Parametric plot using pgfplots package

Question

I want to make a parametric plot using pgfplot in LaTex. My equations are

\rho= K (sinh t-t)
P= 1/3 K (sinh t - 8 sinh (1/2) t +3t) with K=1\pi

I have written the LaTex code for equations , now please help me plot the graph

\documentclass[12pt,a4paper]{article}
\usepackage[a4paper, total={6in, 8in}]{geometry}
    \usepackage[utf8]{inputenc}
    \usepackage[T1]{fontenc}
    \usepackage{fourier}
    \usepackage{mathtools}
\usepackage{amsmath}
\usepackage[english]{babel}
\usepackage{graphicx}
\usepackage{marginnote}
\begin {document}

\underline {Chandrashekhar EOS (1935)} : \\
\begin{align*}
\overline \rho &= K ~\Big( \sinh ~t ~- ~t\Big)\\
\overline P &= \frac {1}{3} K ~ \Big(\sinh~t -~ 8~ \sinh \frac {1}{2} t  +~ 3t\Big)\\
 K &= \frac {1}{4\pi}\\
\end {align*}
\end {document}

Answer 1

Note: Other answers to this question may use other ks for plotting (eg. K=1/4π). In my answer, I have used question's desired k value which is K=1/π. That is why there may be a slight difference in the plots.

---

You can plot your functions with pgfplots. Run the following code with LuaLaTeX, I have used it to avoid probable numerical inaccuracies.

% lualatex
\documentclass{standalone}
    \usepackage{tikz}
    \usepackage{pgfplots}
        \pgfplotsset{compat=1.9}

\begin{document}
\begin{tikzpicture}
\directlua{
    R= function(x,k)
    return k*(math.sinh(x)-x)
    end

    P = function(x,k)
    return k*(math.sinh(x)- 8*math.sinh(x/2)+3*x)/3
    end
        }
        \pgfmathdeclarefunction{R}{2}{%
            \edef\pgfmathresult{\directlua{tex.print(R(\pgfmathfloatvalueof{#1},\pgfmathfloatvalueof{#2}))}}%
        }%
        \pgfmathdeclarefunction{P}{2}{%
            \edef\pgfmathresult{\directlua{tex.print(P(\pgfmathfloatvalueof{#1},\pgfmathfloatvalueof{#2}))}}%
        }%
\begin{axis}
        [
    axis lines = center,
    grid=both,minor tick num=1,
    xlabel=$x$,ylabel=$y$,
    tick align=inside,
        legend style={at={(0.5,-0.1)},
        anchor=north,legend columns=1},
        legend entries={${\rho}= K\Big(\sinh t-t\Big)$\\$P= \frac{1}{3}K\Big(\sinh t-8\sinh\frac {1}{2} t  + 3t\Big)$\\},
        domain=-6:6,
        samples=60,
        ]
        \addplot [solid, thick] {R(x,1/pi)};
       \addplot [dashed, thick] {P(x,1/pi)};
\end{axis}
\end{tikzpicture}
\end{document}

and here is the output:

P.S. Other answers exists for plotting parametric/complicated functions in this question: How to plot a complicated multi-variable function in a tex document automatically, not just importing an image

---

update: If you don't want to use the LuaLaTeX, here is another version of my answer which is suggested in this comment:

% pdflatex
\documentclass{standalone}
    \usepackage{tikz}
    \usepackage{pgfplots}
        \pgfplotsset{compat=1.9}

\begin{document}
\begin{tikzpicture}
       \pgfmathdeclarefunction{R}{2}{\pgfmathparse{#2 * (sinh(#1) - #1)}}
       \pgfmathdeclarefunction{P}{2}{\pgfmathparse{#2 * (sinh(#1) - 8*sinh(#1/2) + 3*#1)/3}}
\begin{axis}
        [
    axis lines = center,
    grid=both,minor tick num=1,
    xlabel=$x$,ylabel=$y$,
    tick align=inside,
        legend style={at={(0.5,-0.1)},
        anchor=north,legend columns=1},
        legend entries={${\rho}= K\Big(\sinh t-t\Big)$\\$P= \frac{1}{3}K\Big(\sinh t-8\sinh\frac {1}{2} t  + 3t\Big)$\\},
        domain=0:10,
        samples=60,
        ]
        \addplot [solid, thick] {R(x,1/pi)};
       \addplot [dashed, thick] {P(x,1/pi)};
\end{axis}
\end{tikzpicture}
\end{document}

Here is the output (it is optimized to show the positive axis only, change the domain to any value you need):

Answer 2

A solution with PSTricks, run with xelatex. The parametricplot P(t)=f(rho(t)) seems to be a line:

\documentclass{article}
\usepackage{mathtools}
\usepackage{pst-plot,pst-math}
\begin {document}

\begin{align*}
\overline{\rho}&= K\left(\sinh t-t\right)\\
\overline{P}   &= \frac{1}{3}K\left(\sinh t-8\sinh\frac{1}{2}t+ 3t\right)\\
             K &= \frac{1}{4\pi}
\end {align*}

\psset{llx=-15mm,lly=-5mm,urx=5mm,ury=5mm,algebraic,yMaxValue=7,
  plotpoints=500,xAxisLabel=$t$,yAxisLabel={},ticks=none}
\begin{psgraph}[labelFontSize=\scriptstyle,psgrid,gridcoor={(7,7)}]{->}(0,0)(7,7){10cm}{10cm}
\pstVerb{ /K 1 4 PI mul div def }
\psplot[linecolor=blue,arrows=-]{0}{7}{ K*(SINH(x)-x) }
\psplot[linecolor=red,arrows=-]{0}{7}{ K/3*(SINH(x)-8*SINH(x/2)+3*x) }
\psparametricplot[linewidth=1.5pt,arrows=-]{0}{5.2}{ K*(SINH(t)-t) | K/3*(SINH(t)-8*SINH(t/2)+3*t) }
\rput[rb](4.5,4.5){\blue$\overline{\rho}=K\left(\sinh t-t\right)$}
\rput[lb](0.5,2.5){\red$\overline{P}=\frac{1}{3}K\left(\sinh t-8\sinh\frac{1}{2}t+3t\right)$}
\uput[0](0,7){$\overline{P}$}\uput[180](0,7){$\overline{\rho}$}
\uput*[90](1.5,0.5){$\overline{P}=f(\overline{\rho})$}
\end{psgraph}

\end{document}

For sharelatex copy this one into a new projekt:

\documentclass{article}
\usepackage{pst-plot,pst-math}
\begin {document}

\psset{llx=-15mm,lly=-5mm,urx=5mm,ury=5mm,algebraic,yMaxValue=7,
  plotpoints=500,xAxisLabel=$t$,yAxisLabel={}}
\begin{psgraph}[labelFontSize=\scriptstyle]{->}(0,0)(7,7){10cm}{10cm}
\pstVerb{ /K 1 4 PI mul div def }
\psplot[linecolor=blue,arrows=-]{0}{7}{ K*(SINH(x)-x) }
\psplot[linecolor=red,arrows=-]{0}{7}{ K/3*(SINH(x)-8*SINH(x/2)+3*x) }
\psparametricplot[linewidth=1.5pt,arrows=-]{0}{5.2}{ K*(SINH(t)-t) | K/3*(SINH(t)-8*SINH(t/2)+3*t) }
\rput[rb](4.5,4.5){\blue$\overline{\rho}=K\left(\sinh t-t\right)$}
\rput[lb](0.5,2.5){\red$\overline{P}=\frac{1}{3}K\left(\sinh t-8\sinh\frac{1}{2}t+3t\right)$}
\uput[0](0,7){$\overline{P}$}\uput[180](0,7){$\overline{\rho}$}
\uput*[90](1.5,0.5){$\overline{P}=f(\overline{\rho})$}
\end{psgraph}

\end{document}

Then choose from the menu (top left) the compiler xelatex. However, sharelatex is not a good choice it doesn't update the TeX distribution.

Answer 3

Another fun with PSTricks. It really uses the parametric approach in question.

\documentclass[pstricks,border=0pt,12pt,dvipsnames]{standalone}
\usepackage{pst-plot,pst-math,amsmath}
\usepackage[nomessages]{fp}

\FPeval\XMin{0-6}
\FPeval\XMax{7}
\FPeval\YMin{0-6}
\FPeval\YMax{6}

\FPeval\XOL{0-1/3} % of DeltaX
\FPeval\XOR{1/3} % of DeltaX
\FPeval\YOB{0-1/3} % of DeltaY
\FPeval\YOT{1/3} % of DeltaY

%\FPset\TrigLabelBase{3}
\FPeval\DeltaX{1}
\FPeval\DeltaY{1}

\FPeval\AxisL{XMin+DeltaX*XOL}
\FPeval\AxisR{XMax+DeltaX*XOR}
\FPeval\AxisB{YMin+DeltaY*YOB}
\FPeval\AxisT{YMax+DeltaY*YOT}

\newlength\Width\Width=12cm


\newlength\llx\llx=-5pt
\newlength\urx\urx=15pt
\newlength\lly\lly=-5pt
\newlength\ury\ury=15pt


\psset
{
    llx=\llx,
    lly=\lly,
    urx=\urx,
    ury=\ury,
    %xtrigLabels,
    %ytrigLabels,
    %trigLabelBase=\TrigLabelBase,
    labelFontSize=\scriptstyle,
    xAxisLabel=$x$,
    yAxisLabel=$y$,
    algebraic,
    plotpoints=500,
    yMaxValue=\YMax,
    yMinValue=\YMin,
}

\pstVerb{/K {.25 Pi div} def}
\def\x{t}
\def\yp{K*(SINH(t)-t)}
\def\yP{K/3*(SINH(t)-8*SINH(t/2)+3*t)}




\begin{document}
\pslegend[bt](80,0)
{
    \color{NavyBlue}\rule{12pt}{3pt} & \color{NavyBlue} $\overline{p} = K (\sinh t -t) $ \\
    \color{Maroon}\rule{12pt}{3pt}   & \color{Maroon} $\overline{P} = \tfrac{1}{3} K (\sinh t -8 \sinh (\tfrac{1}{2}t)+3t) $ \\
                                     & where $K=\frac{1}{4\pi}$
}
\begin{psgraph}
    [
        dx=\DeltaX,
        dy=\DeltaY,
        Dy=\DeltaY,
        Dx=\DeltaX,
        linecolor=gray,
        tickcolor=gray,
        ticksize=-3pt 3pt,
        axespos=top,
    ]{<->}(0,0)(\AxisL,\AxisB)(\AxisR,\AxisT){\dimexpr\Width-\urx+\llx}{!}
    \psaxes
    [
        dx=\DeltaX,
        dy=\DeltaY,
        labels=none,
        subticks=5,
        tickwidth=.4pt,
        subtickwidth=.2pt,
        tickcolor=Red!30,
        subtickcolor=ForestGreen!30,
        xticksize=\YMin\space \YMax,
        yticksize=\XMin\space \XMax,
        subticksize=1,
    ](0,0)(\XMin,\YMin)(\XMax,\YMax)
    \psparametricplot[linecolor=NavyBlue]{\XMin}{7}{\x|\yp}
    \psparametricplot[linecolor=Maroon]{\XMin}{7}{\x|\yP}
\end{psgraph}
\end{document}

Notes for ShareLaTeX users

Go to upper left menu, and you will see the following. Change as pointed by the red arrow. And don't forget to recompile.

The lastest update based on OP's comment

\documentclass[pstricks,border=0pt,12pt,dvipsnames]{standalone}
\usepackage{pst-plot,pst-math,amsmath}
\usepackage[nomessages]{fp}

% Be careful, fp has not defined 
% the unary negate so we have to write 0-8 to represent -8 for example.

\FPeval\XMin{0-0}
\FPeval\XMax{7}
\FPeval\YMin{0-0}
\FPeval\YMax{6}

\FPeval\XOL{0-0} % of DeltaX
\FPeval\XOR{1/3} % of DeltaX
\FPeval\YOB{0-0} % of DeltaY
\FPeval\YOT{1/3} % of DeltaY

%\FPset\TrigLabelBase{3}
\FPeval\DeltaX{1}
\FPeval\DeltaY{1}

\FPeval\AxisL{XMin+DeltaX*XOL}
\FPeval\AxisR{XMax+DeltaX*XOR}
\FPeval\AxisB{YMin+DeltaY*YOB}
\FPeval\AxisT{YMax+DeltaY*YOT}

\newlength\Width\Width=12cm


\newlength\llx\llx=-20pt
\newlength\urx\urx=15pt
\newlength\lly\lly=-20pt
\newlength\ury\ury=15pt


\psset
{
    llx=\llx,
    lly=\lly,
    urx=\urx,
    ury=\ury,
    %xtrigLabels,
    %ytrigLabels,
    %trigLabelBase=\TrigLabelBase,
    labelFontSize=\scriptstyle,
    xAxisLabel=$x$,
    yAxisLabel=$y$,
    algebraic,
    plotpoints=500,
    yMaxValue=\YMax,
    yMinValue=\YMin,
}

\pstVerb{/K {.25 Pi div} def}
\def\x{t}
\def\yp{K*(SINH(t)-t)}
\def\yP{K/3*(SINH(t)-8*SINH(t/2)+3*t)}




\begin{document}
\pslegend[rt]%(80,0)
{
    \color{NavyBlue}\rule{12pt}{3pt} & \color{NavyBlue} $\overline{p} = K (\sinh t -t) $ \\
    \color{Maroon}\rule{12pt}{3pt} & \color{Maroon} $\overline{P} = \tfrac{1}{3} K (\sinh t -8 \sinh (\tfrac{1}{2}t)+3t) $ \\
    & where $K=\frac{1}{4\pi}$
}
\begin{psgraph}
    [
        dx=\DeltaX,
        dy=\DeltaY,
        Dy=\DeltaY,
        Dx=\DeltaX,
        linecolor=gray,
        tickcolor=gray,
        ticksize=-3pt 3pt,
        axespos=top,
    ]{->}(0,0)(\AxisL,\AxisB)(\AxisR,\AxisT){\dimexpr\Width-\urx+\llx}{!}
    \psaxes
    [
        dx=\DeltaX,
        dy=\DeltaY,
        labels=none,
        subticks=5,
        tickwidth=.4pt,
        subtickwidth=.2pt,
        tickcolor=Red!30,
        subtickcolor=ForestGreen!30,
        xticksize=\YMin\space \YMax,
        yticksize=\XMin\space \XMax,
        subticksize=1,
    ](0,0)(\XMin,\YMin)(\XMax,\YMax)
    \psparametricplot[linecolor=NavyBlue]{\XMin}{7}{\x|\yp}
    \psparametricplot[linecolor=Maroon]{\XMin}{7}{\x|\yP}
\end{psgraph}
\end{document}