sphere and points in 3D - tikz or pstricks?

Question

I would like to generate a figure containing points within a cube of dimension 2x2x2. Inside this cube, is a sphere of diameter 2 (both centered on the origin). In this cube, I will draw several points: some of them inside the sphere (in a given color, say orange) and some outside the sphere (in white). I initially thought of using tikz, see the mwe:

\documentclass[]{standalone}
\usepackage{tikz}
\usepackage{ifthen}
\usepackage[nomessages]{fp}
\usepackage{xcolor}
\usetikzlibrary{3d,calc}

% ==========================
% parameters definition:
% ==========================

% colour of the cube
\colorlet{couleurcube}{black!25}
% colour of the sphere
\colorlet{couleursphere}{orange}
% colour of the points outside the sphere
\colorlet{couleurext}{white}
% colour of the points inside the sphere
\colorlet{couleurint}{orange}
% sphere radius
\def\rayonsphere{1}
% grid resolution
\def\resolution{6}
% grid colour
\colorlet{couleurgrille}{black!15}    

% automated adjustment of the point colour depending on its position with respect to the sphere
\newcommand{\tracepoint}[4]{
\FPeval{\somme}{clip(abs(#1)*abs(#1)+abs(#2)*abs(#2)+abs(#3)*abs(#3))}
\FPeval\rayon{clip(\somme^(0.5))}
\FPeval\difference{clip(\rayon-#4)}
\newdimen\ecart
\ecart = \difference pt
\ifthenelse{\ecart>0}{
\node[draw=black!75,shape=circle,fill=couleurext,minimum size=1.5mm,line width=0mm,inner sep=0] (x) at (#1,#2,#3) {};}{
\node[draw=black!75,shape=circle,fill=couleurint,minimum size=1.5mm,line width=0mm,inner sep=0] (x) at (#1,#2,#3) {};}}

\begin{document}
% graphique
\begin{tikzpicture}[background rectangle/.style={ultra thick,draw=none, top color=white, bottom color=white},scale=2]
    % tracé du cube en 3D
    \begin{scope}[x={(.7cm,.4cm)},z={(.9cm,-.4cm)}]    
        \begin{scope}[every path/.style={thick}]
            \node(C) at (0,0,0) {};
            % arêtes du cube derrière la sphère
            \draw[couleurcube,thick] (-1,-1,-1) -- (1,-1,-1);
            \draw[couleurcube,thick] (1,-1,-1) -- (1,1,-1);
            \draw[couleurcube,thick] (1,-1,-1) -- (1,-1,1);
            % 
            % background grid
            \foreach \x in {1,...,\resolution}
                {
                \draw[couleurgrille,thin] (-1+\x*2/\resolution,-1,-1) -- (-1+\x*2/\resolution,1,-1);
                \draw[couleurgrille,thin] (-1,-1+\x*2/\resolution,-1) -- (1,-1+\x*2/\resolution,-1);
                \draw[couleurgrille,thin] (-1+\x*2/\resolution,-1,-1) -- (-1+\x*2/\resolution,-1,1);
                \draw[couleurgrille,thin] (-1,-1,-1+\x*2/\resolution) -- (1,-1,-1+\x*2/\resolution);
                \draw[couleurgrille,thin] (1,-1,-1+\x*2/\resolution) -- (1,1,-1+\x*2/\resolution);
                \draw[couleurgrille,thin] (1,-1+\x*2/\resolution,-1) -- (1,-1+\x*2/\resolution,1);
                }
        \end{scope}
    \end{scope}

    % sphere        
    \filldraw[ball color=couleursphere,draw=none,opacity=0.55] (0,0) circle (\rayonsphere);
    \def\norme{\rayonsphere}

    % 3D cube
    \begin{scope}[x={(.7cm,.4cm)},z={(.9cm,-.4cm)}]    
        \begin{scope}[every path/.style={thick}]
            \node(C) at (0,0,0) {};
            \draw[couleurcube,thick] (1,1,-1) -- (-1,1,-1);
            \draw[couleurcube,thick] (-1,1,-1) -- (-1,-1,-1);
            \draw[couleurcube,thick] (-1,-1,-1) -- (-1,-1,1);
            \draw[couleurcube,thick] (1,1,-1) -- (1,1,1);
            \draw[couleurcube,thick] (-1,1,-1) -- (-1,1,1);
            \draw[couleurcube,thick] (-1,-1,1) -- (1,-1,1);
            \draw[couleurcube,thick] (1,-1,1) -- (1,1,1);
            \draw[couleurcube,thick] (1,1,1) -- (-1,1,1);
            \draw[couleurcube,thick] (-1,1,1) -- (-1,-1,1);
            % 
            % axes
            \draw[black,very thick] (-1,-1,-1) -- (-1,-1,1) node[midway,below=0.5cm] {$\xi_1$};
            \draw[black,very thick] (-1,-1,1) -- (1,-1,1) node[midway,below=0.5cm] {$\xi_2$};
            \draw[black,very thick] (1,-1,1) -- (1,1,1) node[midway,right=0.5cm] {$\xi_3$};
            \node[below=0.25cm,left] at (-1,-1,-1) {-1};
            \node[below=0.25cm,left] at (-1,-1,1) {1};
            \node[below=0.25cm,right] at (-1,-1,1) {-1};
            \node[below=0.25cm,right] at (1,-1,1) {1};
            \node[right=0.15cm] at (1,-1,1) {-1};
            \node[right=0.15cm] at (1,1,1) {1};
            %
            % points
            \tracepoint{0}{0}{0}{\norme};
            \tracepoint{-1}{0}{0}{\norme};
            \tracepoint{0}{-1}{0}{\norme};
            \tracepoint{0}{0}{-1}{\norme};
            \tracepoint{1}{0}{0}{\norme};
            \tracepoint{0}{1}{0}{\norme};
            \tracepoint{0}{0}{1}{\norme};
            \tracepoint{1}{1}{0}{\norme};
            \tracepoint{0}{1}{1}{\norme};
            \tracepoint{1}{0}{1}{\norme};
            \tracepoint{1}{-1}{0}{\norme};
            \tracepoint{0}{1}{-1}{\norme};
            \tracepoint{1}{0}{-1}{\norme};
            \tracepoint{-1}{1}{0}{\norme};
            \tracepoint{0}{-1}{1}{\norme};
            \tracepoint{-1}{0}{1}{\norme};
            \tracepoint{-1}{-1}{0}{\norme};
            \tracepoint{0}{-1}{-1}{\norme};
            \tracepoint{-1}{0}{-1}{\norme};
            %
        \end{scope}
    \end{scope}
\end{tikzpicture}                
\end{document}

This gives:

I am fairly unhappy with the result for two reasons: (1) tikz does not allow for a proper 3D view and, most importantly, I do not like the fact that all the data points seem to be "over" the sphere... So I tried to use pstricks with the following (partial since the automated function for points color is not yet implemented) mwe:

\documentclass[pstricks]{standalone}
\usepackage{pst-solides3d}

\newcommand{\tracepoint}[4]{
\FPeval{\somme}{clip(abs(#1)*abs(#1)+abs(#2)*abs(#2)+abs(#3)*abs(#3))}
\FPeval\rayon{clip(\somme^(0.5))}
\FPeval\difference{clip(\rayon-#4)}
\newdimen\ecart
\ecart = \difference pt
\ifthenelse{\ecart>0}{
\node[draw,shape=circle,fill=white,minimum size=1.5mm,line width=0mm,inner sep=0] (x) at (#1,#2,#3) {};% {\somme / \rayon / \difference};}{
\node[draw,shape=circle,fill=red,minimum size=1.5mm,line width=0mm,inner sep=0] (x) at (#1,#2,#3) {};% {\somme / \rayon / \difference};}}

\begin{document}

\psset{viewpoint=30 40 20 rtp2xyz,Decran=56}
  \begin{pspicture}(-3.5,-3.5)(3.5,3.5)
   \psSolid[object=cube,a=2,opacity=0.2,action=draw*]% ,fillcolor=blue
   \psSolid[object=sphere,r=1,linewidth=0.1pt,ngrid=50 50,fillcolor=red,opacity=0.1,action=draw*]%

   \psPoint(1.25,0,-1.25){x}\rput(x){$\xi_1$}
   \psPoint(0,1.25,-1.25){y}\rput(y){$\xi_2$}
   \psPoint(-1.25,1.25,0){z}\rput(z){$\xi_3$}
   
   \psdot[linecolor=black,fillcolor=orange,dotstyle=o,dotsize=2.5pt](0,0,0)
   %\tracepoint{0}{0}{1}{\norme};
   
  \end{pspicture}

\end{document}

which gives me:

I like the 3D view much better but, here again, I do not know how to draw points inside or outside the sphere...

You have precisely one point which is fully inside the sphere, and a few which sit on its surface. To deal with those properly, you may want to switch to asymptote. However, it is a myth that you cannot change the view with TikZ, there are packages like tikz-3dplot but you can also use [x={(1,0}... etc. ...] to set the view without any additional packages. And in order to have the points inside the sphere, you may simply draw them before the sphere. — user121799, Commented Jul 13, 2018 at 21:36
Many years ago I was insane and foolish enough to plot a 3D scene in tikz as a projection to 2D, because I knew no better. It worked okayish, but with a lot of pain and blood and tears. Just putting my 2¢ as a possible, but not really desirable option. — Oleg Lobachev, Commented Jul 13, 2018 at 23:30
@marmot indeed, I am thinking asymptote might be a way to solve my problem... And, sure, I could plot the sphere above some of the points but I would like to automate the procedure for a figure with several dozens of points for which I cannot easily know beforehand which one are behind the sphere... — Alain, Commented Jul 14, 2018 at 0:39

score 3 · Accepted Answer · 2018-07-14 07:13:34Z

This is too long for a comment but I'll be happy to remove it. You can draw the points twice, but really draw them if they are inside before you draw the sphere and after if they are outside. (I changed the radius because some points are right at surface of a sphere of radius 1.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{ifthen}
\usepackage[nomessages]{fp}
\usepackage{xcolor}
\usetikzlibrary{3d,calc}

% ==========================
% parameters definition:
% ==========================

% colour of the cube
\colorlet{couleurcube}{black!25}
% colour of the sphere
\colorlet{couleursphere}{orange}
% colour of the points outside the sphere
\colorlet{couleurext}{white}
% colour of the points inside the sphere
\colorlet{couleurint}{orange}
% sphere radius
\def\rayonsphere{1.2}
% grid resolution
\def\resolution{6}
% grid colour
\colorlet{couleurgrille}{black!15}  
\def\ImOut{-1}
% automated adjustment of the point colour depending on its position with respect to the sphere
\newcommand{\tracepoint}[4]{
\FPeval{\somme}{clip(abs(#1)*abs(#1)+abs(#2)*abs(#2)+abs(#3)*abs(#3))}
\FPeval\rayon{clip(\somme^(0.5))}
\FPeval\difference{clip(\ImOut*(\rayon-#4))}
\newdimen\ecart
\ecart = \difference pt
\ifthenelse{\ecart>0}{\typeout{#1,#2,#3,#4}
\node[draw=black!75,shape=circle,fill=couleurext,minimum size=1.5mm,line width=0mm,inner sep=0] (x) at (#1,#2,#3) {};}{
%\node[draw=black!75,shape=circle,fill=couleurint,minimum size=1.5mm,line width=0mm,inner sep=0] (x) at (#1,#2,#3) {};
}}

\begin{document}
% graphique
\begin{tikzpicture}[background rectangle/.style={ultra thick,draw=none, top
color=white, bottom color=white},scale=2]
    % tracé du cube en 3D
    \begin{scope}[x={(.7cm,.4cm)},z={(.9cm,-.4cm)}] 
        \begin{scope}[every path/.style={thick}]
            \node(C) at (0,0,0) {};
            % arêtes du cube derrière la sphère
            \draw[couleurcube,thick] (-1,-1,-1) -- (1,-1,-1);
            \draw[couleurcube,thick] (1,-1,-1) -- (1,1,-1);
            \draw[couleurcube,thick] (1,-1,-1) -- (1,-1,1);
            % 
            % background grid
            \foreach \x in {1,...,\resolution}
                {
                \draw[couleurgrille,thin] (-1+\x*2/\resolution,-1,-1) -- (-1+\x*2/\resolution,1,-1);
                \draw[couleurgrille,thin] (-1,-1+\x*2/\resolution,-1) -- (1,-1+\x*2/\resolution,-1);
                \draw[couleurgrille,thin] (-1+\x*2/\resolution,-1,-1) -- (-1+\x*2/\resolution,-1,1);
                \draw[couleurgrille,thin] (-1,-1,-1+\x*2/\resolution) -- (1,-1,-1+\x*2/\resolution);
                \draw[couleurgrille,thin] (1,-1,-1+\x*2/\resolution) -- (1,1,-1+\x*2/\resolution);
                \draw[couleurgrille,thin] (1,-1+\x*2/\resolution,-1) -- (1,-1+\x*2/\resolution,1);
                }
        \end{scope}
    \end{scope}
    \def\norme{\rayonsphere}
    \begin{scope}[x={(.7cm,.4cm)},z={(.9cm,-.4cm)}]
            \tracepoint{0}{0}{0}{\norme};
            \tracepoint{-1}{0}{0}{\norme};
            \tracepoint{0}{-1}{0}{\norme};
            \tracepoint{0}{0}{-1}{\norme};
            \tracepoint{1}{0}{0}{\norme};
            \tracepoint{0}{1}{0}{\norme};
            \tracepoint{0}{0}{1}{\norme};
            \tracepoint{1}{1}{0}{\norme};
            \tracepoint{0}{1}{1}{\norme};
            \tracepoint{1}{0}{1}{\norme};
            \tracepoint{1}{-1}{0}{\norme};
            \tracepoint{0}{1}{-1}{\norme};
            \tracepoint{1}{0}{-1}{\norme};
            \tracepoint{-1}{1}{0}{\norme};
            \tracepoint{0}{-1}{1}{\norme};
            \tracepoint{-1}{0}{1}{\norme};
            \tracepoint{-1}{-1}{0}{\norme};
            \tracepoint{0}{-1}{-1}{\norme};
            \tracepoint{-1}{0}{-1}{\norme};
    \end{scope}

    % sphere        
    \filldraw[ball color=couleursphere,draw=none,opacity=0.55] (0,0) circle (\rayonsphere);

    % 3D cube
    \begin{scope}[x={(.7cm,.4cm)},z={(.9cm,-.4cm)}] 
        \begin{scope}[every path/.style={thick}]
            \node(C) at (0,0,0) {};
            \draw[couleurcube,thick] (1,1,-1) -- (-1,1,-1);
            \draw[couleurcube,thick] (-1,1,-1) -- (-1,-1,-1);
            \draw[couleurcube,thick] (-1,-1,-1) -- (-1,-1,1);
            \draw[couleurcube,thick] (1,1,-1) -- (1,1,1);
            \draw[couleurcube,thick] (-1,1,-1) -- (-1,1,1);
            \draw[couleurcube,thick] (-1,-1,1) -- (1,-1,1);
            \draw[couleurcube,thick] (1,-1,1) -- (1,1,1);
            \draw[couleurcube,thick] (1,1,1) -- (-1,1,1);
            \draw[couleurcube,thick] (-1,1,1) -- (-1,-1,1);
            % 
            % axes
            \draw[black,very thick] (-1,-1,-1) -- (-1,-1,1) node[midway,below=0.5cm] {$\xi_1$};
            \draw[black,very thick] (-1,-1,1) -- (1,-1,1) node[midway,below=0.5cm] {$\xi_2$};
            \draw[black,very thick] (1,-1,1) -- (1,1,1) node[midway,right=0.5cm] {$\xi_3$};
            \node[below=0.25cm,left] at (-1,-1,-1) {-1};
            \node[below=0.25cm,left] at (-1,-1,1) {1};
            \node[below=0.25cm,right] at (-1,-1,1) {-1};
            \node[below=0.25cm,right] at (1,-1,1) {1};
            \node[right=0.15cm] at (1,-1,1) {-1};
            \node[right=0.15cm] at (1,1,1) {1};
            %
            % points
            \def\ImOut{1}
            \typeout{outside}
            \tracepoint{0}{0}{0}{\norme};
            \tracepoint{-1}{0}{0}{\norme};
            \tracepoint{0}{-1}{0}{\norme};
            \tracepoint{0}{0}{-1}{\norme};
            \tracepoint{1}{0}{0}{\norme};
            \tracepoint{0}{1}{0}{\norme};
            \tracepoint{0}{0}{1}{\norme};
            \tracepoint{1}{1}{0}{\norme};
            \tracepoint{0}{1}{1}{\norme};
            \tracepoint{1}{0}{1}{\norme};
            \tracepoint{1}{-1}{0}{\norme};
            \tracepoint{0}{1}{-1}{\norme};
            \tracepoint{1}{0}{-1}{\norme};
            \tracepoint{-1}{1}{0}{\norme};
            \tracepoint{0}{-1}{1}{\norme};
            \tracepoint{-1}{0}{1}{\norme};
            \tracepoint{-1}{-1}{0}{\norme};
            \tracepoint{0}{-1}{-1}{\norme};
            \tracepoint{-1}{0}{-1}{\norme};
            %
        \end{scope}
    \end{scope}
\end{tikzpicture}               
\end{document}

Personally I would use asymptote for this. (Compile with pdflatex -shell-escape .)

\documentclass{standalone}
\usepackage{asypictureB}
\begin{document}
\begin{asypicture}{name=AsySphere}
size(400); // sphere from https://tex.stackexchange.com/q/244771/121799
import three;
import solids;
//unitsize(4cm);
settings.render=8;
pen linestyle1 = rgb(1,1,1)+linewidth(1.5pt)+opacity(1);

//currentprojection=perspective( camera=(1,.4,.9), target = (0,0,0));
//currentlight=nolight;

revolution S=sphere(O,1);
draw(surface(S),surfacepen=brown+opacity(.3));

draw(shift(0,0,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(-1,0,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,-1,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,0,-1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(1,0,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,1,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,0,1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(1,1,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,1,1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(1,0,1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(1,-1,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,1,-1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(1,0,-1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(-1,1,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,-1,1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(-1,0,1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(-1,-1,0)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(0,-1,-1)*scale3(0.03)*unitsphere,linestyle1);
draw(shift(-1,0,-1)*scale3(0.03)*unitsphere,linestyle1);

\end{asypicture} 
\end{document}

thank you ! The 3D effect on the points is exactly what I was looking for. — Alain, Commented Jul 14, 2018 at 13:22

score 5 · Accepted Answer · 2018-07-13 19:54:49Z

Somewhat like so?

\documentclass[pstricks]{standalone}
\usepackage[nomessages]{fp}
\usepackage{pst-solides3d}

\newcommand{\tracepoint}[3]{%
\FPeval{\somme}{clip(abs(#1)*abs(#1)+abs(#2)*abs(#2)+abs(#3)*abs(#3))}
\FPeval{\rayon}{clip(\somme^(0.5))}
\FPiflt\rayon 1
\psPoint(#1,#2,#3){x}\rput(x){\psdot[linecolor=yellow](0,0)}
\else
\psPoint(#1,#2,#3){x}\rput(x){\psdot[linecolor=blue](0,0)}
\fi
}
\begin{document}

\psset{viewpoint=30 40 20 rtp2xyz,Decran=56}
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
\psSolid[object=cube,a=2,opacity=0.2,action=draw*]% ,fillcolor=blue
\psSolid[object=sphere,r=1,linewidth=0.1pt,ngrid=50 50,fillcolor=red,opacity=0.1,action=draw*]%

\psPoint(1.25,0,-1.25){x}\rput(x){$\xi_1$}
\psPoint(0,1.25,-1.25){y}\rput(y){$\xi_2$}
\psPoint(-1.25,1.25,0){z}\rput(z){$\xi_3$}

\tracepoint{0}{0}{0}
\tracepoint{-1}{0}{0}
\tracepoint{0}{-1}{0}
\tracepoint{0}{0}{-1}
\tracepoint{1}{0}{0}
\tracepoint{0}{1}{0}
\tracepoint{0}{0}{1}
\tracepoint{1}{1}{0}
\tracepoint{0}{1}{1}
\tracepoint{1}{0}{1}
\tracepoint{1}{-1}{0}
\tracepoint{0}{1}{-1}
\tracepoint{1}{0}{-1}
\tracepoint{-1}{1}{0}
\tracepoint{0}{-1}{1}
\tracepoint{-1}{0}{1}
\tracepoint{-1}{-1}{0}
\tracepoint{0}{-1}{-1}
\tracepoint{-1}{0}{-1}
\tracepoint{0.5}{0}{0.5}
\end{pspicture}
\end{document}

Almost, the issue I have with the tikz mwe may be found here too: it is hard to see where are the blue and yellow dots since they are all actually plot over the sphere. — Alain, Commented Jul 14, 2018 at 0:37
thanks for the translation of the automated command with pstricks ! — Alain, Commented Jul 14, 2018 at 0:42

user151328user151328 · Accepted Answer · 2018-07-14 05:47:59Z

4

You can do a little trick:

Add some transparency to the points which are inside the sphere and the result looks a little better

\newcommand{\tracepoint}[3]{%
\FPeval{\somme}{clip(abs(#1)*abs(#1)+abs(#2)*abs(#2)+abs(#3)*abs(#3))}
\FPeval{\rayon}{clip(\somme^(0.5))}
\FPifgt\rayon 1
\psPoint(#1,#2,#3){x}\rput(x){\psdot[linecolor=yellow](0,0)}
\else
\psPoint(#1,#2,#3){x}\rput(x){\psdot[linecolor=blue,strokeopacity=0.3,opacity=0.3](0,0)}
\fi
}

answered Jul 14, 2018 at 5:47

user151328

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