Six-Hump Camel Function

Question

I want to achieve esthetically the nearest result compared to:

So far i have only been able to get:

\documentclass[tikz]{standalone}
\usepackage{amsmath}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
        align =center,
        title = {\textbf{Six-hump camel back function}\ ${\scriptsize f(x,y)=\left(4-2.1 x_1^2+\frac{x_1^4}{3}\right) x_1^2+x_1 x_2+\left(-4+4 x_2^2\right) x_2^2}$},
        view={310}{30},
        xtick = {2,1,...,-2},
        ytick = {1,0.5,...,-1},
        ztick = {6,5,...,-2},
        xmin=-2,
        xmax=2,
        ymin=-1,
        ymax=1,
        zmin=-2,
        zmax=6,
        ticklabel style = {font = \scriptsize},
        grid=major,
        xlabel=$x$,
        ylabel=$y$,
        zlabel=$z$

            ]
            %some miminum points
            \addplot3[mark=*,red,point meta=explicit symbolic,nodes near coords] 
        coordinates {(0,0.75,-2)[{\scriptsize A minimum}]};
        \addplot3[mark=*,red,point meta=explicit symbolic,nodes near coords] 
    coordinates {(0,-0.75,-2)[{\scriptsize A minimum}]};

        \addplot3 [
            surf,
            colormap/jet,
            shader=faceted,
            fill opacity=0.75,
            samples=50,
            domain=-2:2,
            y domain=-1:1
            ] {(4-2.1*x^2+x^4/3)*x^2+x*y+(-4+4*y^2)*y^2};

    \end{axis}

\end{tikzpicture} 
\end{document}

That gives:

Thanks for your help

PS: I'm running LaTeX on a Microsoft Windows machine

I did not check, but this looks similar: https://tex.stackexchange.com/a/87900/47927 — Jasper Habicht, Nov 08 '23 at 19:55
Well, I am not sure how to get the colors right with these approaches, I have to admit … — Jasper Habicht, Nov 08 '23 at 20:51

Jasper Habicht · Accepted Answer · 2023-11-09T19:41:59.270

Inspired by the answers to this question, you could try with gnuplot:

\documentclass[border=10pt]{standalone}
\usepackage{amsmath}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            align = center,
            title = {
                \textbf{Six-hump camel back function} \ 
                $\scriptsize f(x,y) = 
                    \left(4 - 2.1 x_1^2 + \frac{x_1^4}{3}\right) x_1^2 + x_1 
                    x_2 + \left(-4 + 4 x_2^2\right) x_2^2$
                },
            view = {310}{30},
            xtick = {2,1,...,-2},
            ytick = {1,0.5,...,-1},
            ztick = {6,5,...,-2},
            xmin = -2, xmax = 2,
            ymin = -1, ymax = 1,
            zmin = -2, zmax = 6,
            ticklabel style = {
                font = \scriptsize
            },
            grid = major,
            xlabel = $x$,
            ylabel = $y$,
            zlabel = $z$

        ]
    \addplot3[
        mark=*,
        red,
        draw = none,
        point meta = explicit symbolic,
        nodes near coords
    ] coordinates {
        (0,0.75,-2) [{\scriptsize A minimum}]
        (0,-0.75,-2) [{\scriptsize A minimum}]
    };

    \addplot3+[
        no markers,
        raw gnuplot,
        z filter/.code={\def\pgfmathresult{-2}},
    ] gnuplot {
        unset surface;
        set contour base;
        set cntrparam levels 20;
        set isosamples 200;
        set samples 51;
        splot [-2:2][-1:1][-2:6] 
            (4 - 2.1*x^2 + x^4/3)*x^2 + x*y + (-4 + 4*y^2)*y^2;
    };

    \addplot3[
        surf,
        colormap/jet,
        shader = faceted,
        fill opacity = 0.75,
        samples = 51,
        domain = -2:2,
        y domain = -1:1
    ] {(4 - 2.1*x^2 + x^4/3)*x^2 + x*y + (-4 + 4*y^2)*y^2};

    \end{axis}

\end{tikzpicture} 
\end{document}

Coloring can be done via contour prepared:

\documentclass[border=10pt]{standalone}
\usepackage{amsmath}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            align = center,
            title = {
                \textbf{Six-hump camel back function} \ 
                $\scriptsize f(x,y) = 
                    \left(4 - 2.1 x_1^2 + \frac{x_1^4}{3}\right) x_1^2 + x_1 
                    x_2 + \left(-4 + 4 x_2^2\right) x_2^2$
                },
            view = {310}{30},
            xtick = {2,1,...,-2},
            ytick = {1,0.5,...,-1},
            ztick = {6,5,...,-2},
            xmin = -2, xmax = 2,
            ymin = -1, ymax = 1,
            zmin = -2, zmax = 6,
            ticklabel style = {
                font = \scriptsize
            },
            grid = major,
            xlabel = $x$,
            ylabel = $y$,
            zlabel = $z$

        ]
    \addplot3[
        mark=*,
        red,
        draw = none,
        point meta = explicit symbolic,
        nodes near coords
    ] coordinates {
        (0,0.75,-2) [{\scriptsize A minimum}]
        (0,-0.75,-2) [{\scriptsize A minimum}]
    };

    \addplot3+[
        contour prepared,
        contour/labels = false,
        colormap/jet,
        point meta = rawz,
        no markers,
        raw gnuplot,
        z filter/.code={\def\pgfmathresult{-2}},
    ] gnuplot {
        unset surface;
        set contour base;
        set cntrparam levels 20;
        set isosamples 200;
        splot [-2:2][-1:1][-2:6] 
            (4 - 2.1*x^2 + x^4/3)*x^2 + x*y + (-4 + 4*y^2)*y^2;
    };

    \addplot3[
        surf,
        colormap/jet,
        shader = faceted,
        fill opacity = 0.75,
        samples = 51,
        domain = -2:2,
        y domain = -1:1
    ] {(4 - 2.1*x^2 + x^4/3)*x^2 + x*y + (-4 + 4*y^2)*y^2};

    \end{axis}

\end{tikzpicture} 
\end{document}

With a few additional options, you'll get quite close to the original:

\documentclass[border=10pt]{standalone}
\usepackage{amsmath}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
    \begin{tikzpicture}
        \begin{axis}[
            align = center,
            title = {
                \textbf{Six-hump camel back function} \ 
                $\scriptsize f(x,y) = 
                    \left(4 - 2.1 x_1^2 + \frac{x_1^4}{3}\right) x_1^2 + x_1 
                    x_2 + \left(-4 + 4 x_2^2\right) x_2^2$
                },
            view = {310}{30},
            xtick = {2,1,...,-2},
            ytick = {1,0.5,...,-1},
            ztick = {6,5,...,-2},
            xmin = -2, xmax = 2,
            ymin = -1, ymax = 1,
            zmin = -2, zmax = 6,
            ticklabel style = {
                font = \scriptsize
            },
            grid = major,
            xlabel = {$x_1$},
            ylabel = {$x_2$},
            zlabel = {$f(x_1,x_2)$},
            axis lines*=left,
            tick align=outside,
            major grid style={
                dotted, gray
            }
        ]
    \addplot3[
        mark=*,
        red,
        draw = none,
        point meta = explicit symbolic,
        nodes near coords
    ] coordinates {
        (0,0.75,-2) [{\scriptsize A minimum}]
        (0,-0.75,-2) [{\scriptsize A minimum}]
    };

    \addplot3+[
        contour prepared,
        contour/labels = false,
        colormap/jet,
        point meta = rawz,
        no markers,
        raw gnuplot,
        z filter/.code={\def\pgfmathresult{-2}},
    ] gnuplot {
        unset surface;
        set contour base;
        set cntrparam levels 20;
        set isosamples 200;
        splot [-2:2][-1:1][-2:6] 
            (4 - 2.1*x^2 + x^4/3)*x^2 + x*y + (-4 + 4*y^2)*y^2;
    };

    \addplot3[
        surf,
        colormap/jet,
        shader = flat,
        draw = black,
        samples = 51,
        domain = -2:2,
        y domain = -1:1
    ] {(4 - 2.1*x^2 + x^4/3)*x^2 + x*y + (-4 + 4*y^2)*y^2};

    \end{axis}

\end{tikzpicture} 
\end{document}

Perfect! Thanks a lot for your help! – Vincent ISOZ Nov 09 '23 at 20:39 — Vincent ISOZ, Nov 09 '23 at 20:39

Six-Hump Camel Function

1 Answers1