Put Gnuplot mutiple surfaces plot in PGFplots axis environment to overcome z-buffer problem

Question

Although PGFplots is the perfect tool for many depictions, it unfortunately does not have a properly working z buffer for intersecting surfaces (at this time, 10-09-2015). I'm trying to overcome this problem by plotting in gnuplot with set terminal tikz and then putting the plots in a PGFplots axis environment. I cannot seem to manage the latter, though.

I use the picture

from http://gnuplot.sourceforge.net/demo_5.0/surface2.9.gnu as an example, where I changed the terminal to tikz:

set terminal tikz standalone
set output 'main.tex'
set dummy u, v
set key bmargin center horizontal Right noreverse enhanced autotitle nobox
set parametric
set view 50, 30, 1, 1
set isosamples 50, 20
set hidden3d back offset 1 trianglepattern 3 undefined 1 altdiagonal bentover
set style data lines
set ticslevel 0
set title "Interlocking Tori" 
set urange [ -3.14159 : 3.14159 ] noreverse nowriteback
set vrange [ -3.14159 : 3.14159 ] noreverse nowriteback
splot cos(u)+.5*cos(u)*cos(v),sin(u)+.5*sin(u)*cos(v),.5*sin(v) with lines,       1+cos(u)+.5*cos(u)*cos(v),.5*sin(v),sin(u)+.5*sin(u)*cos(v) with lines

The resulting tex file main.tex has the following content

\documentclass[10pt]{article}
\usepackage[T1]{fontenc}
\usepackage{textcomp}

\usepackage[utf8x]{inputenc}

\usepackage{gnuplot-lua-tikz}
\pagestyle{empty}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{\gpbboxborder}
\begin{document}

\begin{tikzpicture}[gnuplot]
%% generated with GNUPLOT 4.6p5 (Lua 5.1; terminal rev. 99, script rev. 100)
%% Thu 10 Sep 2015 01:40:19 PM CEST
\path (0.000,0.000) rectangle (12.500,8.750);
\gpcolor{color=gp lt color border}
\node[gp node center] at (6.250,8.163) {Interlocking Tori};
\gpcolor{color=gp lt color 0}
\gpsetlinetype{gp lt plot 0}
\gpsetlinewidth{1.00}
\draw[gp path] (6.400,6.200)--(6.110,6.207);

(...A lot of \draw[gp path] commands...)
%% coordinates of the plot area
\gpdefrectangularnode{gp plot 1}{\pgfpoint{1.800cm}{1.387cm}}{\pgfpoint{10.700cm}{7.979cm}}
\end{tikzpicture}
%% gnuplot variables
\end{document}

from which it is clear that there is no use of a PGFplots axis environment.

Compiling with lualatex -shell-escape main.tex gives , which 'solves' the z buffer problem. But how do I get these drawings in a nice PGFplots axis enviroment?

Answer 1

Before @Christian Feuersänger introduces any good interface, we may parameterize the surfaces carefully and make use of the existing sorting algorithm.

\documentclass[border=9,tikz]{standalone}
\usepackage{pgfplots}\pgfplotsset{compat=newest}
\begin{document}

\pgfmathdeclarefunction{X}{0}{%
    \pgfmathparse{
        y<121?
            cos(x)*(3+cos(3*y))
            :
            (y<139?
                inf
                :
                3+cos(x)*(3+cos(3*y))
            )
    }%
}
\pgfmathdeclarefunction{Y}{0}{%
    \pgfmathparse{
        y<121?
            sin(x)*(3+cos(3*y))
            :
            (y<139?
                inf
                :
                sin(3*y)
            )
    }%
}
\pgfmathdeclarefunction{Z}{0}{%
    \pgfmathparse{
        y<121?
            sin(3*y)
            :
            (y<139?
                inf
                :
                sin(x)*(3+cos(3*y))
            )
    }%
}

\tikz[cap=round,join=round]{
    \begin{axis}[axis equal]
        \addplot3[surf,unbounded coords=jump,
                  domain  =0:360,samples  =27,
                  domain y=0:260,samples y=27,
                  z buffer=sort,point meta=x]
            (X,Y,Z);
    \end{axis}
}

\end{document}

Even more carefully

\documentclass[border=9,tikz]{standalone}
\usepackage{pgfplots}\pgfplotsset{compat=newest}
\begin{document}

\pgfmathdeclarefunction{X}{0}{%
    \pgfmathparse{
        y<121?
            cos(x)*(3+cos(3*y))
            :
            (y<139?
                inf
                :
                (y<261?
                    4+cos(x)*(3+cos(3*y))
                    :
                    (y<279?
                        inf
                        :
                        8+cos(x)*(3+cos(3*y))
                    )
                )
            )
    }%
}
\pgfmathdeclarefunction{Y}{0}{%
    \pgfmathparse{
        y<121?
            sin(x)*(3+cos(3*y))
            :
            (y<139?
                inf
                :
                (y<261?
                    sin(3*y)
                    :
                    (y<279?
                        inf
                        :
                        sin(x)*(3+cos(3*y))
                    )
                )
            )
    }%
}
\pgfmathdeclarefunction{Z}{0}{%
    \pgfmathparse{
        y<121?
            sin(3*y)
            :
            (y<139?
                inf
                :
                (y<261?
                    sin(x)*(3+cos(3*y))
                    :
                    (y<279?
                        inf
                        :
                        sin(3*y)
                    )
                )
            )
    }%
}

\tikz[cap=round,join=round]{
    \begin{axis}[axis equal]
        \addplot3[surf,unbounded coords=jump,
                  domain  =0:360,samples  =27,
                  domain y=0:400,samples y=41,
                  z buffer=sort,point meta=x]
            (X,Y,Z);
    \end{axis}
}

\end{document}

Edit

Similar tricks is used in

• How can I add thickness to a 3D surface plot?
• Pgfplots partial sphere

So basically the setting

                  domain  =0:360,samples  =36,
                  domain y=0:260,samples y=27,

will set up the domain

and the setting

                  domain  =0:360,samples  =36,
                  domain y=0:400,samples y=41,

will set up the domain

By giving that unbounded coords=jump, pgfplots will drop the coordinates that maps to infinity and cut the surface into two pieces. In three-tours case, of course, pgfplots cuts the surface into three pieces.