef{L}{X82E0DD13824DC2C1}{}196{197198199\subsection{\textcolor{Chapter }{IsConvexObject}}200\logpage{[ 3, 1, 1 ]}\nobreak201\hyperdef{L}{X83ACD3DC7C1BE5F8}{}202{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsConvexObject({\mdseries\slshape M})\index{IsConvexObject@\texttt{IsConvexObject}}203\label{IsConvexObject}204}\hfill{\scriptsize (Category)}}\\205\textbf{\indent Returns:\ }206\texttt{true} or \texttt{false}207208209210The \textsf{GAP} category of convex objects, the main category of this package. }211212}213214215\section{\textcolor{Chapter }{Convex objects: Properties}}\label{ConvexObject:Properties}216\logpage{[ 3, 2, 0 ]}217\hyperdef{L}{X85454292847AEBD5}{}218{219220221\subsection{\textcolor{Chapter }{IsFullDimensional}}222\logpage{[ 3, 2, 1 ]}\nobreak223\hyperdef{L}{X7A8A4EF182D275CA}{}224{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsFullDimensional({\mdseries\slshape conv})\index{IsFullDimensional@\texttt{IsFullDimensional}}225\label{IsFullDimensional}226}\hfill{\scriptsize (property)}}\\227\textbf{\indent Returns:\ }228\texttt{true} or \texttt{false}229230231232Checks if the combinatorial dimension of the convex object \mbox{\texttt{\mdseries\slshape conv}} is the same as the dimension of the ambient space. }233234}235236237\section{\textcolor{Chapter }{Convex objects: Attributes}}\label{ConvexObject:Attributes}238\logpage{[ 3, 3, 0 ]}239\hyperdef{L}{X7E20C8697EA9490E}{}240{241242243\subsection{\textcolor{Chapter }{Dimension}}244\logpage{[ 3, 3, 1 ]}\nobreak245\hyperdef{L}{X7E6926C6850E7C4E}{}246{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Dimension({\mdseries\slshape conv})\index{Dimension@\texttt{Dimension}}247\label{Dimension}248}\hfill{\scriptsize (attribute)}}\\249\textbf{\indent Returns:\ }250an integer251252253254Returns the combinatorial dimension of the convex object \mbox{\texttt{\mdseries\slshape conv}}. This is the dimension of the smallest space i which \mbox{\texttt{\mdseries\slshape conv}} can be embedded. }255256257258\subsection{\textcolor{Chapter }{AmbientSpaceDimension}}259\logpage{[ 3, 3, 2 ]}\nobreak260\hyperdef{L}{X791629C67F481601}{}261{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{AmbientSpaceDimension({\mdseries\slshape conv})\index{AmbientSpaceDimension@\texttt{AmbientSpaceDimension}}262\label{AmbientSpaceDimension}263}\hfill{\scriptsize (attribute)}}\\264\textbf{\indent Returns:\ }265an integer266267268269Returns the dimension of the ambient space of the object \mbox{\texttt{\mdseries\slshape conv}}. }270271272273\subsection{\textcolor{Chapter }{ContainingGrid}}274\logpage{[ 3, 3, 3 ]}\nobreak275\hyperdef{L}{X7C4692E0794B126E}{}276{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{ContainingGrid({\mdseries\slshape conv})\index{ContainingGrid@\texttt{ContainingGrid}}277\label{ContainingGrid}278}\hfill{\scriptsize (attribute)}}\\279\textbf{\indent Returns:\ }280a homalg module281282283284Returns the ambient space of the object \mbox{\texttt{\mdseries\slshape conv}} as a homalg module. }285286}287288289\section{\textcolor{Chapter }{Convex objects: Methods}}\label{ConvexObject:Methods}290\logpage{[ 3, 4, 0 ]}291\hyperdef{L}{X7D7E0B658234B893}{}292{293294295\subsection{\textcolor{Chapter }{DrawObject}}296\logpage{[ 3, 4, 1 ]}\nobreak297\hyperdef{L}{X83FA826678EE4C1C}{}298{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{DrawObject({\mdseries\slshape conv})\index{DrawObject@\texttt{DrawObject}}299\label{DrawObject}300}\hfill{\scriptsize (operation)}}\\301\textbf{\indent Returns:\ }3020303304305306Draws a nice picture of the object \mbox{\texttt{\mdseries\slshape conv}}, if your computer supports Java. As a side effect, you might not be able to307exit \textsf{GAP} anymore. }308309310311\subsection{\textcolor{Chapter }{WeakPointerToExternalObject}}312\logpage{[ 3, 4, 2 ]}\nobreak313\hyperdef{L}{X807B4DE27F6BF439}{}314{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{WeakPointerToExternalObject({\mdseries\slshape conv})\index{WeakPointerToExternalObject@\texttt{WeakPointerToExternalObject}}315\label{WeakPointerToExternalObject}316}\hfill{\scriptsize (operation)}}\\317\textbf{\indent Returns:\ }318a pointer319320321322Returns a pointer to an external object which is the basis of \mbox{\texttt{\mdseries\slshape conv}}. This method is not used any more. }323324}325326}327328329\chapter{\textcolor{Chapter }{Fan}}\label{Fan}330\logpage{[ 4, 0, 0 ]}331\hyperdef{L}{X80D0196B80DC94F3}{}332{333334\section{\textcolor{Chapter }{Fan: Category and Representations}}\label{Fan:Category}335\logpage{[ 4, 1, 0 ]}336\hyperdef{L}{X7F4C80A1855F619C}{}337{338339340\subsection{\textcolor{Chapter }{IsFan}}341\logpage{[ 4, 1, 1 ]}\nobreak342\hyperdef{L}{X80B4C7D87A5ECDBF}{}343{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsFan({\mdseries\slshape M})\index{IsFan@\texttt{IsFan}}344\label{IsFan}345}\hfill{\scriptsize (Category)}}\\346\textbf{\indent Returns:\ }347\texttt{true} or \texttt{false}348349350351The \textsf{GAP} category of a fan. Every fan is a convex object. }352353Remember: Every fan is a convex object. }354355356\section{\textcolor{Chapter }{Fan: Properties}}\label{Fan:Properties}357\logpage{[ 4, 2, 0 ]}358\hyperdef{L}{X7A83743785C9E8F1}{}359{360361362\subsection{\textcolor{Chapter }{IsComplete}}363\logpage{[ 4, 2, 1 ]}\nobreak364\hyperdef{L}{X7D689F21828A4278}{}365{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsComplete({\mdseries\slshape fan})\index{IsComplete@\texttt{IsComplete}}366\label{IsComplete}367}\hfill{\scriptsize (property)}}\\368\textbf{\indent Returns:\ }369\texttt{true} or \texttt{false}370371372373Checks if the fan \mbox{\texttt{\mdseries\slshape fan}} is complete, i. e. if it's support is the whole space. }374375376377\subsection{\textcolor{Chapter }{IsPointed}}378\logpage{[ 4, 2, 2 ]}\nobreak379\hyperdef{L}{X843A31A57EAB734C}{}380{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsPointed({\mdseries\slshape fan})\index{IsPointed@\texttt{IsPointed}}381\label{IsPointed}382}\hfill{\scriptsize (property)}}\\383\textbf{\indent Returns:\ }384\texttt{true} or \texttt{false}385386387388Checks if the fan \mbox{\texttt{\mdseries\slshape fan}} is pointed, which means that every cone it contains is strictly convex. }389390391392\subsection{\textcolor{Chapter }{IsSmooth}}393\logpage{[ 4, 2, 3 ]}\nobreak394\hyperdef{L}{X86CBF5497EC15CFC}{}395{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsSmooth({\mdseries\slshape fan})\index{IsSmooth@\texttt{IsSmooth}}396\label{IsSmooth}397}\hfill{\scriptsize (property)}}\\398\textbf{\indent Returns:\ }399\texttt{true} or \texttt{false}400401402403Checks if the fan \mbox{\texttt{\mdseries\slshape fan}} is smooth, i. e. if every cone in the fan is smooth. }404405406407\subsection{\textcolor{Chapter }{IsRegularFan}}408\logpage{[ 4, 2, 4 ]}\nobreak409\hyperdef{L}{X7838A553848AD380}{}410{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsRegularFan({\mdseries\slshape fan})\index{IsRegularFan@\texttt{IsRegularFan}}411\label{IsRegularFan}412}\hfill{\scriptsize (property)}}\\413\textbf{\indent Returns:\ }414\texttt{true} or \texttt{false}415416417418Checks if the fan \mbox{\texttt{\mdseries\slshape fan}} is regular, i. e. if it is the normal fan of a polytope. }419420421422\subsection{\textcolor{Chapter }{IsSimplicial (for a fan)}}423\logpage{[ 4, 2, 5 ]}\nobreak424\hyperdef{L}{X863CBF607A2AD000}{}425{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsSimplicial({\mdseries\slshape fan})\index{IsSimplicial@\texttt{IsSimplicial}!for a fan}426\label{IsSimplicial:for a fan}427}\hfill{\scriptsize (property)}}\\428\textbf{\indent Returns:\ }429\texttt{true} or \texttt{false}430431432433Checks if the fan \mbox{\texttt{\mdseries\slshape fan}} is simplicial, i. e. if every cone in the fan is simplicial. }434435436437\subsection{\textcolor{Chapter }{HasConvexSupport}}438\logpage{[ 4, 2, 6 ]}\nobreak439\hyperdef{L}{X8258DA9E820B9CF5}{}440{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{HasConvexSupport({\mdseries\slshape fan})\index{HasConvexSupport@\texttt{HasConvexSupport}}441\label{HasConvexSupport}442}\hfill{\scriptsize (property)}}\\443\textbf{\indent Returns:\ }444\texttt{true} or \texttt{false}445446447448Checks if the fan \mbox{\texttt{\mdseries\slshape fan}} is simplicial, i. e. if every cone in the fan is simplicial. }449450}451452453\section{\textcolor{Chapter }{Fan: Attributes}}\label{Fan:Attributes}454\logpage{[ 4, 3, 0 ]}455\hyperdef{L}{X81E6FECC824A7C06}{}456{457458459\subsection{\textcolor{Chapter }{Rays}}460\logpage{[ 4, 3, 1 ]}\nobreak461\hyperdef{L}{X831FB73F86E6E4E9}{}462{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Rays({\mdseries\slshape fan})\index{Rays@\texttt{Rays}}463\label{Rays}464}\hfill{\scriptsize (attribute)}}\\465\textbf{\indent Returns:\ }466a list467468469470Returns the rays of the fan \mbox{\texttt{\mdseries\slshape fan}} as a list of cones. }471472473474\subsection{\textcolor{Chapter }{RayGenerators}}475\logpage{[ 4, 3, 2 ]}\nobreak476\hyperdef{L}{X7CC22C4A85B6B51B}{}477{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{RayGenerators({\mdseries\slshape fan})\index{RayGenerators@\texttt{RayGenerators}}478\label{RayGenerators}479}\hfill{\scriptsize (attribute)}}\\480\textbf{\indent Returns:\ }481a list482483484485Returns the generators rays of the fan \mbox{\texttt{\mdseries\slshape fan}} as a list of of list of integers. }486487488489\subsection{\textcolor{Chapter }{RaysInMaximalCones}}490\logpage{[ 4, 3, 3 ]}\nobreak491\hyperdef{L}{X80472C677CB77C5B}{}492{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{RaysInMaximalCones({\mdseries\slshape fan})\index{RaysInMaximalCones@\texttt{RaysInMaximalCones}}493\label{RaysInMaximalCones}494}\hfill{\scriptsize (attribute)}}\\495\textbf{\indent Returns:\ }496a list497498499500Returns a list of lists, which represent an incidence matrix for the501correspondence of the rays and the maximal cones of the fan \mbox{\texttt{\mdseries\slshape fan}}. The ith list in the result represents the ith maximal cone of \mbox{\texttt{\mdseries\slshape fan}}. In such a list, the jth entry is 1 if the jth ray is in the cone, 0502otherwise. }503504505506\subsection{\textcolor{Chapter }{MaximalCones}}507\logpage{[ 4, 3, 4 ]}\nobreak508\hyperdef{L}{X8549BF0C78C9193B}{}509{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{MaximalCones({\mdseries\slshape fan})\index{MaximalCones@\texttt{MaximalCones}}510\label{MaximalCones}511}\hfill{\scriptsize (attribute)}}\\512\textbf{\indent Returns:\ }513a list514515516517Returns the maximal cones of the fan \mbox{\texttt{\mdseries\slshape fan}} as a list of cones. }518519}520521522\section{\textcolor{Chapter }{Fan: Methods}}\label{Fan:Methods}523\logpage{[ 4, 4, 0 ]}524\hyperdef{L}{X8419F1C07A43ACDE}{}525{526527528\subsection{\textcolor{Chapter }{* (for fans)}}529\logpage{[ 4, 4, 1 ]}\nobreak530\hyperdef{L}{X846E545D78D769B8}{}531{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{*({\mdseries\slshape fan1, fan2})\index{*@\texttt{*}!for fans}532\label{*:for fans}533}\hfill{\scriptsize (operation)}}\\534\textbf{\indent Returns:\ }535a fan536537538539Returns the product of the fans \mbox{\texttt{\mdseries\slshape fan1}} and \mbox{\texttt{\mdseries\slshape fan2}}. }540541}542543544\section{\textcolor{Chapter }{Fan: Constructors}}\label{Fan:Constructors}545\logpage{[ 4, 5, 0 ]}546\hyperdef{L}{X7C1E230383F32681}{}547{548549550\subsection{\textcolor{Chapter }{Fan (For Fans)}}551\logpage{[ 4, 5, 1 ]}\nobreak552\hyperdef{L}{X7C3F2E73846549A2}{}553{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Fan({\mdseries\slshape fan})\index{Fan@\texttt{Fan}!For Fans}554\label{Fan:For Fans}555}\hfill{\scriptsize (operation)}}\\556\textbf{\indent Returns:\ }557a fan558559560561Copy constructor for fans. For completeness reasons. }562563564565\subsection{\textcolor{Chapter }{Fan (For a list of rays and a list of cones)}}566\logpage{[ 4, 5, 2 ]}\nobreak567\hyperdef{L}{X79EAB2B5838C6F1A}{}568{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Fan({\mdseries\slshape rays, cones})\index{Fan@\texttt{Fan}!For a list of rays and a list of cones}569\label{Fan:For a list of rays and a list of cones}570}\hfill{\scriptsize (operation)}}\\571\textbf{\indent Returns:\ }572a fan573574575576Constructs the fan out of the given \mbox{\texttt{\mdseries\slshape rays}} and a list of \mbox{\texttt{\mdseries\slshape cones}} given by a lists of numbers of rays. }577578}579580581\section{\textcolor{Chapter }{Fan: Examples}}\label{Fan:Examples}582\logpage{[ 4, 6, 0 ]}583\hyperdef{L}{X874C843E861EB3A6}{}584{585586\subsection{\textcolor{Chapter }{Fan example}}\label{FanExamplePrimary}587\logpage{[ 4, 6, 1 ]}588\hyperdef{L}{X7A5BBAD884D93AD5}{}589{590591\begin{Verbatim}[commandchars=!@B,fontsize=\small,frame=single,label=Example]592!gapprompt@gap>B !gapinput@F := Fan( [[-1,5],[0,1],[1,0],[0,-1]],[[1,2],[2,3],[3,4],[4,1]] );B593<A fan in |R^2>594!gapprompt@gap>B !gapinput@RayGenerators( F );B595[ [ -1, 5 ], [ 0, 1 ], [ 1, 0 ], [ 0, -1 ] ]596!gapprompt@gap>B !gapinput@RaysInMaximalCones( F );B597[ [ 1, 1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 1, 1 ], [ 1, 0, 0, 1 ] ]598!gapprompt@gap>B !gapinput@IsRegularFan( F );B599true600!gapprompt@gap>B !gapinput@IsComplete( F );B601true602!gapprompt@gap>B !gapinput@IsSmooth( F );B603true604!gapprompt@gap>B !gapinput@F1 := MaximalCones( F )[ 1 ];B605<A cone in |R^2>606!gapprompt@gap>B !gapinput@DualCone( F1 );B607<A cone in |R^2>608!gapprompt@gap>B !gapinput@RayGenerators( F1 );B609[ [ -1, 5 ], [ 0, 1 ] ]610!gapprompt@gap>B !gapinput@F2 := StarSubdivisionOfIthMaximalCone( F, 1 );B611<A fan in |R^2>612!gapprompt@gap>B !gapinput@IsSmooth( F2 );B613true614!gapprompt@gap>B !gapinput@RayGenerators( F2 );B615[ [ -1, 5 ], [ -1, 6 ], [ 0, -1 ], [ 0, 1 ], [ 1, 0 ] ]616\end{Verbatim}617}618619}620621}622623624\chapter{\textcolor{Chapter }{Cone}}\label{Cone}625\logpage{[ 5, 0, 0 ]}626\hyperdef{L}{X822975FC7F646FE5}{}627{628629\section{\textcolor{Chapter }{Cone: Category and Representations}}\label{Cone:Category}630\logpage{[ 5, 1, 0 ]}631\hyperdef{L}{X7CAD43A27DB1C2E8}{}632{633634635\subsection{\textcolor{Chapter }{IsCone}}636\logpage{[ 5, 1, 1 ]}\nobreak637\hyperdef{L}{X80DFE6EA8575A9B0}{}638{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsCone({\mdseries\slshape M})\index{IsCone@\texttt{IsCone}}639\label{IsCone}640}\hfill{\scriptsize (Category)}}\\641\textbf{\indent Returns:\ }642\texttt{true} or \texttt{false}643644645646The \textsf{GAP} category of a cone. }647648Remember: Every cone is a convex object. }649650651\section{\textcolor{Chapter }{Cone: Properties}}\label{Cone:Properties}652\logpage{[ 5, 2, 0 ]}653\hyperdef{L}{X82859C047B3C8F5E}{}654{655656657\subsection{\textcolor{Chapter }{IsRay}}658\logpage{[ 5, 2, 1 ]}\nobreak659\hyperdef{L}{X793B0F3E86C039BC}{}660{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsRay({\mdseries\slshape cone})\index{IsRay@\texttt{IsRay}}661\label{IsRay}662}\hfill{\scriptsize (property)}}\\663\textbf{\indent Returns:\ }664\texttt{true} or \texttt{false}665666667668Checks if the cone \mbox{\texttt{\mdseries\slshape cone}} is a ray, i.e. if it has only one ray generator. }669670}671672673\section{\textcolor{Chapter }{Cone: Attributes}}\label{Cone:Attributes}674\logpage{[ 5, 3, 0 ]}675\hyperdef{L}{X79E016FF794B28D0}{}676{677678679\subsection{\textcolor{Chapter }{DualCone}}680\logpage{[ 5, 3, 1 ]}\nobreak681\hyperdef{L}{X8635EC787FEBB3FD}{}682{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{DualCone({\mdseries\slshape cone})\index{DualCone@\texttt{DualCone}}683\label{DualCone}684}\hfill{\scriptsize (attribute)}}\\685\textbf{\indent Returns:\ }686a cone687688689690Returns the dual cone of the cone \mbox{\texttt{\mdseries\slshape cone}}. }691692693694\subsection{\textcolor{Chapter }{HilbertBasis}}695\logpage{[ 5, 3, 2 ]}\nobreak696\hyperdef{L}{X7D549E567C52DCB5}{}697{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{HilbertBasis({\mdseries\slshape cone})\index{HilbertBasis@\texttt{HilbertBasis}}698\label{HilbertBasis}699}\hfill{\scriptsize (attribute)}}\\700\textbf{\indent Returns:\ }701a list702703704705Returns a Hilbert Basis of the cone \mbox{\texttt{\mdseries\slshape cone}}. }706707708709\subsection{\textcolor{Chapter }{RaysInFacets}}710\logpage{[ 5, 3, 3 ]}\nobreak711\hyperdef{L}{X840385CC7ACD01C4}{}712{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{RaysInFacets({\mdseries\slshape cone})\index{RaysInFacets@\texttt{RaysInFacets}}713\label{RaysInFacets}714}\hfill{\scriptsize (attribute)}}\\715\textbf{\indent Returns:\ }716a list717718719720Returns an incidence matrix for the rays in the facets of the cone \mbox{\texttt{\mdseries\slshape cone}}. The ith entry of the result corresponds to the ith facet, the jth entry of721this is 1 if the jth ray is in th ith facet, 0 otherwise. }722723724725\subsection{\textcolor{Chapter }{Facets}}726\logpage{[ 5, 3, 4 ]}\nobreak727\hyperdef{L}{X7AFE6D2C82F73788}{}728{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Facets({\mdseries\slshape cone})\index{Facets@\texttt{Facets}}729\label{Facets}730}\hfill{\scriptsize (attribute)}}\\731\textbf{\indent Returns:\ }732a list733734735736Returns a list of the facets of the cone \mbox{\texttt{\mdseries\slshape cone}} as homalg cones. }737738739740\subsection{\textcolor{Chapter }{GridGeneratedByCone}}741\logpage{[ 5, 3, 5 ]}\nobreak742\hyperdef{L}{X7885EDAB80ED7705}{}743{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{GridGeneratedByCone({\mdseries\slshape cone})\index{GridGeneratedByCone@\texttt{GridGeneratedByCone}}744\label{GridGeneratedByCone}745}\hfill{\scriptsize (attribute)}}\\746\textbf{\indent Returns:\ }747a homalg module748749750751Returns the grid generated by the lattice points of the cone \mbox{\texttt{\mdseries\slshape cone}} as a homalg module. }752753754755\subsection{\textcolor{Chapter }{FactorGrid}}756\logpage{[ 5, 3, 6 ]}\nobreak757\hyperdef{L}{X7B1669747B6CBCAE}{}758{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{FactorGrid({\mdseries\slshape cone})\index{FactorGrid@\texttt{FactorGrid}}759\label{FactorGrid}760}\hfill{\scriptsize (attribute)}}\\761\textbf{\indent Returns:\ }762a homalg module763764765766Returns the factor of the containing grid of the cone \mbox{\texttt{\mdseries\slshape cone}} and the grid generated by \mbox{\texttt{\mdseries\slshape cone}}. }767768769770\subsection{\textcolor{Chapter }{GridGeneratedByOrthogonalCone}}771\logpage{[ 5, 3, 7 ]}\nobreak772\hyperdef{L}{X7FD62BD58783C1D6}{}773{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{GridGeneratedByOrthogonalCone({\mdseries\slshape cone})\index{GridGeneratedByOrthogonalCone@\texttt{GridGeneratedByOrthogonalCone}}774\label{GridGeneratedByOrthogonalCone}775}\hfill{\scriptsize (attribute)}}\\776\textbf{\indent Returns:\ }777a homalg module778779780781Returns the grid generated by the lattice points of the orthogonal cone of the782cone \mbox{\texttt{\mdseries\slshape cone}}. }783784785786\subsection{\textcolor{Chapter }{DefiningInequalities}}787\logpage{[ 5, 3, 8 ]}\nobreak788\hyperdef{L}{X7CB1A6657B3B3550}{}789{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{DefiningInequalities({\mdseries\slshape cone})\index{DefiningInequalities@\texttt{DefiningInequalities}}790\label{DefiningInequalities}791}\hfill{\scriptsize (attribute)}}\\792\textbf{\indent Returns:\ }793a list794795796797Returns a list of the defining inequalities of the cone \mbox{\texttt{\mdseries\slshape cone}}. }798799800801\subsection{\textcolor{Chapter }{IsContainedInFan}}802\logpage{[ 5, 3, 9 ]}\nobreak803\hyperdef{L}{X857893CC7BFDE0E0}{}804{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsContainedInFan({\mdseries\slshape cone})\index{IsContainedInFan@\texttt{IsContainedInFan}}805\label{IsContainedInFan}806}\hfill{\scriptsize (attribute)}}\\807\textbf{\indent Returns:\ }808a fan809810811812If the cone \mbox{\texttt{\mdseries\slshape cone}} is constructed as part of a fan, this method returns the fan. }813814815816\subsection{\textcolor{Chapter }{FactorGridMorphism}}817\logpage{[ 5, 3, 10 ]}\nobreak818\hyperdef{L}{X7AA3F8617E28E7BD}{}819{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{FactorGridMorphism({\mdseries\slshape cone})\index{FactorGridMorphism@\texttt{FactorGridMorphism}}820\label{FactorGridMorphism}821}\hfill{\scriptsize (attribute)}}\\822\textbf{\indent Returns:\ }823a morphism824825826827Returns the morphism to the factor grid of the cone \mbox{\texttt{\mdseries\slshape cone}}. }828829}830831832\section{\textcolor{Chapter }{Cone: Methods}}\label{Cone:Methods}833\logpage{[ 5, 4, 0 ]}834\hyperdef{L}{X7DD2D0EA7EE584AA}{}835{836837838\subsection{\textcolor{Chapter }{IntersectionOfCones}}839\logpage{[ 5, 4, 1 ]}\nobreak840\hyperdef{L}{X803F0640808F0A4A}{}841{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IntersectionOfCones({\mdseries\slshape cone1, cone2})\index{IntersectionOfCones@\texttt{IntersectionOfCones}}842\label{IntersectionOfCones}843}\hfill{\scriptsize (operation)}}\\844\textbf{\indent Returns:\ }845a cone846847848849If the cones \mbox{\texttt{\mdseries\slshape cone1}} and \mbox{\texttt{\mdseries\slshape cone2}} share a face, the method returns their intersection, }850851852853\subsection{\textcolor{Chapter }{Contains}}854\logpage{[ 5, 4, 2 ]}\nobreak855\hyperdef{L}{X851A362E8584EE03}{}856{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Contains({\mdseries\slshape cone1, cone2})\index{Contains@\texttt{Contains}}857\label{Contains}858}\hfill{\scriptsize (operation)}}\\859\textbf{\indent Returns:\ }860\texttt{true} or \texttt{false}861862863864Returns \texttt{true} if the cone \mbox{\texttt{\mdseries\slshape cone1}} contains the cone \mbox{\texttt{\mdseries\slshape cone2}}, \texttt{false} otherwise. }865866867868\subsection{\textcolor{Chapter }{StarFan (for a cone)}}869\logpage{[ 5, 4, 3 ]}\nobreak870\hyperdef{L}{X7C7CF17887D7D27E}{}871{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{StarFan({\mdseries\slshape cone})\index{StarFan@\texttt{StarFan}!for a cone}872\label{StarFan:for a cone}873}\hfill{\scriptsize (operation)}}\\874\textbf{\indent Returns:\ }875a fan876877878879Returns the star fan of the cone \mbox{\texttt{\mdseries\slshape cone}}, as described in cox, 3.2.7 }880881882883\subsection{\textcolor{Chapter }{StarFan (for a cone and a fan)}}884\logpage{[ 5, 4, 4 ]}\nobreak885\hyperdef{L}{X84CFDA0883327BB0}{}886{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{StarFan({\mdseries\slshape cone, fan})\index{StarFan@\texttt{StarFan}!for a cone and a fan}887\label{StarFan:for a cone and a fan}888}\hfill{\scriptsize (operation)}}\\889\textbf{\indent Returns:\ }890a fan891892893894Returns the star fan of the fan \mbox{\texttt{\mdseries\slshape fan}} along the cone \mbox{\texttt{\mdseries\slshape cone}}, as described in cox, 3.2.7 }895896897898\subsection{\textcolor{Chapter }{StarSubdivisionOfIthMaximalCone}}899\logpage{[ 5, 4, 5 ]}\nobreak900\hyperdef{L}{X7E4D3AB37B384638}{}901{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{StarSubdivisionOfIthMaximalCone({\mdseries\slshape fan, numb})\index{StarSubdivisionOfIthMaximalCone@\texttt{StarSubdivisionOfIthMaximalCone}}902\label{StarSubdivisionOfIthMaximalCone}903}\hfill{\scriptsize (operation)}}\\904\textbf{\indent Returns:\ }905a fan906907908909Returns the star subdivision of the fan \mbox{\texttt{\mdseries\slshape fan}} on the \mbox{\texttt{\mdseries\slshape numb}}th maximal cone as in cox, 3.3.13. }910911}912913914\section{\textcolor{Chapter }{Cone: Constructors}}\label{Cone:Constructors}915\logpage{[ 5, 5, 0 ]}916\hyperdef{L}{X7DFBB2A782DCFCCA}{}917{918919920\subsection{\textcolor{Chapter }{Cone (for a ray list)}}921\logpage{[ 5, 5, 1 ]}\nobreak922\hyperdef{L}{X8044339D7E71010B}{}923{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Cone({\mdseries\slshape cone})\index{Cone@\texttt{Cone}!for a ray list}924\label{Cone:for a ray list}925}\hfill{\scriptsize (operation)}}\\926\textbf{\indent Returns:\ }927a cone928929930931Returns a cone generated by the rays in \mbox{\texttt{\mdseries\slshape cone}}. }932933}934935936\section{\textcolor{Chapter }{Cone: Examples}}\label{Cone:Examples}937\logpage{[ 5, 6, 0 ]}938\hyperdef{L}{X84BE1F7279A2C49C}{}939{940941\subsection{\textcolor{Chapter }{Cone example}}\label{ConeExamplePrimary}942\logpage{[ 5, 6, 1 ]}943\hyperdef{L}{X81EAFA247C2687D4}{}944{945946\begin{Verbatim}[commandchars=!@E,fontsize=\small,frame=single,label=Example]947!gapprompt@gap>E !gapinput@C := Cone([[1,2,3],[2,1,1],[1,0,0],[0,1,1]]);E948<A cone in |R^3>949!gapprompt@gap>E !gapinput@Length( RayGenerators( C ) );E9503951!gapprompt@gap>E !gapinput@IsSmooth( C );E952true953!gapprompt@gap>E !gapinput@Length( HilbertBasis( C ) );E9543955!gapprompt@gap>E !gapinput@IsSimplicial( C );E956true957!gapprompt@gap>E !gapinput@DC := DualCone( C );E958<A cone in |R^3>959!gapprompt@gap>E !gapinput@Length( HilbertBasis( DC ) );E9603961\end{Verbatim}962}963964}965966}967968969\chapter{\textcolor{Chapter }{Polytope}}\label{Polytope}970\logpage{[ 6, 0, 0 ]}971\hyperdef{L}{X855106007DE72898}{}972{973974\section{\textcolor{Chapter }{Polytope: Category and Representations}}\label{Polytope:Category}975\logpage{[ 6, 1, 0 ]}976\hyperdef{L}{X86EFB7F37A7256B8}{}977{978979980\subsection{\textcolor{Chapter }{IsPolytope}}981\logpage{[ 6, 1, 1 ]}\nobreak982\hyperdef{L}{X81EA74AA7B4B6DDB}{}983{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsPolytope({\mdseries\slshape M})\index{IsPolytope@\texttt{IsPolytope}}984\label{IsPolytope}985}\hfill{\scriptsize (Category)}}\\986\textbf{\indent Returns:\ }987\texttt{true} or \texttt{false}988989990991The \textsf{GAP} category of a polytope. Every polytope is a convex object. }992993Remember: Every cone is a convex object. }994995996\section{\textcolor{Chapter }{Polytope: Properties}}\label{Polytope:Properties}997\logpage{[ 6, 2, 0 ]}998\hyperdef{L}{X7CBD76CF85B3DD81}{}999{100010011002\subsection{\textcolor{Chapter }{IsNotEmpty}}1003\logpage{[ 6, 2, 1 ]}\nobreak1004\hyperdef{L}{X87705F6D7B129879}{}1005{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsNotEmpty({\mdseries\slshape poly})\index{IsNotEmpty@\texttt{IsNotEmpty}}1006\label{IsNotEmpty}1007}\hfill{\scriptsize (property)}}\\1008\textbf{\indent Returns:\ }1009\texttt{true} or \texttt{false}1010101110121013Checks if the polytope \mbox{\texttt{\mdseries\slshape poly}} is not empty. }1014101510161017\subsection{\textcolor{Chapter }{IsLatticePolytope}}1018\logpage{[ 6, 2, 2 ]}\nobreak1019\hyperdef{L}{X79F588238781B2C9}{}1020{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsLatticePolytope({\mdseries\slshape poly})\index{IsLatticePolytope@\texttt{IsLatticePolytope}}1021\label{IsLatticePolytope}1022}\hfill{\scriptsize (property)}}\\1023\textbf{\indent Returns:\ }1024\texttt{true} or \texttt{false}1025102610271028Checks if the polytope \mbox{\texttt{\mdseries\slshape poly}} is a lattice polytope, i.e. all its vertices are lattice points. }1029103010311032\subsection{\textcolor{Chapter }{IsVeryAmple}}1033\logpage{[ 6, 2, 3 ]}\nobreak1034\hyperdef{L}{X80A58559802BB02E}{}1035{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsVeryAmple({\mdseries\slshape poly})\index{IsVeryAmple@\texttt{IsVeryAmple}}1036\label{IsVeryAmple}1037}\hfill{\scriptsize (property)}}\\1038\textbf{\indent Returns:\ }1039\texttt{true} or \texttt{false}1040104110421043Checks if the polytope \mbox{\texttt{\mdseries\slshape poly}} is very ample. }1044104510461047\subsection{\textcolor{Chapter }{IsNormalPolytope}}1048\logpage{[ 6, 2, 4 ]}\nobreak1049\hyperdef{L}{X7C3C14CB83C98EFD}{}1050{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsNormalPolytope({\mdseries\slshape poly})\index{IsNormalPolytope@\texttt{IsNormalPolytope}}1051\label{IsNormalPolytope}1052}\hfill{\scriptsize (property)}}\\1053\textbf{\indent Returns:\ }1054\texttt{true} or \texttt{false}1055105610571058Checks if the polytope \mbox{\texttt{\mdseries\slshape poly}} is normal. }1059106010611062\subsection{\textcolor{Chapter }{IsSimplicial (for a polytope)}}1063\logpage{[ 6, 2, 5 ]}\nobreak1064\hyperdef{L}{X7AB9716B7DFE7CCF}{}1065{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsSimplicial({\mdseries\slshape poly})\index{IsSimplicial@\texttt{IsSimplicial}!for a polytope}1066\label{IsSimplicial:for a polytope}1067}\hfill{\scriptsize (property)}}\\1068\textbf{\indent Returns:\ }1069\texttt{true} or \texttt{false}1070107110721073Checks if the polytope \mbox{\texttt{\mdseries\slshape poly}} is simplicial. }1074107510761077\subsection{\textcolor{Chapter }{IsSimplePolytope}}1078\logpage{[ 6, 2, 6 ]}\nobreak1079\hyperdef{L}{X7F0DF19F82E6DEBD}{}1080{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{IsSimplePolytope({\mdseries\slshape poly})\index{IsSimplePolytope@\texttt{IsSimplePolytope}}1081\label{IsSimplePolytope}1082}\hfill{\scriptsize (property)}}\\1083\textbf{\indent Returns:\ }1084\texttt{true} or \texttt{false}1085108610871088Checks if the polytope \mbox{\texttt{\mdseries\slshape poly}} is simple. }10891090}109110921093\section{\textcolor{Chapter }{Polytope: Attributes}}\label{Polytope:Attributes}1094\logpage{[ 6, 3, 0 ]}1095\hyperdef{L}{X87D8FC34790A474E}{}1096{109710981099\subsection{\textcolor{Chapter }{Vertices}}1100\logpage{[ 6, 3, 1 ]}\nobreak1101\hyperdef{L}{X79E4BB4F849AC8A1}{}1102{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Vertices({\mdseries\slshape poly})\index{Vertices@\texttt{Vertices}}1103\label{Vertices}1104}\hfill{\scriptsize (attribute)}}\\1105\textbf{\indent Returns:\ }1106a list1107110811091110Returns the vertices of the polytope \mbox{\texttt{\mdseries\slshape poly}}. For reasons, the corresponding tester is HasVerticesOfPolytopes }1111111211131114\subsection{\textcolor{Chapter }{LatticePoints}}1115\logpage{[ 6, 3, 2 ]}\nobreak1116\hyperdef{L}{X7FFECA277E47A55B}{}1117{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{LatticePoints({\mdseries\slshape poly})\index{LatticePoints@\texttt{LatticePoints}}1118\label{LatticePoints}1119}\hfill{\scriptsize (attribute)}}\\1120\textbf{\indent Returns:\ }1121a list1122112311241125Returns the lattice points of the polytope \mbox{\texttt{\mdseries\slshape poly}}. }1126112711281129\subsection{\textcolor{Chapter }{FacetInequalities}}1130\logpage{[ 6, 3, 3 ]}\nobreak1131\hyperdef{L}{X78D14B178577BFB1}{}1132{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{FacetInequalities({\mdseries\slshape poly})\index{FacetInequalities@\texttt{FacetInequalities}}1133\label{FacetInequalities}1134}\hfill{\scriptsize (attribute)}}\\1135\textbf{\indent Returns:\ }1136a list1137113811391140Returns the facet inequalities for the polytope \mbox{\texttt{\mdseries\slshape poly}}. }1141114211431144\subsection{\textcolor{Chapter }{VerticesInFacets}}1145\logpage{[ 6, 3, 4 ]}\nobreak1146\hyperdef{L}{X7E31AE1886051099}{}1147{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{VerticesInFacets({\mdseries\slshape poly})\index{VerticesInFacets@\texttt{VerticesInFacets}}1148\label{VerticesInFacets}1149}\hfill{\scriptsize (attribute)}}\\1150\textbf{\indent Returns:\ }1151a list1152115311541155Returns the incidence matrix of vertices and facets of the polytope \mbox{\texttt{\mdseries\slshape poly}}. }1156115711581159\subsection{\textcolor{Chapter }{AffineCone}}1160\logpage{[ 6, 3, 5 ]}\nobreak1161\hyperdef{L}{X7C3748B8878B799A}{}1162{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{AffineCone({\mdseries\slshape poly})\index{AffineCone@\texttt{AffineCone}}1163\label{AffineCone}1164}\hfill{\scriptsize (attribute)}}\\1165\textbf{\indent Returns:\ }1166a cone1167116811691170Returns the affine cone of the polytope \mbox{\texttt{\mdseries\slshape poly}}. }1171117211731174\subsection{\textcolor{Chapter }{NormalFan}}1175\logpage{[ 6, 3, 6 ]}\nobreak1176\hyperdef{L}{X7D7E33B97A7B4039}{}1177{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{NormalFan({\mdseries\slshape poly})\index{NormalFan@\texttt{NormalFan}}1178\label{NormalFan}1179}\hfill{\scriptsize (attribute)}}\\1180\textbf{\indent Returns:\ }1181a fan1182118311841185Returns the normal fan of the polytope \mbox{\texttt{\mdseries\slshape poly}}. }1186118711881189\subsection{\textcolor{Chapter }{RelativeInteriorLatticePoints}}1190\logpage{[ 6, 3, 7 ]}\nobreak1191\hyperdef{L}{X7E82C1C483269893}{}1192{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{RelativeInteriorLatticePoints({\mdseries\slshape poly})\index{RelativeInteriorLatticePoints@\texttt{RelativeInteriorLatticePoints}}1193\label{RelativeInteriorLatticePoints}1194}\hfill{\scriptsize (attribute)}}\\1195\textbf{\indent Returns:\ }1196a list1197119811991200Returns the lattice points in the relative interior of the polytope \mbox{\texttt{\mdseries\slshape poly}}. }12011202}120312041205\section{\textcolor{Chapter }{Polytope: Methods}}\label{Polytope:Methods}1206\logpage{[ 6, 4, 0 ]}1207\hyperdef{L}{X82806E0786AB09E5}{}1208{120912101211\subsection{\textcolor{Chapter }{* (for polytopes)}}1212\logpage{[ 6, 4, 1 ]}\nobreak1213\hyperdef{L}{X87DA13AA8305F283}{}1214{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{*({\mdseries\slshape polytope1, polytope2})\index{*@\texttt{*}!for polytopes}1215\label{*:for polytopes}1216}\hfill{\scriptsize (operation)}}\\1217\textbf{\indent Returns:\ }1218a polytope1219122012211222Returns the Cartesian product of the polytopes \mbox{\texttt{\mdseries\slshape polytope1}} and \mbox{\texttt{\mdseries\slshape polytope2}}. }1223122412251226\subsection{\textcolor{Chapter }{\#}}1227\logpage{[ 6, 4, 2 ]}\nobreak1228\hyperdef{L}{X8123456781234567}{}1229{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{\#({\mdseries\slshape polytope1, polytope2})\index{#@\texttt{\#}}1230\label{#}1231}\hfill{\scriptsize (operation)}}\\1232\textbf{\indent Returns:\ }1233a polytope1234123512361237Returns the Minkowski sum of the polytopes \mbox{\texttt{\mdseries\slshape polytope1}} and \mbox{\texttt{\mdseries\slshape polytope2}}. }12381239}124012411242\section{\textcolor{Chapter }{Polytope: Constructors}}\label{Polytope:Constructors}1243\logpage{[ 6, 5, 0 ]}1244\hyperdef{L}{X87A9DA5083C07E1E}{}1245{124612471248\subsection{\textcolor{Chapter }{Polytope (for lists of points)}}1249\logpage{[ 6, 5, 1 ]}\nobreak1250\hyperdef{L}{X86B877E378DF5E25}{}1251{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{Polytope({\mdseries\slshape points})\index{Polytope@\texttt{Polytope}!for lists of points}1252\label{Polytope:for lists of points}1253}\hfill{\scriptsize (operation)}}\\1254\textbf{\indent Returns:\ }1255a polytope1256125712581259Returns a polytope that is the convex hull of the points \mbox{\texttt{\mdseries\slshape points}}. }1260126112621263\subsection{\textcolor{Chapter }{PolytopeByInequalities}}1264\logpage{[ 6, 5, 2 ]}\nobreak1265\hyperdef{L}{X7E8849CF87B77402}{}1266{\noindent\textcolor{FuncColor}{$\triangleright$\ \ \texttt{PolytopeByInequalities({\mdseries\slshape ineqs})\index{PolytopeByInequalities@\texttt{PolytopeByInequalities}}1267\label{PolytopeByInequalities}1268}\hfill{\scriptsize (operation)}}\\1269\textbf{\indent Returns:\ }1270a polytope1271127212731274Returns a polytope defined by the inequalities \mbox{\texttt{\mdseries\slshape ineqs}}. }12751276}127712781279\section{\textcolor{Chapter }{Polytope: Examples}}\label{Polytope:Examples}1280\logpage{[ 6, 6, 0 ]}1281\hyperdef{L}{X854CE5FA7BD81060}{}1282{12831284\subsection{\textcolor{Chapter }{Polytope example}}\label{PolytopeExamplePrimary}1285\logpage{[ 6, 6, 1 ]}1286\hyperdef{L}{X83A852617A774F16}{}1287{12881289\begin{Verbatim}[commandchars=!@B,fontsize=\small,frame=single,label=Example]1290!gapprompt@gap>B !gapinput@P := Polytope( [ [ 2, 0 ], [ 0, 2 ], [ -1, -1 ] ] );B1291<A polytope in |R^2>1292!gapprompt@gap>B !gapinput@IsVeryAmple( P );B1293true1294!gapprompt@gap>B !gapinput@LatticePoints( P );B1295[ [ -1, -1 ], [ 0, 0 ], [ 0, 1 ],1296[ 0, 2 ], [ 1, 0 ], [ 1, 1 ], [ 2, 0 ] ]1297!gapprompt@gap>B !gapinput@NFP := NormalFan( P );B1298<A complete fan in |R^2>1299!gapprompt@gap>B !gapinput@C1 := MaximalCones( NFP )[ 1 ];B1300<A cone in |R^2>1301!gapprompt@gap>B !gapinput@RayGenerators( C1 );B1302[ [ -1, -1 ], [ -1, 3 ] ]1303!gapprompt@gap>B !gapinput@IsRegularFan( NFP );B1304true1305\end{Verbatim}1306}13071308}13091310}13111312\def\indexname{Index\logpage{[ "Ind", 0, 0 ]}1313\hyperdef{L}{X83A0356F839C696F}{}1314}13151316\cleardoublepage1317\phantomsection1318\addcontentsline{toc}{chapter}{Index}131913201321\printindex13221323\newpage1324\immediate\write\pagenrlog{["End"], \arabic{page}];}1325\immediate\closeout\pagenrlog1326\end{document}132713281329