Scale, rotate and transform ellipse in TikZ picture

Question

I want to draw some ellipses and then transform the ellipse without changing its area.

I want to be able to transform the ellipse according to the following two equations;

First transformation:

∆y = constant * x and ∆x = 0

where x is the horizontal position of the coordinate and ∆y the change of the vertical (y) position

Second transformation

∆x = constant * y and ∆y = 0

where y is the vertical position of the coordinate and ∆x the change of the horizontal (x) position

In the first picture I start with an horizontal ellipse (see attached figure) which contains some red and blue lines which also need to be transformed.

This ellipse is transformed using the first transformation (got this working...)

I cannot get the second transformation to work (see Image 3...) I use a rotation but this is not correct because it does not hold equation 2, I do not know how to use scope for this...

The fourth image is again the first transformation but this does not want to work anymore....

Can anyone help ?

Thanks.

here is my latex code

\documentclass[11pt]{article}
\usepackage{tikz} 
\usepackage{verbatim}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{5pt}%
\usetikzlibrary{calc}


\begin{document}



\newcommand{\addphasespace}[5]{
\begin{scope}[xshift=#1]
\draw (-1.5,-1.5) rectangle (1.5,1.5);
\draw (-1.5,0) -- (1.5,0);
\draw (0,-1.5) -- (0,1.5);
\node[inner sep=1pt,font=\large] at (0,-1.75) {$x$};
\node[inner sep=1pt,font=\large] at (1.75,0) {$y$};
\node[inner sep=1pt,font=\large] at (0,1.9) {#5};
\begin{scope}[#4]
\fill[black!20,yshift=0cm] (0,0) circle (#2 and #3);
\draw[yshift=0cm,red] (-#2,0) -- (#2,0);
\draw[yshift=0cm,blue] (0,-#3) -- (0,#3);
\draw[yshift=0cm,blue] (0.1,-0.195) -- (0.1,0.195);
\draw[yshift=0cm,blue] (-0.1,-0.195) -- (-0.1,0.195);
\draw[yshift=0cm,blue] (0.2,-0.19) -- (0.2,0.19);
\draw[yshift=0cm,blue] (-0.2,-0.19) -- (-0.2,0.19);
\draw[yshift=0cm,blue] (0.3,-0.18) -- (0.3,0.18);
\draw[yshift=0cm,blue] (-0.3,-0.18) -- (-0.3,0.18);
\draw[yshift=0cm,blue] (0.4,-0.17) -- (0.4,0.17);
\draw[yshift=0cm,blue] (-0.4,-0.17) -- (-0.4,0.17);
\draw[yshift=0cm,blue] (0.5,-0.158) -- (0.5,0.158);
\draw[yshift=0cm,blue] (-0.5,-0.158) -- (-0.5,0.158);
\draw[yshift=0cm,blue] (0.6,-0.135) -- (0.6,0.135);
\draw[yshift=0cm,blue] (-0.6,-0.135) -- (-0.6,0.135);
\draw[yshift=0cm,blue] (0.7,-0.1) -- (0.7,0.1);
\draw[yshift=0cm,blue] (-0.7,-0.1) -- (-0.7,0.1);
\draw[yshift=0cm] (0,0) circle (#2 and #3);
\end{scope}
\end{scope}
}

%%%%%%%%%%%%%%%


\begin{figure}
\begin{tikzpicture}
\draw[help lines] (-3,-3) grid (15,3);

% area ellipse  = pi*0.8*0.2 = 0.5026

%Plot1
\addphasespace{0cm}{0.8}{0.2}{}{1}

%Plot2
\draw[->] (4.8,0) -- (4.8,-0.85);
\draw[->] (3.2,0) -- (3.2,0.85);
\addphasespace{4cm}{0.8}{0.2}{opacity = 0.3}{2}
\addphasespace{4cm}{0.8}{0.2}{x={(1,-1.3)}}{2}


%%Plot3
\draw[->] (7.2,1.05) -- (7.9,1.05);
\draw[->] (8.8,-1.05) -- (8.1,-1.05);
\addphasespace{8cm}{0.8}{0.2}{opacity = 0.3,x={(1,-1.3)}}{3}
\addphasespace{8cm}{0.8}{0.2}{x={(1,-1.3)},rotate=-37.57}{3}


%%Plot3
\draw[->]  (12.82,1.05) -- (12.82,0.1);
\draw[->]  (11.18,-1.05) -- (11.18,0.1);
\addphasespace{12cm}{0.8}{0.2}{opacity = 0.3 ,x={(1,-1.3)},rotate=-75.14}{4}
\addphasespace{12cm}{0.8}{0.2}{x={(1,-1.3)},rotate=-127.57,x={(1,-1.5)}}{4}





\end{tikzpicture}
\end{figure}

\end{document}

Answer 1

You are right when you say that the second transformation does not match its description, yet strictly speaking that's already the case for the first one. (Note: in a previous version I was claiming that the transformations are nonlinear, which they are not, but the transformations I implemented were those following your prescription. They coincide with what follows, except that the following is simpler, of course.)

\documentclass[11pt]{article}
\usepackage[top=0.7in,bottom=0.7in,left=0.5in,right=0.5in]{geometry}
\usepackage{tikz} 
\usepackage{mathtools}
% \usepackage[active,tightpage]{preview}
% \PreviewEnvironment{tikzpicture}
% \setlength\PreviewBorder{5pt}%
\usetikzlibrary{calc}
\newcommand{\MyConst}{-0.8}




\newcommand{\addphasespace}[2]{
\begin{scope}[xshift=#1]
\draw (-1.5,-1.5) rectangle (1.5,1.5);
\draw (-1.5,0) -- (1.5,0);
\draw (0,-1.5) -- (0,1.5);
\node[inner sep=1pt,font=\large] at (0,-1.75) {$x$};
\node[inner sep=1pt,font=\large] at (1.75,0) {$y$};
\node[inner sep=1pt,font=\large] at (0,1.9) {#2};
\end{scope}
}

\newcommand{\drawellipse}[4][]{
\begin{scope}[#4]
\fill[black!20,yshift=0cm] (0,0) circle (#2 and #3);
\draw[yshift=0cm,red] (-#2,0) -- (#2,0);
\foreach \X in {0,0.1,...,#2}
{
\pgfmathsetmacro{\Y}{#3*sin(acos(\X/#2))}
\draw[yshift=0cm,blue] (\X,-\Y) -- (\X,\Y);
\draw[yshift=0cm,blue] (-\X,-\Y) -- (-\X,\Y);
}
\draw[yshift=0cm] (0,0) circle (#2 and #3);
\end{scope}
}

\newcommand{\trafoone}[1]{
\pgftransformcm{1}{#1}{0}{1}{\pgfpoint{0pt}{0pt}}
}
\newcommand{\trafotwo}[1]{
\pgftransformcm{1}{0}{#1}{1}{\pgfpoint{0pt}{0pt}}
}
\newcommand{\trafooneaftertwo}[2]{
\pgftransformcm{1}{#1}{#2}{1+#1*#2}{\pgfpoint{0pt}{0pt}}
}
\newcommand{\trafotwoafterone}[2]{
\pgftransformcm{1+#1*#2}{#2}{#1}{1}{\pgfpoint{0pt}{0pt}}
}

%%%%%%%%%%%%%%%

\begin{document}


You are looking at transformations of the type
\[
\begin{pmatrix} x\\ y\end{pmatrix}\xmapsto{~f_1~}
\begin{pmatrix} x\\ y+c_1\, x\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix} x\\ y\end{pmatrix}\xmapsto{~f_2~}
\begin{pmatrix} x+c_2\,y\\ y\end{pmatrix}\;,
\]
which can be written in matrix form as
\[
\begin{pmatrix} x\\ y\end{pmatrix}\xmapsto{~f_1~}
\underbrace{\begin{pmatrix} 1 & 0\\ c_1 & 1\end{pmatrix}}_{=A_1}\cdot
\begin{pmatrix} x\\ y\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix} x\\ y\end{pmatrix}\xmapsto{~f_2~}
\underbrace{\begin{pmatrix} 1 & c_2\\ 0 & 1\end{pmatrix}}_{=A_2}\cdot
\begin{pmatrix} x\\ y\end{pmatrix}\;.
\]
These transformations do not commute,
\[
 A_1\cdot A_2=
 \begin{pmatrix} 1 & c_2\\ c_1 & 1+c_1\,c_2\end{pmatrix}
 \ne
 \begin{pmatrix} 1+c_1\,c_2 & c_2\\ c_1 & 1\end{pmatrix}=A_2\cdot A_1\;.
\]
You can implement these transformations with \verb|\pgftransformcm|, which is
equivalent to \texttt{[cm=\dots]} in a scope. This version comes with four commands
\[ \verb|\trafoone{#1}|\;,\quad\verb|\trafotwo{#1}|\;,\quad
\verb|\trafooneaftertwo{#1}{#2}|\quad\text{and}\quad \verb|\trafotwoafterone{#1}{#2}|\;,
\] 
which correspond to the transformation matrices
\[A_1\;,\quad A_2\;,\quad A_1\cdot A_2 \quad\text{and}\quad
A_2\cdot A_1\;, %@DavidCarlisle @barbarabeeton please don't kill me for the spaces ;-)
\]
respectively. See Figure \ref{fig:trafos} for some examples, which are chosen to resemble
your screenshot.
\begin{figure}
% @egreg please don't kill me for using \centerline ;-)
\centering
\begin{tikzpicture}
\draw[help lines] (-3,-3) grid (15,3);
% area ellipse  = pi*0.8*0.2 = 0.5026
%Plot1
\addphasespace{0cm}{1}
\drawellipse{0.8}{0.2}{}
%Plot2
\addphasespace{4cm}{2}
\begin{scope}[xshift=4cm]
\drawellipse{0.8}{0.2}{opacity = 0.3}
\begin{scope}
\trafoone{\MyConst}
\drawellipse{0.8}{0.2}{}
\end{scope}
\end{scope}
%%Plot3
\addphasespace{8cm}{3}
\begin{scope}[xshift=8cm]
\trafoone{\MyConst}
\drawellipse{0.8}{0.2}{opacity = 0.3}
\begin{scope}
\trafotwoafterone{\MyConst}{-\MyConst}
\drawellipse{0.8}{0.2}{};
\end{scope}
\end{scope}
%%Plot4
\addphasespace{12cm}{4}
\begin{scope}[xshift=12cm]
\trafotwo{\MyConst}
\drawellipse{0.8}{0.2}{opacity = 0.3}
\begin{scope}
\trafooneaftertwo{-\MyConst}{\MyConst}
\drawellipse{0.8}{0.2}{};
\end{scope}
\end{scope}
\end{tikzpicture}
\caption{$f_1$, $f_2$, $f_2\circ f_1$ and $f_1\circ f_2$.}
\label{fig:trafos}
\end{figure}
\end{document}

And an animation, like in @J Leon V.'s answer.

\documentclass[tikz,border=3.14]{standalone}


\newcommand{\addphasespace}[2]{
\begin{scope}[xshift=#1]
\draw (-1.5,-1.5) rectangle (1.5,1.5);
\draw (-1.5,0) -- (1.5,0);
\draw (0,-1.5) -- (0,1.5);
\node[inner sep=1pt,font=\large] at (0,-1.75) {$x$};
\node[inner sep=1pt,font=\large] at (1.75,0) {$y$};
\node[inner sep=1pt,font=\large] at (0,1.9) {#2};
\end{scope}
}

\newcommand{\drawellipse}[4][]{
\begin{scope}[#4]
\fill[black!20,yshift=0cm] (0,0) circle (#2 and #3);
\draw[yshift=0cm,red] (-#2,0) -- (#2,0);
\foreach \X in {0,0.1,...,#2}
{
\pgfmathsetmacro{\Y}{#3*sin(acos(\X/#2))}
\draw[yshift=0cm,blue] (\X,-\Y) -- (\X,\Y);
\draw[yshift=0cm,blue] (-\X,-\Y) -- (-\X,\Y);
}
\draw[yshift=0cm] (0,0) circle (#2 and #3);
\end{scope}
}

%%%%%%%%%%%%%%%
\begin{document}
\foreach \MyConst [count=\Z] in {0,0.1,...,2}
{\begin{tikzpicture}
\draw[help lines] (-3,-3) grid (15,3);
% area ellipse  = pi*0.8*0.2 = 0.5026
%Plot1
\addphasespace{0cm}{1}
\drawellipse{0.8}{0.2}{}
%Plot2
\addphasespace{4cm}{2}
\begin{scope}[xshift=4cm]
\drawellipse{0.8}{0.2}{opacity = 0.3}
\begin{scope}
\pgftransformcm{1}{\MyConst}{0}{1}{\pgfpoint{0pt}{0pt}}
\drawellipse{0.8}{0.2}{}
\coordinate (l1) at (-0.8,0);
\coordinate (r1) at (0.8,0);
\end{scope}
\ifnum\Z=1
\else
\draw[-latex] (-0.8,0) -- (l1);
\draw[-latex] (0.8,0) -- (r1);
\fi
\end{scope}
%%Plot3
\addphasespace{8cm}{3}
\begin{scope}[xshift=8cm]
\drawellipse{0.8}{0.2}{opacity = 0.3}
\begin{scope}
\pgftransformcm{1}{0}{\MyConst}{1}{\pgfpoint{0pt}{0pt}}
\drawellipse{0.8}{0.2}{};
\coordinate (l2) at (-0.8,0);
\coordinate (r2) at (0.8,0);
\end{scope}
\end{scope}
%%Plot4
\addphasespace{12cm}{4}
\begin{scope}[xshift=12cm]
\drawellipse{0.8}{0.2}{opacity = 0.3}
\begin{scope}
\pgftransformcm{1}{\MyConst}{\MyConst}{{1+\MyConst*\MyConst}}{\pgfpoint{0pt}{0pt}}
\drawellipse{0.8}{0.2}{};
\coordinate (l3) at (-0.8,0);
\coordinate (r3) at (0.8,0);
\end{scope}
\ifnum\Z=1
\else
\draw[-latex] (-0.8,0) -- (l3);
\draw[-latex] (0.8,0) -- (r3);
\fi
\end{scope}
\end{tikzpicture}}
\end{document}

As you can see, the transformations do preserve the area, as they should (since they have determinant 1).

Answer 2

Although it is not very clear to me, one way to change and scale according to the arrows that you place, is using the option /tikz/cm={<a>,<b>,<c>,<d>,<coordinate>}, the first figure simply rotating the shape without changing the area, the following are scaling according to the arrows, let me know if it is what you are looking for, you can experiment with the code.

UPDATE: A modification to visualize an arbitrary variation in the area, which simulates that it conserves the constant area.

%Plot1
\addphasespace{0cm}{0.8}{0.2}{rotate=360/20*\X}{1}
%Plot2
\draw[->] (4.8,0) -- ++(0,-0.85);
\draw[->] (3.2,0) -- ++(0,0.85);
\addphasespace{4cm}{0.8}{0.2}{opacity = 0.3}{2}
\addphasespace{4cm}{0.8}{0.2}{cm={1,-1*\X/20,0.5*\X/20,1-0.5*\X/20,(0,0)}}{2}
%Plot3
\draw[->] (7.2,0.8) -- ++(0.85,0);
\draw[->] (8.8,-0.8) -- ++(-0.85,0);
\addphasespace{8cm}{0.8}{0.2}{opacity = 0.3,cm={1,-1,0.5,0.5,(0,0)}}{3}
\addphasespace{8cm}{0.8}{0.2}{cm={1-1*\X/20,-1,0.5+0.5*\X/20,0.5-0.5*\X/20,(0,0)}}{3}
%Plot4
\draw[->]  (12.82,0.8) -- ++(0,-0.85);
\draw[->]  (11.18,-0.8) -- ++(0,0.85);
\addphasespace{12cm}{0.8}{0.2}{opacity = 0.3,cm={1,1,-0.5,0.5,(0,0)}}{4}
\addphasespace{12cm}{0.8}{0.2}{cm={1,1-1*\X/20,-0.5+0.5*\X/20,0.5+0.5*\X/20,(0,0)}}{4}

UPDATE RESULT:

RESULT:

MWE:

% arara: pdflatex: {synctex: yes, action: nonstopmode}
% arara: animate: {density: 150, delay: 8, other: -background white -alpha remove}
% arara: showanimate

\documentclass[tikz,border=1pt]{standalone}
\usepackage{tikz} 
\usetikzlibrary{calc}
\begin{document}
    \foreach \X in {1,...,20}
    {\begin{tikzpicture}
        \newcommand{\addphasespace}[5]{
        \begin{scope}[xshift=#1]
        \draw (-1.5,-1.5) rectangle (1.5,1.5);
        \draw (-1.5,0) -- (1.5,0);
        \draw (0,-1.5) -- (0,1.5);
        \node[inner sep=1pt,font=\large] at (0,-1.75) {$x$};
        \node[inner sep=1pt,font=\large] at (1.75,0) {$y$};
        \node[inner sep=1pt,font=\large] at (0,1.9) {#5};
        \begin{scope}[#4]
        \fill[black!20] (0,0) circle (#2 and #3);
        \draw[red] (-#2,0) -- (#2,0);
        \draw[blue] (0,-#3) -- (0,#3);
        \draw[blue] (0.1,-0.195) -- (0.1,0.195);
        \draw[blue] (-0.1,-0.195) -- (-0.1,0.195);
        \draw[blue] (0.2,-0.19) -- (0.2,0.19);
        \draw[blue] (-0.2,-0.19) -- (-0.2,0.19);
        \draw[blue] (0.3,-0.18) -- (0.3,0.18);
        \draw[blue] (-0.3,-0.18) -- (-0.3,0.18);
        \draw[blue] (0.4,-0.17) -- (0.4,0.17);
        \draw[blue] (-0.4,-0.17) -- (-0.4,0.17);
        \draw[blue] (0.5,-0.158) -- (0.5,0.158);
        \draw[blue] (-0.5,-0.158) -- (-0.5,0.158);
        \draw[blue] (0.6,-0.135) -- (0.6,0.135);
        \draw[blue] (-0.6,-0.135) -- (-0.6,0.135);
        \draw[blue] (0.7,-0.1) -- (0.7,0.1);
        \draw[blue] (-0.7,-0.1) -- (-0.7,0.1);
        \draw(0,0) circle (#2 and #3);
        \end{scope}
        \end{scope}
    }
    %%%%%%%%%%%%%%%
    \draw[help lines] (-3,-3) grid (15,3);
    % area ellipse  = pi*0.8*0.2 = 0.5026
    %Plot1
    \addphasespace{0cm}{0.8}{0.2}{rotate=360/20*\X}{1}
    %Plot2
    %Plot2
    \draw[->] (4.8,0) -- ++(0,-0.85);
    \draw[->] (3.2,0) -- ++(0,0.85);
    \addphasespace{4cm}{0.8}{0.2}{opacity = 0.3}{2}
    \addphasespace{4cm}{0.8}{0.2}{cm={1,-1*\X/20,1*\X/20,1,(0,0)}}{2}


    %%Plot3
    \draw[->] (7.2,0.8) -- ++(0.85,0);
    \draw[->] (8.8,-0.8) -- ++(-0.85,0);
    \addphasespace{8cm}{0.8}{0.2}{opacity = 0.3,cm={1,-1,1,1,(0,0)}}{3}
    \addphasespace{8cm}{0.8}{0.2}{cm={1-1*\X/20,-1,1,1-1*\X/20,(0,0)}}{3}


    %%Plot4
    \draw[->]  (12.82,0.8) -- ++(0,-0.85);
    \draw[->]  (11.18,-0.8) -- ++(0,0.85);
    \addphasespace{12cm}{0.8}{0.2}{opacity = 0.3,cm={1,1,-1,1,(0,0)}}{4}
    \addphasespace{12cm}{0.8}{0.2}{cm={1,1-1*\X/20,-1+\X/20,1,(0,0)}}{4}
    \end{tikzpicture}}
\end{document}

PSD: For animation I use imagemagic, that converts pdf files to gif documents.