How to use Tikz/Pgfplots to plot an algebraic curve on a restricted domain?

Question

I want to plot for example equation y^2 x + x^3-3y^2-2x^2+x=0. But at the same time I need to gray out some parts of the plane (see shaded area). Is there a way to do this?

Answer 1

I have a tikz/pgfplot solution:

\documentclass[border=15pt]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}\pgfplotsset{compat=1.13}
\usetikzlibrary{arrows,positioning,calc,patterns}
\begin{document}
\begin{tikzpicture}[
    thick]
        \begin{axis}[
            xmin=-3, xmax=6, 
            ymin=-5, ymax=5,
            axis x line = center, 
            axis y line = center,
            xtick = \empty,
            ytick = \empty,
            xlabel = {$x$},
            ylabel = {$y$},
            legend style = {nodes=right},
            legend pos = north east,
            clip mode = individual,
            ]
            \addplot[red, samples=100, domain=0:2.6] {(x-1)*sqrt(x/(3-x))};
            \addplot[red, samples=100, domain=0:2.6] {-(x-1)*sqrt(x/(3-x))};
            \draw [dashed] (3,-5) -- (3,5);
            \path [pattern=north west lines, pattern color = gray] 
                (3.1, -5+0.1) rectangle (5.5, 5-0.1);
            \path [pattern=north west lines, pattern color = gray] 
                (-2.9, -5+0.1) rectangle (-0.1, 5-0.1);
            \path (0,0) node [below left] {$O$};
            \path (1,0) node [below] {$A$};
            \filldraw [fill=white] (3,0) circle(1pt) node [below right]{3};
            \path (2.6, 5-.5) node[fill=white, above] {$tg_{Y_\infty}$};
        \end{axis}
\end{tikzpicture}
\end{document}

You can adjust colors, labels etc as you want. Notice that if you use a pgfplots older than 1.11 (I think) you have to add axis cs: to all the drawing and path coordinates in the axis environment.

If you want to use the implicit form of the equation, you can use the contour gnuplot option (see this example on the gnuplot side and the section 4.6 of the pgfplots manual):

\documentclass[border=15pt]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}\pgfplotsset{compat=1.13}
\usetikzlibrary{arrows,positioning,calc,patterns}
\begin{document}
\begin{tikzpicture}[
    thick]
        \begin{axis}[
            xmin=-3, xmax=6, 
            ymin=-5, ymax=5,
            axis x line = center, 
            axis y line = center,
            axis line style = {thick, gray},
            xtick = \empty,
            ytick = \empty,
            xlabel = {$x$},
            every axis x label/.append style = {below, gray},
            ylabel = {$y$},
            every axis y label/.append style = {above left, gray},
            legend style = {nodes=right},
            legend pos = north east,
            clip mode = individual,
            view = {0}{90},    
        ]
        \addplot3[contour gnuplot={
                levels=0, labels=false,
            }, 
            samples=100,
            ] 
            {y*y*x + x*x*x -3*y*y-2*x*x+x};
            \draw [dashed] (3,-5) -- (3,5);
            \path [pattern=north west lines, pattern color = gray] 
                (3.1, -5+0.1) rectangle (5.5, 5-0.1);
            \path [pattern=north west lines, pattern color = gray] 
                (-2.9, -5+0.1) rectangle (-0.1, 5-0.1);
            \path (0,0) node [below left] {$O$};
            \path (1,0) node [below] {$A$};
            \filldraw [fill=white] (3,0) circle(1pt) node [below right]{3};
            \path (2.6, 5-.5) node[fill=white, above] {$tg_{Y_\infty}$};
        \end{axis}
\end{tikzpicture}
\end{document}

You have to run it with -shell-escape (or --enable-write18) flag to pdflatex, and be patient... (it's really slow):

Answer 2

As there's no solution so far with Tikz/pgf, I propose an easy solution with pstricks. It's compilable with pdflatex if you launch it with the --enable-write18 switch (for MiKTeX) or -shell-escape (TeX Live, MacTeX). Alternatively, you can compile with XeLaTeX.

\documentclass[x11names, border=3pt]{standalone}
\usepackage{fourier} 
\usepackage{pst-plot}
\usepackage{auto-pst-pdf}

\begin{document}

\psset{algebraic, unit=1.5, arrowinset=0.12, arrowsize=4pt, linejoin=1, plotstyle=curve, plotpoints=200}

\begin{pspicture*}(-3, -4)(6, 4.5)
\psaxes[linecolor=LightSteelBlue3, ticks=none,  labels=none,  arrows=-> ](0,0)(-3, -4)(6, 4)[$x$,-135] [$y$,120]
\psset{linewidth=1.2pt, linecolor=IndianRed3, plotpoints=100}
\begin{psclip}{\psframe[linestyle=none](0,-3.9)(3,3.9)}
\psplot{0}{2.6}{(x-1)*sqrt(x/(3-x))}
\psplot{0}{2.6}{(1-x)*sqrt(x/(3-x))}
\end{psclip}
\psdot(1,0)\uput[d](1,0){$ 1 $}
\psset{linestyle=dashed, dash=4pt 4pt, linecolor=LightSteelBlue3}
\psdot[dotstyle=o](3,0)
\psline[linewidth=0.4pt](3, -4)(3, 4) % asymptote
\uput[u](3,4){$tg_{Y_\infty}$}
\psset{linestyle=none,  fillstyle=vlines,  hatchwidth=0.4pt, framesep=1pt}
\psframe(-3,-3.9)(-0.1,3.9)
\psframe(3.1,-3.9)(5.7,3.9)
\uput[dr](3,0){\psframebox*{$ 3 $}}
\uput[dl](0,0){\psframebox*{$O$}}
\end{pspicture*}

\end{document}