text:
Q: Tikz coordinates on top of a picture (unit size)? → make into rulers → Make into macro (evenly sized pictures), 1 picture per page only cropped The following code by @cis does something wonderfully, it measures the distance between two points on a picture, and then creates a ruler around the picture.
\documentclass[a4paper, landscape]{article}
\usepackage[margin=2cm, showframe=false]{geometry}
\usepackage{mwe} % Dummy images

\usepackage{tikz}
\usetikzlibrary{calc}
\pgfkeys{/tikz/savevalue/.code 2 args={\global\edef#1{#2}}}

\def\ShowHelps{1} %    1 'yes'        0   'no'
\begin{document}
\centering\begin{tikzpicture}[font=\sffamily,
Help/.style={font=\Huge, red}
]
\node[anchor=south west,inner sep=0pt] (myimage) at (0,0) {
\IfFileExists{cmHgI.jpg}
  {\includegraphics[scale=0.15]{cmHgI.jpg}}
    {\includegraphics[width=0.75\textwidth]{example-image.jpg}}
};
\begin{scope}[x={(myimage.south east)},y={(myimage.north west)}]
% Restrict the draw area
\clip (0,0) rectangle (1,1);
% Frame around image -  optional
\draw[] (0,0) rectangle (1,1); % optional
% CoSy inside node
\newcommand\ShowHelpCoSy{
  \draw[help lines,xstep=.1,ystep=.1] (0,0) grid (1,1);
  \foreach \x in {0,1,...,9} { \node[anchor=north] at (\x/10,0) {0.\x}; }
  \foreach \y in {0,1,...,9} { \node[anchor=east] at (0,\y/10) {0.\y}; }
}
\ifnum\ShowHelps=1 \ShowHelpCoSy \fi%
\ifnum\ShowHelps=1\fill[Help] (0.5,0.75) circle(3pt) node[above]{(0.5,0.75)};\fi% <-- Position Test

% Now the stuff:
\path[] (0.705,0.4) coordinate(U1) -- (0.705,0.519) coordinate(U2);
\ifnum\ShowHelps=1 \draw[Help] (U1) circle(2pt) node[label=left:U1]{} -- (U2) circle(2pt) node[label=left:U2]{};\fi % show unitlength line

\path let \p1=($(U2)-(U1)$) in
\pgfextra{  \pgfmathsetlengthmacro{\unitlength}{veclen(\x1,\y1)}   }
[savevalue={\u}{\unitlength}];
\ifnum\ShowHelps=1 \node[Help] at (0.725,0.25) {unitlength U1U2 = \u}; \fi%<--- show unitlength
\ifnum\ShowHelps=1 \draw[blue, transform canvas={xshift=2mm}] (U1) -- +(0,\u);\fi  % Test

% Small Ruler
\draw[very thick] (0.3,0.35) coordinate(A) -- +(0,3*\u);
\foreach \n in {0,1,...,3}{%%
\draw[very thick] ([yshift=\n*\u]A) -- +(-3mm,0) node[left]{\n}
\ifnum\n=0 node[anchor=west, right=3mm]{cm}\fi;
}%%
\foreach \n in {0.1,0.2,...,3}{%%
\draw[] ([yshift=\n*\u]A) -- +(-1.5mm,0);
}%%

% Image Ruler y
\draw[very thick] (0,0) coordinate(B) -- +(0,10*\u);
\foreach \n in {0,1,...,10}{%%
\draw[very thick] ([yshift=\n*\u]B) -- +(3mm,0) node[right]{
\ifnum\n=0 \else \ifnum\n=1 \n\,cm \else \n \fi\fi};
}%%
\foreach \n in {0.1,0.2,...,10}{%%
\draw[] ([yshift=\n*\u]B) -- +(1.5mm,0);
}%%

% Image Ruler x
\draw[very thick] (0,1) coordinate(C) -- (15*\u,1);
\foreach \n in {0,1,...,15}{%%
\pgfmathsetmacro\Color{\n<9 ? "black" : "white"}
\draw[very thick, \Color] ([xshift=\n*\u]C) -- +(0,-3mm) node[below, text=\Color, inner sep=1pt]{
\ifnum\n=0 \else \ifnum\n=1 \n\,cm \else \n \fi\fi};
}%%
\foreach \n in {0.1,0.2,...,15}{%%
\pgfmathsetmacro\Color{\n<9 ? "black" : "white"}
\draw[\Color] ([xshift=\n*\u]C) -- +(0,-1.5mm);
}%%
\end{scope}
\end{tikzpicture}
\end{document}

My question is: how can we turn this into a macro, or simplify this snippet, such that it can be used over and over and over.
Namely, such that there is:

*

*1 picture per page (nothing else on the page, such that the page is 'cropped' tightly around the picture);

*the distance is measured only on the first picture (only 1 time, the user is going to determine U1 and U2, and the distance between them);

*the size of the virtual rulers should thus be saved, and should be shown on every subsequent picture (this is good, because all of the pictures are sized evenly).

A: I am not sure, whether I understand, but to apply a code to several images you can use a foreach-loop:
\foreach \image/\rulerpos in {
{example-image-a}/{0.3,0.25},%
{example-image-b}/{0.8,0.25},%
{example-image-c}/{0.3,0.4},%
{AnotherPicture}/{0.1,0.1}%
}{
..........
..........
..........
}

First answer:

\documentclass[a4paper, landscape]{article}
\usepackage[margin=2cm, showframe=false]{geometry}
\usepackage{mwe} % Dummy images

\usepackage{tikz}
\usetikzlibrary{calc}
\pgfkeys{/tikz/savevalue/.code 2 args={\global\edef#1{#2}}}

\def\ShowHelps{0} %    1 'yes'        0   'no'
\begin{document}
\foreach \image/\rulerpos in {
{example-image-a}/{0.3,0.25},%
{example-image-b}/{0.8,0.25},%
{example-image-c}/{0.3,0.4},%
{AnotherPicture}/{0.1,0.1}%
}{%%%%%%%%%%%%%%%
\begin{center}
\begin{tikzpicture}[font=\sffamily,
Help/.style={font=\Huge, red}
]
\node[anchor=south west,inner sep=0pt] (myimage) at (0,0) {
\IfFileExists{\image.jpg}
  {\includegraphics[width=0.75\textwidth]{\image.jpg}}
    {\includegraphics[width=0.75\textwidth]{example-image.jpg}}
};
\begin{scope}[x={(myimage.south east)},y={(myimage.north west)}]
% Restrict the draw area
\clip (0,0) rectangle (1,1);
% Frame around image -  optional
\draw[] (0,0) rectangle (1,1); % optional
% CoSy inside node
\newcommand\ShowHelpCoSy{
  \draw[help lines,xstep=.1,ystep=.1] (0,0) grid (1,1);
  \foreach \x in {0,1,...,9} { \node[anchor=north] at (\x/10,0) {0.\x}; }
  \foreach \y in {0,1,...,9} { \node[anchor=east] at (0,\y/10) {0.\y}; }
}
\ifnum\ShowHelps=1 \ShowHelpCoSy \fi%
\ifnum\ShowHelps=1\fill[Help] (0.5,0.75) circle(3pt) node[above]{(0.5,0.75)};\fi% <-- Position Test

% Now the stuff:
\path[] (0.705,0.4) coordinate(U1) -- (0.705,0.519) coordinate(U2);
\ifnum\ShowHelps=1 \draw[Help] (U1) circle(2pt) node[label=left:U1]{} -- (U2) circle(2pt) node[label=left:U2]{};\fi % show unitlength line

\path let \p1=($(U2)-(U1)$) in
\pgfextra{  \pgfmathsetlengthmacro{\unitlength}{veclen(\x1,\y1)}   }
[savevalue={\u}{\unitlength}];
\ifnum\ShowHelps=1 \node[Help] at (0.725,0.25) {unitlength U1U2 = \u}; \fi%<--- show unitlength
\ifnum\ShowHelps=1 \draw[blue, transform canvas={xshift=2mm}] (U1) -- +(0,\u);\fi  % Test

% Small Ruler
\draw[very thick] (\rulerpos) coordinate(A) -- +(0,3*\u);
\foreach \n in {0,1,...,3}{%%
\draw[very thick] ([yshift=\n*\u]A) -- +(-3mm,0) node[left]{\n}
\ifnum\n=0 node[anchor=west, right=3mm]{cm}\fi;
}%%
\foreach \n in {0.1,0.2,...,3}{%%
\draw[] ([yshift=\n*\u]A) -- +(-1.5mm,0);
}%%

% Image Ruler x
\draw[very thick] (0,0) coordinate(B) -- +(0,10*\u);
\foreach \n in {0,1,...,10}{%%
\draw[very thick] ([yshift=\n*\u]B) -- +(3mm,0) node[right]{
\ifnum\n=0 \else \ifnum\n=1 \n\,cm \else \n \fi\fi};
}%%
\foreach \n in {0.1,0.2,...,10}{%%
\draw[] ([yshift=\n*\u]B) -- +(1.5mm,0);
}%%

% Image Ruler y
\draw[very thick] (0,1) coordinate(C) -- (15*\u,1);
\foreach \n in {0,1,...,15}{%%
\pgfmathsetmacro\Color{\n==\n ? "black" : "white"}
\draw[very thick, \Color] ([xshift=\n*\u]C) -- +(0,-3mm) node[below, text=\Color, inner sep=1pt]{
\ifnum\n=0 \else \ifnum\n=1 \n\,cm \else \n \fi\fi};
}%%
\foreach \n in {0.1,0.2,...,15}{%%
\pgfmathsetmacro\Color{\n==\n ? "black" : "white"}
\draw[\Color] ([xshift=\n*\u]C) -- +(0,-1.5mm);
}%%
\end{scope}
\end{tikzpicture}
\end{center}
\newpage
}%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

Second Answer: In this version the images move to the page center.

\documentclass[a4paper, landscape]{article}
\usepackage[margin=2cm, showframe=false]{geometry}
\usepackage{mwe} % Dummy images

\usepackage{tikz}
\usepackage{tikzpagenodes} % current page text area.center
\usetikzlibrary{calc}
\pgfkeys{/tikz/savevalue/.code 2 args={\global\edef#1{#2}}}

\def\ShowHelps{1} %    1 'yes'        0   'no'
\begin{document}
\foreach \image/\rulerpos/\captiontext in {
{example-image-a}/{0.3,0.25},%
%{example-image-b}/{0.8,0.25},%
%{example-image-c}/{0.3,0.4},%
%{cmHgI}/{0.1,0.1},%
{AnotherPicture}/{0.2,0.1}%
}{%%%%%%%%%%%%%%%
%\centering
\begin{tikzpicture}[
remember picture, overlay, % page center 1/3
shift={(current page.center)}, % page center 2/3
font=\sffamily,
Help/.style={font=\Huge, red}
]
\node[anchor=center, %south west,
inner sep=0pt] (myimage) at (0,0) {
\IfFileExists{\image.jpg}
  {\includegraphics[width=0.75\textwidth]{\image.jpg}}
    {\includegraphics[width=0.75\textwidth]{example-image.jpg}}
};
\begin{scope}[x={($2*(myimage.east)$)},y={($2*(myimage.north)$)},
shift={(myimage.south west)} % page center 3/3
]
% Restrict the draw area
\clip (0,0) rectangle (1,1);
% Frame around image -  optional
\draw[] (0,0) rectangle (1,1); % optional
% CoSy inside node
\newcommand\ShowHelpCoSy{
  \draw[help lines,xstep=.1,ystep=.1] (0,0) grid (1,1);
  \foreach \x in {0,1,...,9} { \node[anchor=north] at (\x/10,0) {0.\x}; }
  \foreach \y in {0,1,...,9} { \node[anchor=east] at (0,\y/10) {0.\y}; }
}
\ifnum\ShowHelps=1 \ShowHelpCoSy \fi%
\ifnum\ShowHelps=1\fill[Help] (0.5,0.75) circle(3pt) node[above]{(0.5,0.75)};\fi% <-- Position Test

% Now the stuff:
\path[] (0.705,0.4) coordinate(U1) -- (0.705,0.519) coordinate(U2);
\ifnum\ShowHelps=1 \draw[Help] (U1) circle(2pt) node[label=left:U1]{} -- (U2) circle(2pt) node[label=left:U2]{};\fi % show unitlength line

\path let \p1=($(U2)-(U1)$) in
\pgfextra{  \pgfmathsetlengthmacro{\unitlength}{veclen(\x1,\y1)}   }
[savevalue={\u}{\unitlength}];
\ifnum\ShowHelps=1 \node[Help] at (0.725,0.25) {unitlength U1U2 = \u}; \fi%<--- show unitlength
\ifnum\ShowHelps=1 \draw[blue, transform canvas={xshift=2mm}] (U1) -- +(0,\u);\fi  % Test

% Small Ruler
\draw[very thick] (\rulerpos) coordinate(A) -- +(0,3*\u);
\foreach \n in {0,1,...,3}{%%
\draw[very thick] ([yshift=\n*\u]A) -- +(-3mm,0) node[left]{\n}
\ifnum\n=0 node[anchor=west, right=3mm]{cm}\fi;
}%%
\foreach \n in {0.1,0.2,...,3}{%%
\draw[] ([yshift=\n*\u]A) -- +(-1.5mm,0);
}%%

% Image Ruler x
\draw[very thick] (0,0) coordinate(B) -- +(0,10*\u);
\foreach \n in {0,1,...,10}{%%
\draw[very thick] ([yshift=\n*\u]B) -- +(3mm,0) node[right]{
\ifnum\n=0 \else \ifnum\n=1 \n\,cm \else \n \fi\fi};
}%%
\foreach \n in {0.1,0.2,...,10}{%%
\draw[] ([yshift=\n*\u]B) -- +(1.5mm,0);
}%%

% Image Ruler y
\draw[very thick] (0,1) coordinate(C) -- (15*\u,1);
\foreach \n in {0,1,...,15}{%%
\pgfmathsetmacro\Color{\n==\n ? "black" : "white"}
\draw[very thick, \Color] ([xshift=\n*\u]C) -- +(0,-3mm) node[below, text=\Color, inner sep=1pt]{
\ifnum\n=0 \else \ifnum\n=1 \n\,cm \else \n \fi\fi};
}%%
\foreach \n in {0.1,0.2,...,15}{%%
\pgfmathsetmacro\Color{\n==\n ? "black" : "white"}
\draw[\Color] ([xshift=\n*\u]C) -- +(0,-1.5mm);
}%%
\end{scope}
\end{tikzpicture}
%\vspace*{\fill}
\newpage
}%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}


text:
Q: How plot a bifurcation diagram ? or show find bifurcation points I have a function
$rx(3-x^2)$
How do I find the points it bifurcates and what does it mean ?
I know how to find fixed points and check them for stability, how can I use that to answer this question ?

A: Answer to first question (in case where x is a 1D variable ):

How plot a bifurcation diagram ?

*

*for r from $r_{min}$ to $r_{max}$ compute  $i_{max}$ iterations : $x_{i+1} =
f_r(x_i) = rx_i(3-x_i^2)$. Then you have a list of 2D points : [r,x], so you can plot it on the 2D diagram with r on horizontal axis and x on vertical one.

*Make new diagram : try to clear diagram by removing preperiodic points = first make some iterations without drawing it, then make more iterations and draw each point

More is here ( images , code )
HTH

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