Connect nodes with arc and arrow

Question

In the following MWE: I am trying to connect the nodes C, A, S and T with arcs and arrow heads. I am able to connect them with straight lines but reading through some documentation here I am still unable to do it. I have drawn a circle that should show the path I am look to get but with arrow in an anti-clockwise direction.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document}
    \begin{tikzpicture}[scale=4,cap=round,>=latex]
        \draw[->] (-1.125cm,0cm) -- (1.125cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.125cm) -- (0cm,1.125cm) node[above,fill=white] {$y$};
        \draw[thick] (0cm,0cm) circle(1cm);
            \foreach \x/\xtext/\xcoordinate in {
                45/QI/{(x>0,y>0)}, 
                135/QII/{(x<0,y>0)}, 
                225/QIII/{(x<0,y<0)}, 
                315/QIV/{(x>0,y<0)}} 
                {%
                    \draw (\x:0.65cm) node[fill=white,text width=2.5cm,align=center]
                    {\xtext \\ $\xcoordinate$};
                }
            \draw[thick] (0cm,0cm) circle(0.3cm);
            \foreach \y/\ytext in {
                45/A, 
                135/S, 
                225/T, 
                315/C} 
                {%
                    \draw (\y:0.3cm) node[shape=circle,fill=white,text width=1cm,align=center,font=\bfseries\large,text=black,inner sep=0pt]
                    {\ytext}coordinate (\ytext);
                }
            \draw[-latex,shorten <= 10pt,shorten >= 10pt] (S) -- (T);
            \draw[-latex,shorten <= 10pt,shorten >= 10pt] (A) -- (S);
            \draw[-latex,shorten <= 10pt,shorten >= 10pt] (C) -- (A);
            \draw[-latex,shorten <= 10pt,shorten >= 10pt] (T) -- (C);
    \end{tikzpicture}
\end{document}

This is what I have so far:

My closest attempt is to use:

\draw[shorten <= 10pt,shorten >= 10pt,bend right,->]  (C) to (A);
\draw[shorten <= 10pt,shorten >= 10pt,bend right,->]  (A) to (S);
\draw[shorten <= 10pt,shorten >= 10pt,bend right,->]  (S) to (T);
\draw[shorten <= 10pt,shorten >= 10pt,bend right,->]  (T) to (C);

This is what I am looking for:

Answer 1

Let me recycle this answer and adjust it to your settings. The problem is always the same: connect nodes on a circular arc. One answer is to construct an actual circle, and compute the intersections with the node boundary. Luckily you have already taken them to be circle shapes, so all that really was to do is to name the paths, that is, to add the key name path=\ytext to the circular nodes. (BTW, you can name the nodes, there is no need to add extra coordinates.) This yields:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning,arrows.meta,bending,calc,intersections}
\tikzset{pics/circular arc/.style args={from #1 to #2}{code={
  \def\pv##1{\pgfkeysvalueof{/tikz/circular arc/##1}} 
  \path[name path=arc,draw=none] 
   let \p1=(#1),\p2=(#2),\n1={Mod(720+atan2(\y1,\x1),360)},
   \n2={Mod(720+atan2(\y2,\x2),360)},
     \n3={ifthenelse(abs(\n1-\n2)<180,\n2,\n2+360)}
   in    (\n1:\pv{r}) arc(\n1:\n3:\pv{r});
 \draw[pic actions,
    name intersections={of=#1 and arc,by=arcstart},
    name intersections={of=#2 and arc,by=arcend}] 
    let \p1=(arcstart),\p2=(arcend),\n1={Mod(720+atan2(\y1,\x1),360)},
     \n2={Mod(720+atan2(\y2,\x2),360)},
     \n3={ifthenelse(abs(\n1-\n2)<180,\n2,\n2+360)}
  in (\n1:\pv{r}) arc(\n1:\n3:\pv{r});
 }},circular arc/.cd,r/.initial=0.3cm}
\begin{document}
    \begin{tikzpicture}[scale=4,line cap=round,>=latex]
        \draw[->] (-1.125cm,0cm) -- (1.125cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.125cm) -- (0cm,1.125cm) node[above,fill=white] {$y$};
        \draw[thick] (0cm,0cm) circle(1cm);
            \foreach \x/\xtext/\xcoordinate in {
                45/QI/{(x>0,y>0)}, 
                135/QII/{(x<0,y>0)}, 
                225/QIII/{(x<0,y<0)}, 
                315/QIV/{(x>0,y<0)}} 
                {%
                    \draw (\x:0.65cm) node[fill=white,text width=2.5cm,align=center]
                    {\xtext \\ $\xcoordinate$};
                }
            \draw[thick] (0cm,0cm) circle(0.3cm);
            \foreach \y/\ytext in {
                45/A, 
                135/S, 
                225/T, 
                315/C} 
                {%
                    \draw (\y:0.3cm) 
                    node[shape=circle,name path=\ytext,
                    fill=white,text width=1cm,align=center,
                    font=\bfseries\large,text=black,inner sep=0pt]
                    (\ytext){\ytext};
                }
            \path[shorten >=-2pt,-{Latex[bend]},transform shape] pic{circular arc=from S to T}
             pic{circular arc=from T to C} 
             pic{circular arc=from C to A}
              pic{circular arc=from A to S};
            %\draw[red,ultra thin] (0,0) circle[radius=0.3cm];  
    \end{tikzpicture}
\end{document}

If you uncomment the line

\draw[red,ultra thin] (0,0) circle[radius=0.3cm];  

you get

showing that the arcs really follow a circle.

If you replace text width=1cm by text width=1.75em, you get

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning,arrows.meta,bending,calc,intersections}
\tikzset{pics/circular arc/.style args={from #1 to #2}{code={
  \def\pv##1{\pgfkeysvalueof{/tikz/circular arc/##1}} 
  \path[name path=arc,draw=none] 
   let \p1=(#1),\p2=(#2),\n1={Mod(720+atan2(\y1,\x1),360)},
   \n2={Mod(720+atan2(\y2,\x2),360)},
     \n3={ifthenelse(abs(\n1-\n2)<180,\n2,\n2+360)}
   in    (\n1:\pv{r}) arc(\n1:\n3:\pv{r});
 \draw[pic actions,
    name intersections={of=#1 and arc,by=arcstart},
    name intersections={of=#2 and arc,by=arcend}] 
    let \p1=(arcstart),\p2=(arcend),\n1={Mod(720+atan2(\y1,\x1),360)},
     \n2={Mod(720+atan2(\y2,\x2),360)},
     \n3={ifthenelse(abs(\n1-\n2)<180,\n2,\n2+360)}
  in (\n1:\pv{r}) arc(\n1:\n3:\pv{r});
 }},circular arc/.cd,r/.initial=0.3cm}
\begin{document}
    \begin{tikzpicture}[scale=4,line cap=round,>=latex]
        \draw[->] (-1.125cm,0cm) -- (1.125cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.125cm) -- (0cm,1.125cm) node[above,fill=white] {$y$};
        \draw[thick] (0cm,0cm) circle(1cm);
            \foreach \x/\xtext/\xcoordinate in {
                45/QI/{(x>0,y>0)}, 
                135/QII/{(x<0,y>0)}, 
                225/QIII/{(x<0,y<0)}, 
                315/QIV/{(x>0,y<0)}} 
                {%
                    \draw (\x:0.65cm) node[fill=white,text width=2.5cm,align=center]
                    {\xtext \\ $\xcoordinate$};
                }
            \draw[thick] (0cm,0cm) circle(0.3cm);
            \foreach \y/\ytext in {
                45/A, 
                135/S, 
                225/T, 
                315/C} 
                {%
                    \draw (\y:0.3cm) 
                    node[shape=circle,name path global=\ytext,
                    fill=white,text width=1.75em,align=center,
                    font=\bfseries\large,text=black,inner sep=0pt]
                    (\ytext){\ytext};
                }
            \path[shorten >=-2pt,-{Latex[bend]},transform shape] pic{circular arc=from S to T}
             pic{circular arc=from T to C} 
             pic{circular arc=from C to A}
              pic{circular arc=from A to S};
            %\draw[red,ultra thin] (0,0) circle[radius=0.3cm];  
    \end{tikzpicture}
\end{document}

This code also uses name path global instead of name path. (On my updated TeXLive 2019 installation I do not need global.)

Finally, if all the nodes have really the same size, you can just use simple arcs.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning,arrows.meta,bending}
\begin{document}
    \begin{tikzpicture}[scale=4,line cap=round,>=latex]
        \draw[->] (-1.125cm,0cm) -- (1.125cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.125cm) -- (0cm,1.125cm) node[above,fill=white] {$y$};
        \draw[thick] (0cm,0cm) circle(1cm);
            \foreach \x/\xtext/\xcoordinate in {
                45/QI/{(x>0,y>0)}, 
                135/QII/{(x<0,y>0)}, 
                225/QIII/{(x<0,y<0)}, 
                315/QIV/{(x>0,y<0)}} 
                {%
                    \draw (\x:0.65cm) node[fill=white,text width=2.5cm,align=center]
                    {\xtext \\ $\xcoordinate$};
                }
            \draw[thick] (0cm,0cm) circle(0.3cm);
            \foreach \y/\ytext in {
                45/A, 
                135/S, 
                225/T, 
                315/C} 
                {%
                    \draw (\y:0.3cm) 
                    node[shape=circle,
                    fill=white,text width=1.75em,align=center,
                    font=\bfseries\large,text=black,inner sep=0pt]
                    (\ytext){\ytext};
                }
            \foreach \X in {0,...,3} 
            {\draw[-{Latex[bend]}] (\X*90-30:0.3cm) arc(\X*90-30:\X*90+30:0.3cm);}
            %\draw[red,ultra thin] (0,0) circle[radius=0.3cm];  
    \end{tikzpicture}
\end{document}

You can adjust the value 30 to your needs to make the arrows longer or shorter. This works fine but the arrows do not adjust their length to the size of the nodes. In you setting with fixed text width you prevent nodes from growing, so this solution should be OK.

Answer 2

I believe the correct way of doing this is using the angles package. See if the following is what you want or not:

Edit: created "auxiliar" nodes C2, A2 , S2and T2

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
\usetikzlibrary{angles}
\begin{document}
    \begin{tikzpicture}[scale=4,cap=round,>=latex]
        \draw[->] (-1.125cm,0cm) -- (1.125cm,0cm) node[right,fill=white] {$x$};
        \draw[->] (0cm,-1.125cm) -- (0cm,1.125cm) node[above,fill=white] {$y$};
        \draw[thick] (0cm,0cm) circle(1cm);
            \foreach \x/\xtext/\xcoordinate in {
                45/QI/{(x>0,y>0)}, 
                135/QII/{(x<0,y>0)}, 
                225/QIII/{(x<0,y<0)}, 
                315/QIV/{(x>0,y<0)}} 
                {%
                    \draw (\x:0.65cm) node[fill=white,text width=2.5cm,align=center]
                    {\xtext \\ $\xcoordinate$};
                }

            \foreach \y/\ytext in {
                45/A, 
                135/S, 
                225/T, 
                315/C} 
                {%
                    \draw (\y:0.3cm) node[]
                    {\ytext}coordinate (\ytext);
                }
            \node[] at (0cm,0cm) (O) {};
            \node[] at ([yshift=1, xshift=1]S) (S2) {};
            \node[] at ([yshift=-1, xshift=1]A) (A2) {};
            \node[] at ([yshift=1,xshift=-1]T) (T2) {};
            \node[] at ([xshift=-1]C) (C2) {};

            \pic [draw, ->, angle radius=1.5cm] {angle = A--O--S2};
            \pic [draw, ->, angle radius=1.5cm] {angle = S--O--T2};
            \pic [draw, ->, angle radius=1.5cm] {angle = T--O--C2};
            \pic [draw, ->, angle radius=1.5cm] {angle = C--O--A2};


    \end{tikzpicture}
\end{document}