8

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....

enter image description here

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}
3

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.)

enter image description here

\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}

enter image description here

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

  • Dear marmot and J Leon v., Thanks for the help. This is exactly what I am looking for !! The transformations are basically lens and drift matrixes in paraxial beam approximation, see here en.wikipedia.org/wiki/Ray_transfer_matrix_analysis. They do not commute as you prove. And both transformations have determinant 1 indicating that the area of the ellipse is not changed. – jfrans1 Jun 23 '18 at 10:09
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: enter image description here

RESULT: enter image description here

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.

  • Nice animations. I originally claimed that these are nonlinear transformations, but this was wrong. Unfortunately, I do not think that your transformations are entirely correct. The OP says that " I want to draw some ellipses and then transform the ellipse without changing its area. ", and I believe that her or his transformations do not change the area as they have determinant 1. Your transformations do change the area. What am I missing? – user121799 Jun 23 '18 at 1:21
  • ;-) Please feel free ;-) I'll remove my comment ;-) – user121799 Jun 23 '18 at 3:30
  • Of course, my answer do not take care of the area and maths, I only found this modifier for transformation shapes that respects the blue lines relative to the shapes and then make a code to test it, an then ask for corrections like you done. So I can improve it using some code from other great answers and explanations like yours and from the ducks examples link. – J Leon V. Jun 23 '18 at 3:41
  • Dear marmot and J Leon v., Thanks for the help. This is exactly what I am looking for !! The transformations are basically lens and drift matrix transformations in paraxial beam approximation, see here en.wikipedia.org/wiki/Ray_transfer_matrix_analysis. They do not commute as you prove. And both transformations have determinant 1 indicating that the area of the ellipse is not changed. The problem is solved so I suppose we can close the topic and mark it as solved. Thanks for the help ! Cheers, Jim – jfrans1 Jun 23 '18 at 10:15

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