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\documentclass[tikz,border=5mm]{standalone}

\begin{document}
\begin{tikzpicture}[scale=2]
   % Koordinatenachsen
   \draw[latex-latex] 
      (-3,-2) node[above left] {$y$} |-
      (-2,-3) node[below right] {$x$}
   ;
   % waagerechte Linien
   \foreach \x in {-2,-1,+1}{
      \draw (-2.5,\x) node[left] {$y_{i\x}$} --+ (4,0);
   }
   \draw (-2.5,0) node[left] {$y_{i}$} --+ (4,0);
   % senkrechte Linien
   \foreach \x in {-2,-1,+1}{
      \draw (\x,-2.5) node[below] {$x_{i\x}$} --+ (0,4);
   }
   \draw (0,-2.5) node[below] {$x_{i}$} --+ (0,4);
   % Punkte
   \fill (-.5,-.5) coordinate(P) circle (1pt) node[above] {P};
   % Ausrichtung als weitere Schleifenvariable hinzugefügt
   \foreach \x/\y/\direction in {0/E/east,90/N/north,180/W/west,270/S/south}{
      \path (P) --+ (\x:1) coordinate(\y) coordinate[midway] (P\y);
      \fill (\y) circle (1pt) node[above] {\y};
      \draw[-latex] (P\y) node [anchor=\direction] {\MakeLowercase{\y}} --+ (\x:.25);  
   }
   % Ausrichtung als weitere Schleifenvariable hinzugefügt
   \foreach \x/\y/\direction in {
      N/E/north east,N/W/north west,S/E/south east,S/W/south west%
   }{
      \path 
          (\x) -| (\y) coordinate[midway] (\x\y)
          (P) -- (\x\y) node[midway, anchor=\direction] {\MakeLowercase{\x\y}}
      ;
      \fill (\x\y) circle (1pt) node[above] {\x\y};
   }
\end{tikzpicture}
\end{document}

\documentclass[tikz,border=5mm]{standalone}
\begin{document}
\begin{tikzpicture}[scale=2,execute at begin node=$, execute at end node=$]
	%Koordinatenachsen
	\draw[-latex] (-3,-3) --++ (0,1) node[above left] {y};
	\draw[-latex] (-3,-3) --++ (1,0) node[below right] {x};
	%grid waagerechte Linien
	\foreach \x in {-2,-1,+1}{
		\draw (-2.5,\x) node[below] {y_{i\x}} --++ (4,0);
	}
	\draw (-2.5,0) node[below] {y_{i}} --++ (4,0);
	%grid senkrechte Linien
	\foreach \x in {-2,-1,+1}{
		\draw (\x,-2.5) node[below] {x_{i\x}} --++ (0,4);
	}
	\draw (0,-2.5) node[below] {x_{i}} --++ (0,4);
	%Punkte
	\fill (-.5,-.5) coordinate(P) circle (1pt) node[above] {P};
	\foreach \x/\y in {0/E,90/N,180/W,270/S}{
		\path (P) --++ (\x:1) coordinate(\y) coordinate[pos=.5] (P\y);
		\fill (\y) circle (1pt) node[above] {\y};
		\draw[-latex] (P\y) --++ (\x:.25) node[pos=-.375] {\MakeLowercase{\y}};	
	}
	\foreach \x/\y in {N/E,N/W,S/E,S/W}{
		\path (\x) -| (\y) coordinate[pos=.5] (\x\y);
		\fill (\x\y) circle (1pt)  node[above] {\x\y};
		\path (P) -- (\x\y) node[pos=.375] {\MakeLowercase{\x\y}};
	}
\end{tikzpicture}
\end{document}