Coordinate Transformations

\tikzmath{\a=1;};
\tikzmath{\b=0.5;};

\tikzmath{\aa=2;};
\tikzmath{\bb=-1;};

\coordinate (A) at (0,0);
\coordinate (B) at (1,0);
\coordinate (C) at (1,1);
\coordinate (D) at (0,1);


\coordinate (T) at (-1,1);

\newcommand{\Draw}[1]{
}


\filldraw (A) circle(1pt);
\filldraw (B) circle(1pt);
\filldraw (C) circle(1pt);
\filldraw (D) circle(1pt);
\filldraw (T) circle(2pt);

%\draw[style=help lines,thin,color=lightgray] (0,0) grid (2*\a,2*\b);


%\draw (A) rectangle (\a,\b);

%\draw[rotate=45] [orange] (A) rectangle (\a,\b);
%\draw[xshift=1 cm] [orange] (A) rectangle (\a,\b);
%\draw[xshift=5 cm] [orange] (-3,3) rectangle (\a,\b);

%\draw[xscale=1] [brown] (A) rectangle (\a,\b);
%\draw[yscale=2] [brown] (A) rectangle (\a,\b);

%\draw[scale=3] [brown] (A) rectangle (\a,\b);

%\draw
%   [rotate around={60:(-2,-2)}] [magenta]
%   (A)
%   ellipse [x radius=\a, y radius=\b];
   
%\draw
%   [red]
%   (A)
%   ellipse [x radius=\a, y radius=\b];

%\draw
%   [   cm={1,0,0,1,(2,3)}   ] [yellow]
%   (\b,\b) -- (\b+1,\b+1) -- (\b+1,\b+0);

   
%\draw
%   [blue]
%   (C) circle (\b);
%\draw
%   [shx={(0cm,1cm)},y={(1cm,0cm)]   [blue]
%   (C) circle (\b);


\draw
   [red]
   (B)
   ellipse [x radius=\a, y radius=\b];

\draw
   [x={(0 cm,1 cm)},y={(1 cm,0 cm)},blue]
   (B)
   ellipse [x radius=\a, y radius=\b];

\begin{scope}
   [rotate=120]
   \draw
       [x={(0 cm,1 cm)},y={(1 cm,0 cm)},blue]
        [green]
       (B)
       ellipse [x radius=\a, y radius=\b];

\end{scope}