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I'm trying to have the a silimiar plot to the one Wolfram Alpha provides me here to the equation

$24.6347 \left(246.347 x + \sqrt{(246.347 x + 20)^2 + 1} + 20\right)^{-a} =\ 30.2879 \left(-33.6532 x + \sqrt{x (1132.54 x - 20191.9) + 90001} + 300\right)^{-a}$

for $a>0$. This equation however has no closed form solution for x in terms of a so I don't know how to achieve such a plot.

Minimal "Working" Example:

\documentclass[11pt,a4paper]{scrartcl}
\usepackage{pgfplots}
 \pgfplotsset{compat=newest}
 \begin{document}
 \begin{tikzpicture}


\begin{axis}[xlabel=a ,
    ylabel = x,
     xmin=0, xmax=10,
    ymin=0,ymax=1]
  \addplot [
    domain=0:10, 
    samples=500, 
    color=blue,
    ]
    {24.6347*(246.347*x+sqrt((246.347*x+20)^2+1)+20)^(-a)=30.2879*(-33.6532*x+ sqrt(x*(1132.54*x-20191.9)+90001)+300)^(-a)}; 
%switched the variables because gnuplot requires the variable x as input as far as i know
\end{axis}
\end{tikzpicture}
\end{document}
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3 Answers 3

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4

AFAIK you cannot really produce implicit plots but you can produce contour plots, which have the same information. So instead of plotting the solutions of A=B one plots the contours of A-B=0. You need to run with -shell-escape since this needs gnuplot (which your above example didn't need).

\documentclass[11pt,a4paper]{scrartcl}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
\begin{tikzpicture}
\begin{axis}[view={0}{90},ylabel=$a$,
    xlabel=$x$,xmin=0, xmax=1,ymin=0,ymax=1]
    \addplot3 [contour gnuplot={levels=0,labels=false},
        domain=0:1,domain y=0:1] 
        {24.6347*(246.347*x+sqrt((246.347*x+20)^2+1)+20)^(-y)
    -30.2879*(-33.6532*x+ sqrt(x*(1132.54*x-20191.9)+90001)+300)^(-y)}; 
\end{axis}
\end{tikzpicture}
\end{document}

enter image description here

In order to reproduce the Wolfram alpha plot you need to switch x and y back, and adjust the domains.

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  • This is not working for me. I get the error message ..._contourtmp0.table could not be opened. Normal gnuplots are working fine though. Commented Jul 9, 2019 at 15:02
  • Your example is now working (no idea what fixed it). How would I go about adding a straight line at a=0.8 in your example? Commented Jul 9, 2019 at 16:02
  • @PaulOstmann I do not know what caused your problems but the straight line is as easy as saying \draw (0,0.8) -- (1,0.8); inside the axis environment. (It is a bit of a cheat since these are technically 3d coordinates but works.) Commented Jul 9, 2019 at 17:01
  • I found another solution myself: I simply added another contour plot {y-1}(I initially said 0.8 because your solution has the standard domain with y<=1) but now im stuck on trying to add a third plot: the difference of the 2 functions by which I mean 1 - the function in my post. It's supposed to look like this: imgur link Commented Jul 9, 2019 at 17:08
  • @PaulOstmann With your function you mean the difference the contour plot is using? And yes, \addplot also works. Does ` \addplot3 [contour gnuplot={levels=0,labels=false}, domain=0:1,domain y=0:1] {1-24.6347*(246.347*x+sqrt((246.347*x+20)^2+1)+20)^(-y) +30.2879*(-33.6532*x+ sqrt(x*(1132.54*x-20191.9)+90001)+300)^(-y)};` work for you? Commented Jul 9, 2019 at 17:14
3

Actually, it does have a closed form solution, at least for a as a function of x.

latex equations

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  • Honestly I'd prefer your solution. This is just a closed-form plot. Commented Jul 9, 2019 at 22:36
2

run with xelatex:

\documentclass{article}
\usepackage{pst-func}
\begin{document}

\psset{xunit=0.75}
\begin{pspicture}(-2,-1)(20,6)
\psaxes[Dx=2]{->}(0,0)(-2,-0.5)(18,5)[$a$,0][$x$,90]
\rput[lb](0,0){%
  \pspicture*(0.1,0.1)(16,6)
  \psplotImp[algebraic,linecolor=red,linewidth=1.5pt](0.05,0.05)(20,6){
    24.6347*(246.347*y + sqrt((246.347*y + 20)^2 + 1) + 20)^(-x)- 
    30.2879*(-33.6532*y + sqrt(y*(1132.54*y - 20191.9) + 90001) + 300)^(-x)}
  \endpspicture}
\end{pspicture}

\end{document}

enter image description here

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