Zeichnen einer Ebene und eines Orthonormalsystems mit Tikz

Question

Ich möchte folgendes Bild zeichnen und habe vor allem in der linken Ebene Probleme:

\documentclass[tikz]{standalone}    
\begin{document} 
\begin{tikzpicture} 
\draw[thick,->] (52,0) -- (60,0) node[anchor=north west] {$x'$}; 
\draw[thick,->] (56,-04) -- (56,04) node[anchor=south east] {$x_N$};
\draw[red,thick,dashed] (56,0) circle (2cm); 
\draw[thick,->] (57.5,1.55) -- (58,2) node[anchor=north west] {$I^{+}(1)$}; 
\draw[thick,->] (57.5,-1.55) -- (58,-2) node[anchor=north west] {$I(1)$}; 
\draw (56,0) node[below right] {$O$} node{$\bullet$}; 
\draw (58,0) node[below right] {$1$} node{}; 
\draw (54,0) node[below left] {$-1$} node{}; 
\end{tikzpicture} 
\end{document} 

ich zeichne das Orthonormalsystem mithilfe von tikzpictureund weiß nicht, wie ich weitermachen soll. Irgendwelche Hilfe?

Answer 1

Hier ist ein Versuch mit hobby.

\documentclass[tikz]{standalone}
\usetikzlibrary{hobby,patterns}
\begin{document}
\begin{tikzpicture}
\draw[thick,->] (52,0) -- (60,0) node[anchor=north west] {$x'$};
\draw[thick,->] (56,-04) -- (56,04) node[anchor=south east] {$x_N$};
\node[draw=red,dashed,thick,circle,minimum width=4cm] (n) at (56,0) {};
\fill[pattern=north east lines,opacity=.6] (58,0)arc(0:180:2cm) -- (58,0);
\draw[thick,->] (57.5,1.55) -- (58,2) node[anchor=north west] {$I^{+}(1)$};
\draw[thick,->] (57.5,-1.55) -- (58,-2) node[anchor=north west] {$I(1)$};
\draw (56,0) node[below right] {$O$} node{$\bullet$};
\draw (58,0) node[below right] {$1$} node{};
\draw (54,0) node[below left] {$-1$} node{};

\node[draw,circle,minimum width=2cm] (n1) at (50,3) {};
\begin{scope}
\clip[draw](48,1) to[closed,curve through={(50,3) .. (52,4) .. (51,6) ..(47,6) .. (46.6,5) .. (45,3.8) }] (48,1);
\node[draw,circle,fill=magenta,minimum width=2cm] (n1) at (50,3) {};
\end{scope}
\node[inner sep=1pt,fill,circle,label={below right: $x_0$}] at (50,3){};
\draw[->,shorten <= -4mm] (n1.90) -- +(0.7,0.7) node[pos=1.3]{$\Omega_1 \cap \Omega$};
\draw[->] (51,6) -- +(0.7,0.3) node[pos=1.3]{$\Omega$};
\draw[->] (n1.300) -- +(0.7,-0.3) node[pos=1.3]{$\Omega_1$};
\draw[->] (n1.10) to[bend left] node[midway,below]{8*} (n.120);   %% 8* = ???
\node at (50,-1){ Some text???};
\end{tikzpicture}
\end{document}

Einige Textinhalte waren mir nicht klar, welche ich ???im Code mit gekennzeichnet habe.