%
% lemniskate.tex -- Lemniskate
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\documentclass[tikz]{standalone}
\usepackage{amsmath}
\usepackage{times}
\usepackage{txfonts}
\usepackage{pgfplots}
\usepackage{csvsimple}
\usetikzlibrary{arrows,intersections,math}
\begin{document}
\def\skala{4}
\begin{tikzpicture}[>=latex,thick,scale=\skala]

\draw[color=red,line width=1.0pt]
	plot[domain=-45:45,samples=100] ({\x}:{sqrt(2*cos(2*\x))});
\draw[color=red,line width=1.0pt]
	plot[domain=135:225,samples=100] ({\x}:{sqrt(2*cos(2*\x))});

\def\a{18}
\def\b{39}

\draw[->,color=blue,line width=1pt] (0,0) -- (\a:{sqrt(2*cos(2*\a))});

\draw[color=red,line width=2.0pt]
	plot[domain=45:\a,samples=100] ({\x}:{sqrt(2*cos(2*\x))});

\draw[->] (-1.5,0) -- (1.7,0) coordinate[label={$X$}];
\draw[->] (0,-0.7) -- (0,0.7) coordinate[label={right:$Y$}];

\fill[color=white] (1,0) circle[radius=0.02];
\draw (1,0) circle[radius=0.02];
\node at ({1},0) [below] {$\displaystyle a\mathstrut$};

\fill[color=white] (-1,0) circle[radius=0.02];
\draw (-1,0) circle[radius=0.02];
\node at ({-1},0) [below] {$\displaystyle\llap{$-$}a\mathstrut$};

\node[color=blue] at (\a:{0.6*sqrt(2*cos(2*\a))}) [below] {$r$};
\node[color=red] at ({\b}:{sqrt(2*cos(2*\b))}) [above] {$s$};

\fill[color=white] (\a:{sqrt(2*cos(2*\a))}) circle[radius=0.02];
\draw[color=red] (\a:{sqrt(2*cos(2*\a))}) circle[radius=0.02];

\draw ({sqrt(2)},{-0.1/\skala}) -- ({sqrt(2)},{0.1/\skala});
\node at ({sqrt(2)},0) [below right]
	{$\displaystyle a\mathstrut\sqrt{2}$};
\draw ({-sqrt(2)},{-0.1/\skala}) -- ({-sqrt(2)},{0.1/\skala});
\node at ({-sqrt(2)},0) [below left]
	{$\displaystyle -a\mathstrut\sqrt{2}$};


\end{tikzpicture}
\end{document}