How can I draw nomograms in LaTeX?

Question

What is a nomogram

“A nomogram, also called a nomograph, alignment chart or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a function.”

Example

The rocket equation, given at the top of this nomogram, can be solved by drawing a line from the scale on the left to the scale on the right. The solution can be read off from the scale in the middle.

Question

How can I reproduce the figure above in LaTeX? Is there a TikZ approach?

Bonus question

How can I create arbitrary nomograms with LaTeX?

Additional info

The figure above has been created with: http://pynomo.org/wiki/index.php?title=Main_Page

More about nomography: http://www.slideshare.net/arulalan/nomography

Answer 1

Disclaimer

Take this as a starting point: there are a few things to be done to make the solution really working in all cases. Namely:

• handle cases where remainder(max/min)!=0
• handle min!=0
• handle scale from max to min besides min to max

A possibility:

\documentclass[tikz,border=10pt]{standalone}
\usepackage{pdftexcmds,etoolbox}


\makeatletter
\newif\ifroundedenabled%
\newif\ifscalemaxtomin%
\newif\ifscalefromzero%
\pgfkeys{/nomogram/.cd,
  % keys for a single diagram
  part 1/.style={},
  part 2/.style={},
  part 3/.style={},
  single diagram/.cd,
  scale max to min/.is if=scalemaxtomin,
  scale max to min=false,
  scale from zero/.is if=scalefromzero,
  scale from zero=true,
  at pos/.store in=\dgrposition,
  at pos={(0,0)},
  height/.store in=\dgrheight,
  height=10cm,
  min value/.store in=\minval,
  min value=0,
  max value/.store in=\maxval,
  max value=10,
  step/.store in=\incstep,
  step=1,
  horizontal rule width/.store in=\horulewidth,
  horizontal rule width=1cm,
  min step/.store in=\minstep,
  min step={\incstep/2},
  minor tick rule width/.store in=\minortickrulewidth,
  minor tick rule width=3mm,
  little tick rule width/.store in=\litteltickrulewidth,
  little tick rule width=1mm,
  tick direction/.store in=\tickpos,
  tick direction=left,
  label above/.store in=\lababove,
  label above={},
  label sloped/.store in=\labsloped,
  label sloped={},
  label above rotation/.store in=\rotation,
  label above rotation=0,
  diagram/.code={
    \path \dgrposition node(A){};
    \pgfgetlastxy{\xA}{\yA};
    \ifscalemaxtomin%
      \pgfmathsetmacro\sndval{\maxval-\incstep}
      \foreach \y[count=\yi from 0] in {\maxval,\sndval,...,\minval}{
        \global\let\maxitems\yi%
      }%
    \else%
      \pgfmathsetmacro\sndval{\minval+\incstep}
      \foreach \y[count=\yi from 0] in {\minval,\sndval,...,\maxval}{
        \global\let\maxitems\yi%
      }%
    \fi  
    \draw(\xA,\yA)--++(0,\dgrheight)
     node[pos=0.5,sloped,above]{\labsloped}
     node[above,rotate=\rotation,transform shape]{\lababove};% vertical line+above label
    \pgfmathsetmacro\actualstep{\dgrheight/\maxitems}%
        % little ticks
        \pgfmathsetmacro\littletickstep{\actualstep/10}
    \foreach \y in {0,\littletickstep,...,\dgrheight}{
     \ifnum\pdf@strcmp{\tickpos}{left}=\z@%
       \draw(\xA,\yA+\y pt) --++(-\litteltickrulewidth,0);
     \fi%
     \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
        \draw(\xA,\yA+\y pt) --++(\litteltickrulewidth,0);
     \fi%
    }%
    % min step
    \pgfmathsetmacro\mintickstep{\actualstep/2}
    \ifscalemaxtomin%
      \foreach \y [count=\yi from 0,
             evaluate=\yi as \ytext using ((\maxval-\yi*\incstep+\yi*\incstep/2))]
              in {0,\mintickstep,...,\dgrheight}{%
        \ifnumodd{\yi}{% true
        \ifnum\pdf@strcmp{\tickpos}{left}=\z@%  
          \draw (\xA,\yA+\y pt) --++(-\minortickrulewidth,0) 
            node[left,font=\footnotesize]{\pgfmathprintnumber{\ytext}};
        \fi%
        \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
          \draw (\xA,\yA+\y pt) --++(\minortickrulewidth,0) 
            node[right,font=\footnotesize]{\pgfmathprintnumber{\ytext}};
        \fi%
        }{}%
      }%
    \else%
      \foreach \y [count=\yi from 0,
         evaluate=\yi as \ytext using ((\yi*\incstep+2*\minval)/2)] in 
         {0,\mintickstep,...,\dgrheight}{%
        \ifnumodd{\yi}{% true
          \ifnum\pdf@strcmp{\tickpos}{left}=\z@%  
            \draw (\xA,\yA+\y pt) --++(-\minortickrulewidth,0) 
              node[left,font=\footnotesize]{\pgfmathprintnumber{\ytext}};
          \fi%
         \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
            \draw (\xA,\yA+\y pt) --++(\minortickrulewidth,0) 
              node[right,font=\footnotesize]{\pgfmathprintnumber{\ytext}};
         \fi%
        }{}%
      }%
    \fi%  
    % main step
    \ifscalemaxtomin%
      \ifscalefromzero%
          \foreach \y[count=\yi from 0,
              evaluate=\yi as \ytext using (\maxval-\yi*\incstep)] in 
              {0,\actualstep,...,\dgrheight}{%
            \ifnum\pdf@strcmp{\tickpos}{left}=\z@%  
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(-\horulewidth,0) 
                node[left]{\pgfmathprintnumber{\ytext}};
            \fi%
            \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(\horulewidth,0) 
                node[right]{\pgfmathprintnumber{\ytext}};
            \fi%
              }%
      \else%
          \foreach \y[count=\yi from 0,
              evaluate=\yi as \ytext using (\maxval-\yi*\incstep)] in 
              {0,\actualstep,...,\dgrheight}{%
            \ifnum\pdf@strcmp{\tickpos}{left}=\z@%  
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(-\horulewidth,0) 
                node[left]{\pgfmathprintnumber{\ytext}};
            \fi%
            \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(\horulewidth,0) 
                node[right]{\pgfmathprintnumber{\ytext}};
            \fi%
              }%
          \fi%
    \else%
      \ifscalefromzero%
          \foreach \y[count=\yi from 0,
              evaluate=\yi as \ytext using (\yi*\sndval+\minval)] in 
              {0,\actualstep,...,\dgrheight}{%
            \ifnum\pdf@strcmp{\tickpos}{left}=\z@%  
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(-\horulewidth,0) 
                node[left]{\pgfmathprintnumber{\ytext}};
            \fi%
            \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(\horulewidth,0) 
                node[right]{\pgfmathprintnumber{\ytext}};
            \fi%        
              }%
      \else%
          \foreach \y[count=\yi from 0,
              evaluate=\yi as \ytext using (\yi*\incstep+\minval)] in 
              {0,\actualstep,...,\dgrheight}{%
            \ifnum\pdf@strcmp{\tickpos}{left}=\z@%  
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(-\horulewidth,0) 
                node[left]{\pgfmathprintnumber{\ytext}};
            \fi%
            \ifnum\pdf@strcmp{\tickpos}{right}=\z@%
              \pgfkeys{/pgf/number format/.cd,fixed,precision=2}
              \draw (\xA,\yA+\y pt) --++(\horulewidth,0) 
                node[right]{\pgfmathprintnumber{\ytext}};
            \fi%        
              }%
          \fi%
    \fi%
  }%
}

% that's just an alias for \node
\def\drawnomogrampart{\tikz@path@overlay{node}}
\makeatother

\begin{document}

\begin{tikzpicture}
\drawnomogrampart[/nomogram/single diagram/.cd,
height=7cm,
min value=50,
max value=130,
step=10,
label above=$\Delta v$,
scale from zero=false,
diagram]{};
\begin{scope}[rotate=-30,transform shape]
\drawnomogrampart[/nomogram/single diagram/.cd,
horizontal rule width=0.5cm,
height=8cm,
min value=0,
max value=0.901,% for rounding purposes
step=0.1,
label above rotation=30,
label above={$M_f=1-\textrm{e}^{-\Delta v/9.81*I_{sp}}$},
label sloped={Mass Fraction $(M_f)$},
tick direction=right,
diagram]{};
\end{scope}
\drawnomogrampart[/nomogram/single diagram/.cd,
scale from zero=false,
height=7cm,
scale max to min=true,
min value=400,
max value=2000,
step=200,
at pos={(7,0)},
tick direction=right,
label above=IPS (s),
diagram]{};
\end{tikzpicture}
\end{document}

The result:

Answer 2

Due to precision problems with calculation with TikZ, I change the approach and use a small pearl script to do this calculations. Run with pdflatex -shell-script

I use decorations.markings library to do graduations on every type of path. (No more accumulations of marks problem).

I tried to put some explainations in the code.

The code :

\documentclass[tikz,margin=2pt]{standalone}
\usetikzlibrary{calc}
\usetikzlibrary{decorations.markings,intersections}


    %%%%                        ---- Use path several times
    %%%%                        ---- thanks to Andrew Stacey
    \makeatletter
    \tikzset{
      use path for main/.code={%
        \tikz@addmode{%
          \expandafter\pgfsyssoftpath@setcurrentpath\csname tikz@intersect@path@name@#1\endcsname
        }%
      },
      use path for actions/.code={%
        \expandafter\def\expandafter\tikz@preactions\expandafter{\tikz@preactions\expandafter\let\expandafter\tikz@actions@path\csname tikz@intersect@path@name@#1\endcsname}%
      },
      use path/.style={%
        use path for main=#1,
        use path for actions=#1,
      }
    }

\pgfkeys{/MonoG/.cd,
    % foot of the scale
    path/.store in=\p@th,
    path={(0,0)--(0,10)},
    % minimale value
    min/.store in=\Min,
    min=0,
    % maximale value 
    max/.store in=\Max,
    max=10,
    % Major graduation Step
    step/.store in=\Step,
    step=1,
    % Big ticks
    % tick length
    big ticks/.store in=\BTick,
    big ticks=.7,
    % font (whatever compatible with tikz font parameter)
    big ticks font/.store in=\BFont,
    big ticks font=\small,
    % line width
    big ticks width/.store in=\BWidth,
    big ticks width=thick,
    % medium ticks (same choices)
    med ticks/.store in=\MTick,
    med ticks=.4,
    med ticks font/.store in=\MFont,
    med ticks font=\footnotesize,
    med ticks width/.store in=\MWidth,
    med ticks width=thin,
    % small ticks
    small ticks/.store in=\STick,
    small ticks=.2,
    small ticks width/.store in=\SWidth,
    small ticks width=very thin,
    % number of sublevel of small tick
    small ticks level/.store in=\STlevel,
    small ticks level=1,
    % tick direction
    tick direction/.store in=\TickPos,
    tick direction=right,
    % bottom label
    bottom label/.store in=\BotLabel,
    bottom label={},
    % midway label
    midway label/.store in=\MidLabel,
    midway label={},
    % top label
    top label/.store in=\TopLabel,
    top label={},
    % Reverse graduation    
    reverse/.store in=\Reverse,
    reverse=false,
    diagram/.code={%

    % draw the path and place the nodes for labelling
    \draw[\BWidth,name path=Path] \p@th 
        node[pos=1,above] (MnAbove) {\TopLabel}
        node[midway,sloped,above] (MnMid) {\MidLabel}
        node[pos=0,below] (MnBelow) {\BotLabel} ;

    % left right position flag
    \ifnum\pdf@strcmp{\TickPos}{left}=\z@%
        \def\TPos{1}
    \fi%
    \ifnum\pdf@strcmp{\TickPos}{right}=\z@%
        \def\TPos{-1}
    \fi%

    % reverse scale flag    
    \def\Rev{1}
    \ifnum\pdf@strcmp{\Reverse}{true}=\z@%
        \def\Rev{-1}
    \fi%

% set the style for the tick labels
\tikzset{Big/.style={font=\BFont,\TickPos,transform shape,rotate=-90},
    Med/.style={font=\MFont,\TickPos,transform shape,rotate=-90},
    /pgf/decoration/reset marks % Reset marks each time
        % to avoid accumulation problems
    } ;

% Call of external calculation script
    \immediate\write18{%
    ./script.pl     \Min\space%
        \Max\space%
        \Step\space%
        \STlevel\space\STick\space > Sortie.tex}

    \input{Sortie}
    }
}
\makeatother



\begin{document}

\begin{tikzpicture}
\path[/MonoG/.cd,
    step=5,
    min=5,
    max=25.01,
    path={(4,0) parabola (8,10)},
    diagram
    ] ;

\path[/MonoG/.cd,
    step=1000,
    max=4500,
    tick direction=left,
    small ticks=.2,     
    small ticks level=2,
    top label=$\Delta_V$,
    diagram
    ] ;

% to avoid 5.10e-2
\pgfkeys{/pgf/number format/.cd,fixed,precision=3}

\path[/MonoG/.cd,
    path={(11,0)--(11,10)},
    step=100,
    max=800,
    min=199.9,
    reverse=true,
    small ticks level=2,
    small ticks = .3,
    top label=IPS (s),
    diagram
    ] ;

\path[/MonoG/.cd,
    step=.1,
    max=.9,
    path={(0,0)--(4,10)},
    midway label=Mass Fraction $(M_f)$,
    top label={$M_f=1-\textrm{e}^{-\Delta v/9.81*I_{sp}}$},
    diagram
    ] ;

\end{tikzpicture}
    \begin{tikzpicture}
    \path[/MonoG/.cd,
        step=10,
        max=12,
        small ticks level=2,
        diagram
        ] ;

    \path[/MonoG/.cd,
        step=1,
        max=10.01,
        min=0,
        tick direction=left,  
        diagram
        ] ;


    \path[/MonoG/.cd,
        path={(3,0)--(3,10)},
        min=.2,
        step=5,
        max=9.7,
        diagram
        ] ;

    \path[/MonoG/.cd,
        tick direction=left,
        path={(3,0)--(3,10)},
        min=-5,
        step=5,
        max=15,
        diagram
        ] ;

    \path[/MonoG/.cd,
        path={(12,0) arc (0:180:3)},
        min=0,
        step=30,
        small ticks level=2,        
        max=180.01,
        diagram
        ] ;
    \end{tikzpicture}

\end{document}

The pearl script

#!/usr/bin/perl -w

use POSIX "fmod" ;

$Min = $ARGV[0] ;       # Minimum
$Max = $ARGV[1] ;       # Maximum
$Step = $ARGV[2] ;      # Step of graduation
$Level =  $ARGV[3] ;    # level of small ticks default 1
$TickL =  $ARGV[4] ;    # length of small ticks (1st level)

$Coeff = $Max-$Min ;  # Scale on the path from 0 to 1 

$Begin = $Min - fmod($Min,$Step) ;  # Calculation of the place  
if ( $Begin < $Min ) {              # of the first Label
    $Begin += $Step ; }

$X = $Begin ;
$Y =  ($X - $Min) / $Coeff ;

print "\\begin{scope}[decoration={markings," ;
while ($X <= $Max) {    
    print "mark=at position $Y*\\Rev with {\\draw[\\BWidth] (0,0) --++ (0,\\BTick*\\TPos) node[Big] {\\pgfmathprintnumber{$X}} ;},\n" ;
    $X += $Step ; 
    $Y = ($X - $Min) / $Coeff ; 
    }
print "}]\n" ;
print "\\draw[postaction={decorate},use path=Path];\n" ;
print "\\end{scope}\n" ;


## half step
$X = $Begin - $Step/2 ;             # Calculation of            
if ( $X < $Min ) { $X += $Step ; }  # the first Label
$Y = ($X - $Min) / $Coeff ;

print "\\begin{scope}[decoration={markings," ;
while ($X <= $Max) {    
    print "mark=at position $Y*\\Rev with {\\draw[\\MWidth] (0,0) --++ (0,\\MTick*\\TPos) node[Med] {\\pgfmathprintnumber{$X}} ;},\n" ;
    $X += $Step ; 
    $Y = ($X - $Min) / $Coeff ; 
    }
print "}]\n" ;
print "\\draw[postaction={decorate},use path=Path];\n" ;
print "\\end{scope}\n" ;


## 10e ...
$Step /= 10 ;

$Begin = $Min - fmod($Min,$Step) ;  # Calculation of            
if ( $Begin < $Min ) {              # the first Label
    $Begin += $Step ; }
if ( $Begin < $Min ) { $Begin += $Step ; }  # the first Label

for (my $i = 1; $i <= $Level ; $i++) {
    $X = $Begin ;
    $Y = ($X - $Min) / $Coeff ;
print "\\begin{scope}[decoration={markings," ;
    while ($X <= $Max) {    
        print "mark=at position $Y*\\Rev with {\\draw[very thin] (0,0) --++ (0,$TickL*\\TPos) ;},\n" ;      
        $X += $Step ; 
        $Y = ($X - $Min) / $Coeff ; 
        }
print "}]\n" ;
print "\\draw[postaction={decorate},use path=Path];\n" ;
print "\\end{scope}\n" ;
    $Step /= 2 ;
    $TickL /= 2 ;
    }