
diff -r b77b9fce62d6 -r 571226041e79 presentations/basic-plot.tex
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+++ b/presentations/basic-plot.tex	Tue Apr 06 17:02:41 2010 +0530
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%Tutorial slides on Python.
+%
+% Author: FOSSEE 
+% Copyright (c) 2009, FOSSEE, IIT Bombay
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\documentclass[14pt,compress]{beamer}
+%\documentclass[draft]{beamer}
+%\documentclass[compress,handout]{beamer}
+%\usepackage{pgfpages} 
+%\pgfpagesuselayout{2 on 1}[a4paper,border shrink=5mm]
+
+% Modified from: generic-ornate-15min-45min.de.tex
+\mode<presentation>
+{
+  \usetheme{Warsaw}
+  \useoutertheme{infolines}
+  \setbeamercovered{transparent}
+}
+
+\usepackage[english]{babel}
+\usepackage[latin1]{inputenc}
+%\usepackage{times}
+\usepackage[T1]{fontenc}
+
+% Taken from Fernando's slides.
+\usepackage{ae,aecompl}
+\usepackage{mathpazo,courier,euler}
+\usepackage[scaled=.95]{helvet}
+
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+
+\usepackage{listings}
+\lstset{language=Python,
+    basicstyle=\ttfamily\bfseries,
+    commentstyle=\color{red}\itshape,
+  stringstyle=\color{darkgreen},
+  showstringspaces=false,
+  keywordstyle=\color{blue}\bfseries}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Macros
+\setbeamercolor{emphbar}{bg=blue!20, fg=black}
+\newcommand{\emphbar}[1]
+{\begin{beamercolorbox}[rounded=true]{emphbar} 
+      {#1}
+ \end{beamercolorbox}
+}
+\newcounter{time}
+\setcounter{time}{0}
+\newcommand{\inctime}[1]{\addtocounter{time}{#1}{\tiny \thetime\ m}}
+
+\newcommand{\typ}[1]{\lstinline{#1}}
+
+\newcommand{\kwrd}[1]{ \texttt{\textbf{\color{blue}{#1}}}  }
+
+%%% This is from Fernando's setup.
+% \usepackage{color}
+% \definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
+% % Use and configure listings package for nicely formatted code
+% \usepackage{listings}
+% \lstset{
+%    language=Python,
+%    basicstyle=\small\ttfamily,
+%    commentstyle=\ttfamily\color{blue},
+%    stringstyle=\ttfamily\color{orange},
+%    showstringspaces=false,
+%    breaklines=true,
+%    postbreak = \space\dots
+% }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Title page
+\title[Basic Plotting]{Python for Scientific Computing : Basic Plotting}
+
+\author[FOSSEE] {FOSSEE}
+
+\institute[IIT Bombay] {Department of Aerospace Engineering\\IIT Bombay}
+\date{}
+%% \date[] {7 November, 2009\\Day 1, Session 1}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%\pgfdeclareimage[height=0.75cm]{iitmlogo}{iitmlogo}
+%\logo{\pgfuseimage{iitmlogo}}
+
+
+%% Delete this, if you do not want the table of contents to pop up at
+%% the beginning of each subsection:
+%% \AtBeginSubsection[]
+%% {
+%%   \begin{frame}<beamer>
+%%     \frametitle{Outline}
+%%     \tableofcontents[currentsection,currentsubsection]
+%%   \end{frame}
+%% }
+
+%% \AtBeginSection[]
+%% {
+%%   \begin{frame}<beamer>
+%%     \frametitle{Outline}
+%%     \tableofcontents[currentsection,currentsubsection]
+%%   \end{frame}
+%% }
+
+% If you wish to uncover everything in a step-wise fashion, uncomment
+% the following command: 
+%\beamerdefaultoverlayspecification{<+->}
+
+%%\includeonlyframes{current,current1,current2,current3,current4,current5,current6}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% DOCUMENT STARTS
+\begin{document}
+
+\begin{frame}
+  \maketitle
+\end{frame}
+
+%% \begin{frame}
+%%   \frametitle{Outline}
+%%   \tableofcontents
+%%   % You might wish to add the option [pausesections]
+%% \end{frame}
+
+%% \begin{frame}
+%%   \frametitle{Workshop Schedule: Day 1}
+%%   \begin{description}
+%% 	\item[Session 1] Sat 09:00--10:00
+%% 	\item[Session 2] Sat 10:05--11:05
+%% 	\item[Session 3] Sat 11:20--12:20
+%% 	\item[Session 4] Sat 12:25--13:25
+%%         \item[Quiz 1] Sat 14:25--14:40
+%%         \item[Session 5] Sat 14:40--15:25
+%%         \item[Session 6] Sat 15:40--16:40
+%%         \item[Quiz 2] Sat 16:45--17:00
+%%   \end{description}
+%% \end{frame}
+
+%% \begin{frame}
+%%   \frametitle{Workshop Schedule: Day 2}
+%%   \begin{description}
+%% 	\item[Session 1] Sun 09:00--10:00
+%% 	\item[Session 2] Sun 10:05--11:05
+%% 	\item[Session 3] Sun 11:20--12:20
+%% 	\item[Session 4] Sun 12:25--13:25
+%%         \item[Quiz 1] Sun 14:25--14:40
+%%         \item[Session 5] Sun 14:40--15:25
+%%         \item[Session 6] Sun 15:40--16:40
+%%         \item[Quiz 2] Sun 16:45--17:00
+%%   \end{description}
+%% \end{frame}
+
+\begin{frame}
+  \frametitle{About the Workshop}
+  \begin{block}{Intended Audience}
+  \begin{itemize}
+       \item Engg., Mathematics and Science teachers.
+       \item Interested students from similar streams.
+  \end{itemize}
+  \end{block}  
+
+  \begin{block}{Goal: Successful participants will be able to}
+    \begin{itemize}
+      \item Use Python as a basic Plotting tool.
+      \item Understand how to use Python as a scripting and problem solving language.
+
+    \end{itemize}
+  \end{block}
+\end{frame}
+
+\section{Getting started}
+\begin{frame}
+\frametitle{Checklist}
+   \begin{itemize}
+    \item IPython
+    \item Pylab
+    %% \item Editor: We recommend scite.
+    %% \item Data files: 
+    %%   \begin{itemize}
+    %%   \item \typ{sslc1.txt}
+    %%   \item \typ{pendulum.txt}
+    %%   \item \typ{points.txt}
+    %%   \item \typ{pos.txt}
+    %%   \end{itemize}
+    %% \item Python scripts: 
+    %%   \begin{itemize}
+    %%   \item \typ{sslc_allreg.py}
+    %%   \item \typ{sslc_science.py}
+    %%   \end{itemize}
+    %% \item Images
+    %%   \begin{itemize}
+    %%   \item \typ{lena.png}
+    %%   \item \typ{smoothing.gif}
+  \end{itemize}
+  %% \end{enumerate}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Starting up \ldots}
+\begin{block}{}
+\begin{verbatim}
+  $ ipython -pylab  
+\end{verbatim}
+\end{block}
+\begin{lstlisting}     
+  In []: print "Hello, World!"
+  Hello, World!
+\end{lstlisting}
+Exiting
+\begin{lstlisting}     
+  In []: ^D(Ctrl-D)
+  Do you really want to exit([y]/n)? y
+\end{lstlisting}
+\end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Loops}
+%% Breaking out of loops
+%% \begin{lstlisting}     
+%%   In []: while True:
+%%     ...:     print "Hello, World!"
+%%     ...:     
+%%   Hello, World!
+%%   Hello, World!^C(Ctrl-C)
+%%   ------------------------------------
+%%   KeyboardInterrupt                   
+
+%% \end{lstlisting}
+%% \end{frame}
+
+\section{Plotting}
+
+\subsection{Drawing plots}
+\begin{frame}[fragile]
+\frametitle{First Plot}
+\begin{columns}
+    \column{0.20\textwidth}
+    \hspace*{-0.12in}
+  \includegraphics[height=1.2in, interpolate=true]{data/firstplot}
+    \column{0.7\textwidth}
+    \begin{block}{}
+    \begin{small}
+\begin{lstlisting}
+In []: x=lins<tab>
+In []: x=linspace(
+ ... : (Ctrl-C)
+In []: x = linspace(0, 2*pi, 50)
+In []: plot(x, sin(x))
+\end{lstlisting}
+    \end{small}
+    \end{block}
+ \begin{block}{Ipython Feature}
+    \begin{itemize}
+      \item Use Tab for auto-suggestions.
+      \item In []: (Ctrl-C) to get back In[] prompt from ...
+  \end{itemize}
+  \end{block}
+\end{columns}
+\end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{Function Documentation}
+\begin {block}{}
+\begin{lstlisting}
+In []: linspace?
+\end{lstlisting}
+\end{block}
+\begin{block}{Ipython Feature}
+    \begin{itemize}
+      \item linspace? , ? mark after a function shows its documentation
+      \item q to exit help  
+  \end{itemize}
+  \end{block}
+\end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{Walkthrough}
+\begin{block}{\typ{x = linspace(start, stop, num)} }
+returns \typ{num} evenly spaced points, in the interval [\typ{start}, \typ{stop}].
+\end{block}
+\begin{lstlisting}
+x[0] = start
+x[num - 1] = end
+\end{lstlisting}
+\vspace*{.35in}
+\begin{block}{\typ{plot(x, y)}}
+plots \typ{x} and \typ{y} using default line style and color
+\end{block}
+%\inctime{10}
+\end{frame}
+
+\subsection{Decoration}
+\begin{frame}[fragile]
+\frametitle{Adding Labels and title}
+\begin{columns}
+  \column{0.25\textwidth}
+  \hspace*{-0.45in}
+  \includegraphics[height=2in, interpolate=true]{data/label}  
+  \hspace*{0.5in}
+  \column{0.55\textwidth}
+  \begin{block}{}
+  \small
+  \begin{lstlisting}
+In []: xlabel('x')
+In []: ylabel('sin(x)')
+In []: title('Sinusoids')
+
+  \end{lstlisting}
+  \small
+%  \end{lstlisting}
+%\typ{xlabel(s)} sets the label of the \typ{x}-axis to \typ{s}
+
+%  \begin{lstlisting}
+  \end{block}
+%\typ{ylabel(s)} sets the label of the \typ{y}-axis to \typ{s}
+\end{columns}
+\end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Another example}
+%%   \begin{lstlisting}
+%% In []: clf()
+%%   \end{lstlisting}
+%% \emphbar{Clears the plot area.}
+%%   \begin{lstlisting}
+%% In []: y = linspace(0, 2*pi, 50)
+%% In []: plot(y, sin(2*y))
+%% In []: xlabel('y')
+%% In []: ylabel('sin(2y)')
+%%   \end{lstlisting}
+%% \end{frame}
+
+\subsection{More decoration}
+\begin{frame}[fragile]
+\frametitle{ Legends}
+\vspace*{-0.15in}
+%  \begin{block}{}
+%  \small
+\begin{lstlisting}
+In []: legend(['sin(x)'])
+\end{lstlisting}
+%  \small
+%  \end{block}
+  \vspace*{-0.1in}
+  \begin{center}
+  \includegraphics[height=2in, interpolate=true]{data/legend}  
+  \end{center}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Legend Placement}
+\begin{block}{}
+    \small
+\begin{lstlisting}
+In []: legend(['sin(x)'], loc = 'center')
+\end{lstlisting}
+\end{block}
+
+\begin{columns}
+    \column{0.6\textwidth}
+ \includegraphics[height=1.5in, interpolate=true]{data/position}  
+\column{0.35\textwidth}
+\vspace{-0.15in}
+\begin{lstlisting}
+'best' 
+'right'
+'center'
+\end{lstlisting}
+\end{columns}
+\begin{block}{Ipython Feature}
+    \begin{itemize}
+      \item Use up arrow and down arrow to get old commands .
+  \end{itemize}
+  \end{block}
+
+\end{frame}
+
+
+
+\begin{frame}[fragile]
+\frametitle{Annotate a point }
+
+\begin{lstlisting}
+In []:annotate('origin', xy = (0, 0))
+\end{lstlisting}
+
+
+\begin{columns}
+    \column{0.6\textwidth}
+ \includegraphics[height=2in, interpolate=true]{data/annotate}  
+\column{0.45\textwidth}
+\vspace{-0.2in}
+%% \begin{lstlisting}
+%% 'best' 
+%% 'right'
+%% 'center'
+%% \end{lstlisting}
+\end{columns}
+\end{frame}
+
+
+
+%% \begin{frame}[fragile]
+%%   \frametitle{For arbitrary location}
+%% \vspace*{-0.1in}
+%% \begin{lstlisting}
+%% In []: legend(['sin(2y)'], loc=(.8,.1)) 
+%% \end{lstlisting}
+%% \emphbar{Specify south-east corner position}
+%% %\vspace*{-0.2in}
+%% \begin{center}
+%%   \includegraphics[height=2in, interpolate=true]{data/loc}  
+%% \end{center}
+%% %\inctime{10}
+%% \end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Saving \& Closing}
+\begin{lstlisting}
+In []: savefig('sin.png')
+
+\end{lstlisting}
+\end{frame}
+%%   \begin{lstlisting}
+%% In []: clf()
+%%   \end{lstlisting}
+%% \emphbar{Clears the plot area.}
+%%   \begin{lstlisting}
+%% In []: y = linspace(0, 2*pi, 50)
+%% In []: plot(y, sin(2*y))
+%% In []: xlabel('y')
+%% In []: ylabel('sin(2y)')
+%%   \end{lstlisting}
+%% \end{frame}
+
+
+
+
+
+
+\section{Multiple plots}
+\begin{frame}[fragile]
+\frametitle{Overlaid Plots}
+\begin{lstlisting}
+In []: plot(x, cos(x))
+In []: xlabel('x')
+In []: ylabel('f(x)')
+In []: legend(['sin(x)', 'cos(x)']) 
+In []: clf()
+\end{lstlisting}
+\emphbar{By default plots would be overlaid!}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Plotting separate figures}
+\begin{lstlisting}
+In []: figure(1)
+In []: plot(x, sin(x))
+In []: figure(2)
+In []: plot(x, cos(x))
+In []: figure(1)
+In []: title('sin(x)')
+In []: close()
+In []: close()
+\end{lstlisting}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Showing it better}
+\vspace{-0.15in}
+\begin{lstlisting}
+In []: plot(x, sin(x), 'g' ,linewidth=2)
+
+In []: clf()
+
+\end{lstlisting}
+\vspace*{-0.2in}
+\begin{center}
+  \includegraphics[height=2.2in, interpolate=true]{data/green}  
+\end{center}
+%\inctime{10}
+\end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{Showing it better , Using Dots }
+\vspace{-0.15in}
+\begin{lstlisting}
+In []: plot(x, sin(x), '.')
+
+In []: clf()
+
+\end{lstlisting}
+\vspace*{-0.2in}
+\begin{center}
+  \includegraphics[height=2.2in, interpolate=true]{data/dash}  
+\end{center}
+%\inctime{10}
+\end{frame}
+
+
+
+
+
+%% \begin{frame}[fragile]
+%% \frametitle{Review Ipython }
+%% \vspace{-0.15in}
+%% \begin{lstlisting}
+%% In []: lins<tab>
+
+%% In []: function?
+%%      :q 
+%% (Ctrl-D)
+%% In []: ^C(Ctrl-C)
+
+
+
+%% \end{lstlisting}
+%% %\inctime{10}
+%% \end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{Review Ipython Features }
+\vspace{-0.1in}
+\begin{itemize}
+\item Entering Ipython.
+\begin{verbatim}
+  $ ipython -pylab  
+\end{verbatim}
+\item Seeing the documentation . 
+\begin{lstlisting}
+In []: linspace?
+\end{lstlisting}
+\item Quitting the documentation .
+\begin{lstlisting}
+:q 
+\end{lstlisting}
+\item Quitting from the ... prompt.
+\begin{lstlisting}
+Ctrl-C
+\end{lstlisting}
+\end{itemize}
+\begin{itemize}
+\item Quitting Ipython
+\begin{lstlisting}
+Ctrl-D
+\end{lstlisting}
+\end{itemize}
+
+
+%\inctime{10}
+\end{frame}
+
+\begin{frame}[fragile]
+\frametitle{Review Plotting }
+\vspace{-0.15in}
+\begin{itemize}
+\item Outputting things.
+\end{itemize}
+\begin{lstlisting}
+In []: print ''hello world'' 
+\end{lstlisting}
+\begin{itemize}
+\item Create equally spaced points.
+\end{itemize}
+\begin{lstlisting}
+In []: x=linspace(0,2*pi,50)
+\end{lstlisting}
+\begin{itemize}
+\item Simple Plotting
+\end{itemize}
+\begin{lstlisting}
+In []: plot(x,sin(x),'.',linewidth=2)
+\end{lstlisting}
+\end{frame}
+
+
+\begin{frame}[fragile]
+\frametitle{Review Plotting }
+\vspace{-0.1in}
+\begin{itemize}
+\item   label axis
+\end{itemize}
+\begin{lstlisting}
+In []: xlabel('x')
+\end{lstlisting}
+\begin{itemize}
+\item Title The plot  
+\end{itemize}
+\begin{lstlisting}
+In []: title('sinusoid')
+\end{lstlisting}
+\begin{itemize}
+\item Place legend at a proper place
+\end{itemize}
+\begin{lstlisting}
+In []: legend(['sin(x)'])
+\end{lstlisting}
+\begin{itemize}
+\item Annotate the plot
+\end{itemize}
+\begin{lstlisting}
+In []:annotate('origin' , xy=(0,0)) 
+\end{lstlisting}
+%\inctime{10}
+\end{frame}
+
+
+
+
+\begin{frame}[fragile]
+\frametitle{Review Plotting }
+\vspace{-0.15in}
+\begin{itemize}
+\item Save a plot
+\end{itemize}
+\begin{lstlisting}
+In []:savefig('sine.png')
+\end{lstlisting}
+\begin{itemize}
+\item Managing multiple plots using figure
+\end{itemize}
+\begin{lstlisting}
+In []:figure(1)
+\end{lstlisting}
+\begin{itemize}
+\item Clearing plot 
+\end{itemize}
+\begin{lstlisting}
+In []: clf()
+\end{lstlisting}
+\begin{itemize}
+\item Closing plot 
+\end{itemize}
+\begin{lstlisting}
+In []: close()
+\end{lstlisting}
+
+%\inctime{10}
+\end{frame}
+
+
+
+
+
+
+
+%% \begin{frame}[fragile]
+%% \frametitle{Review  , Function and Commands }
+%% \vspace{-0.15in}
+%% \begin{lstlisting}
+
+%% In []: print 'hello world'
+%% In []: x=linspace(0,2*pi,50)
+%% In []: plot(x, sin(x), 'g',linewidth=2)
+%% In []: xlabel('x')
+%% In []: ylabel('sin(x)')
+%% In []: title('sinusoid')
+%% In []: legend(['sin(x)','cos(x)'])
+%% In []: annotate('origin', xy=(0,0))
+%% In []: savefig('sine.png')
+%% In []: clf()
+%% In []: figure(1)
+%% \end{lstlisting}
+%% %\inctime{10}
+%% \end{frame}
+
+
+
+%% \begin{frame}[fragile]
+%% \frametitle{Annotating}
+%% \vspace*{-0.15in}
+%% \begin{lstlisting}
+%% In []: annotate('local max', xy=(1.5, 1))
+%% \end{lstlisting}
+%% \vspace*{-0.2in}
+%% \begin{center}
+%%   \includegraphics[height=2in, interpolate=true]{data/annotate}  
+%% \end{center}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Axes lengths}
+%% \emphbar{Get the axes limits}
+%%   \begin{lstlisting}
+%% In []: xmin, xmax = xlim() 
+%% In []: ymin, ymax = ylim() 
+%%   \end{lstlisting}
+%% \emphbar{Set the axes limits}
+%%   \begin{lstlisting}
+%% In []: xmax = 2*pi
+%% In []: xlim(xmin, xmax) 
+%% In []: ylim(ymin-0.2, ymax+0.2) 
+%%   \end{lstlisting}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Review Problem}
+%% \begin{enumerate}
+%% \item Plot x, -x, sin(x), xsin(x) in range $-5\pi$ to $5\pi$
+%% \item Add a legend
+%% \item Annotate the origin
+%% \item Set axes limits to the range of x
+%% \end{enumerate}
+%% \begin{lstlisting}
+%% In []: x=linspace(-5*pi, 5*pi, 500)
+%% In []: plot(x, x, 'b')
+%% In []: plot(x, -x, 'b')
+%% \end{lstlisting}
+%% $\vdots$
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Review Problem \ldots}
+%% \begin{lstlisting}
+%% In []: plot(x, sin(x), 'g', linewidth=2)
+%% In []: plot(x, x*sin(x), 'r', 
+%%             linewidth=3)
+%% \end{lstlisting}
+%% \begin{lstlisting}
+%% In []: legend(['x', '-x', 'sin(x)', 
+%%                'xsin(x)'])
+%% In []: annotate('origin', xy = (0, 0))
+%% In []: xlim(-5*pi, 5*pi)
+%% In []: ylim(-5*pi, 5*pi)
+%% \end{lstlisting}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%% \frametitle{Saving Commands}
+%% Save commands of review problem into file
+%% \begin{itemize}
+%% \item Use \typ{\%hist} command of IPython 
+%% \item Identify the required line numbers
+%% \item Then, use \typ{\%save} command of IPython
+%% \end{itemize}
+%% \typ{In []: \%hist}\\
+%% \typ{In []: \%save four_plot.py} \alert{\typ{16 18-27}} 
+%% \begin{block}{Careful about errors!}
+%%   \kwrd{\%hist} will contain the errors as well,\\
+%%   so be careful while selecting line numbers.
+%% \end{block}
+%% \end{frame}
+
+%% \begin{frame}
+%% \frametitle{Python Scripts\ldots}
+%%  This is called a Python Script.
+%%  \begin{itemize}
+%%  \item run the script in IPython using \typ{\%run -i four_plot.py}\\
+%%  \end{itemize}
+%% \end{frame}
+
+%% \begin{frame}[fragile]
+%%   \frametitle{What did we learn?}
+%%   \begin{itemize}
+%%     \item \kwrd{\%hist}
+%%     \item Saving commands to a script
+%%     \item Running a script using \kwrd{\%run -i}
+%%     \item Creating simple plots.
+%%     \item Adding labels and legends.
+%%     \item Annotating plots.
+%%     \item Changing the looks: size, linewidth
+%%   \end{itemize}
+%% \end{frame}
+
+\end{document}
+




