Csdrhrt

In one of my previous answers, my first proposal

documentclass[tikz]{standalone}

usetikzlibrary{quotes,angles}

begin{document}

begin{tikzpicture}

draw (0,-4)--(0,4) node[above] {$Im$} (-4,0)--(4,0) node[right] {$Re$};

draw[dashed] (0,0) circle (3) circle (2);

coordinate (a) at (80:3);

coordinate (b) at (3,0);

coordinate (m) at (25:2);

coordinate (n) at (-95:2);

coordinate (p) at (145:2);

coordinate (o) at (0,0);

fill[black] (a) circle (2pt) (b) circle (2pt) (m) circle (2pt) (n) circle (2pt) (p) circle (2pt) (2,0) circle (2pt);

draw (a) node[above right] {$z=|z|e^{itheta}$};

draw (b) node[below] {$|z|$};

draw (2,0) node[below] {$|z|^{1/3}$};

draw (m) node[right] {$|z|^{1/3}e^{itheta/3}$};

draw (n) node[below] {$|z|^{1/3}e^{i(theta+4pi)/3}$};

draw (p) node[above] {$|z|^{1/3}e^{i(theta+2pi)/3}$};

draw (.1,1.5)--(0,1.5) node[left] {$i$};

draw (1.5,.1)--(1.5,0) node[below] {$1$};

draw (0,0)--(a) (0,0)--(m) (0,0)--(n) (0,0)--(p);

draw[dashed] (m)--(n)--(p)--cycle;

pic[draw,dashed,thick,"$theta$",angle radius=0.8cm,angle eccentricity=1.3] {angle=b--o--a};

end{tikzpicture}

end{document}

gives this output

Seeing that it is a bit difficult to read some nodes (like the |z|^1/3e^{i (θ + 2 π)/3} one), I redefine the inner sep and set the fill color of the nodes. Eventually I get

documentclass[tikz]{standalone}

usetikzlibrary{quotes,angles,positioning}

begin{document}

begin{tikzpicture}

begin{scope}[every node/.style={fill=white,inner sep=2pt}]

draw (0,-4)--(0,4) node[above] {$Im$} (-4,0)--(4,0) node[right] {$Re$};

draw[dashed] (0,0) circle (3) circle (2);

coordinate (a) at (80:3);

coordinate (b) at (3,0);

coordinate (m) at (80/3:2);

coordinate (n) at ({80/3-120}:2);

coordinate (p) at ({80/3+120}:2);

coordinate (o) at (0,0);

draw (a) node[above right] {$z=|z|e^{itheta}$};

draw (b) node[below right] {$|z|$};

draw (2,0) node[below left=0cm and -2em] {$|z|^{1/3}$};

draw (m) node[right] {$|z|^{1/3}e^{itheta/3}$};

draw (n) node[below] {$|z|^{1/3}e^{i(theta+4pi)/3}$};

draw (p) node[above] {$|z|^{1/3}e^{i(theta+2pi)/3}$};

draw (.1,1.5)--(0,1.5) node[left] {$i$};

draw (1.5,.1)--(1.5,0) node[below] {$1$};

draw (0,0)--(a) (0,0)--(m) (0,0)--(n) (0,0)--(p);

draw[dashed] (m)--(n)--(p)--cycle;

end{scope}

pic[draw,dashed,thick,"$theta$",angle radius=0.8cm,angle eccentricity=1.3] {angle=b--o--a};

fill[black] (a) circle (2pt) (b) circle (2pt) (m) circle (2pt) (n) circle (2pt) (p) circle (2pt) (2,0) circle (2pt);

end{tikzpicture}

end{document}

The separation of the nodes and the paths are now quite good IMHO, except some cases where the separation is over-made:

A solution to it is to change the default rectangle node shape

to something like this

^{Sorry, I am not good at drawing, especially drawing with a computer mouse.}

In other word, I think I have to create a new TikZ node shape which depends on the maximum and the minimum "y-coordinates" of all characters.

It is way too complicated to me, and I haven't found a hint on this.

Can you help me? Any help is much appreciated.

Maybe the contour package is a solution for you (See also this answer)?

documentclass[tikz, border=2mm]{standalone}



usepackage[outline]{contour}

contourlength{2pt}  % increase the white space



usetikzlibrary{patterns}



begin{document}



begin{tikzpicture}

draw [step=.25] (-2,-.5) grid (2,.5);



node at (0,0) {contour{white}{$|z|^{1/3}e^{i(theta+2pi)/3}$}};



end{tikzpicture}



end{document}

Note: The above does not work with XeLaTeX or LuaLaTeX. In this case you need to remove the outline option and add for example contournumber{60} (or a higher number, depending on the desired smoothness of the edges of the white space, see the documentation of the package).