Remarks
The paths with the arrows inside needed some extra attention, but on this great site I found a way to do it, of course :-)
. For more see this post: TikZ: How to draw an arrow in the middle of the line?
The picture in your post
I recreated the picture in your post with TikZ.
Implementation
\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.markings}
\tikzset{
arrow inside/.style = {
postaction = {
decorate,
decoration={
markings,
mark=at position 0.5 with {\arrow{>}}
}
}
}
}
\begin{document}
\begin{tikzpicture}[>=latex,scale=1.5]
\begin{scope}
% Axes
\draw (0,0) node[below left] {$O$}
(-0.5,0) -- (4,0) node[below] {$x$}
(0,-0.5) -- (0,3) node[left] {$y$};
% Ticks
\draw (1,0) -- (1,-0.1) node[below] {$a$}
(3,0) -- (3,-0.1) node[below] {$b$}
(0,1) -- (-0.1,1) node[left] {$c$}
(0,2) -- (-0.1,2) node[left] {$d$};
% Square
\draw[thick] (1,1) node[below left] {$A$} --
(3,1) node[below right] {$B$} --
(3,2) node[above right] {$C$} --
(1,2) node[above left] {$D$} -- cycle;
\draw[arrow inside] (1.5,1) -- (1.5,2);
\end{scope}
\begin{scope}[xshift=6cm]
% Axes
\draw (0,0) node[below left] {$O$}
(-0.5,0) -- (4,0) node[below] {$u$}
(0,-0.5) -- (0,3) node[left] {$v$};
%Help Lines
\draw (0,0) -- (30:3) (0,0) -- (70:3);
% Angles
\draw[->] (0.6,0) arc[start angle=0, end angle=70, radius=0.6] node[above right] {\small $\phi = d$};
\draw[->] (0.8,0) node[above right] {\small$\phi = c$} arc[start angle=0, end angle=30, radius=0.8];
% Transformation
\draw[thick] (30:1.5) node[right] {$A'$} --
(30:3) node[below right] {$B'$} arc[start angle=30, end angle=70, radius=3]
(70:3) node[above right] {$C'$} --
(70:1.5) node[above left] {$D'$} arc[start angle=70, end angle=30, radius=1.5];
\draw[arrow inside] (30:1.9) arc[start angle=30, end angle=70, radius=1.9];
\end{scope}
\end{tikzpicture}
\end{document}
Output
I hope, you are satisfied.

From your description
I don't know, if I got your description right, but based on it I did the following
Implementation
\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.markings}
\tikzset{
arrow inside/.style = {
postaction = {
decorate,
decoration={
markings,
mark=at position #1 with {\arrow{>}}
}
}
},
arrow inside/.default = 0.5
}
\begin{document}
\begin{tikzpicture}[>=latex]
\begin{scope}
% Axes
\draw (0,0) node[below left] {$O$}
(-0.5,0) -- (5,0) node[below] {$x$}
(0,-0.5) -- (0,4) node[left] {$y$};
% Ticks
\draw (0,pi) -- (-0.1,pi) node[left] {$\pi$};
% Square
\draw[thick] (4.5,0) --
(0,0) node[above left] {$A$} --
(0,pi) node[above right] {$D$} --
(4.5,pi);
\draw[arrow inside] (1.5,0) -- (1.5,pi);
\end{scope}
\begin{scope}[xshift=10cm]
% Axes
\draw (0,0) node[below left] {$O$}
(-4,0) -- (4,0) node[below] {$u$}
(0,-0.5) -- (0,3.5) node[left] {$v$};
% Ticks
\draw (e,0) -- (e,-0.1) node[below] {$\mathrm{e}$}
(-e,0) -- (-e,-0.1) node[below] {$-\mathrm{e}$};
% Transformation
\draw[thick] (3.8,0) --
(e,0) node[above left] {$A'$}
arc[start angle=0, end angle=180, radius=e]
(-e,0) node[above right] {$D'$} --
(-3.8,0);
\draw[arrow inside=0.4] (3.2,0) arc[start angle=0, end angle=180, radius=3.2];
\end{scope}
\end{tikzpicture}
\end{document}
Output
