# Drawing intersecting 3D surface plots

I'm attempting to draw six semi-transparent intersecting spheres of different colors---two on each axis x, y, z and all joining at the origin---as shown in the figure below. Note how all the spheres intersect.

Seeing as this figure is created in Matlab, I tried converting the figure to pgfplots code, using matlab2tikz, but this did not work. Below is the Matlab code for the figure.

Theta=linspace(0,2*pi,200);
Phi=linspace(-pi/2,pi/2,200);
[theta,phi]=meshgrid(Theta,Phi);

rho1=cos(theta).*cos(phi);
rho2=-cos(theta).*cos(phi);
rho3=sin(theta).*cos(phi);
rho4=-sin(theta).*cos(phi);
rho5=sin(phi);
rho6=-sin(phi);

[x1,y1,z1]=sph2cart(theta,phi,rho1);
[x2,y2,z2]=sph2cart(theta,phi,rho2);
[x3,y3,z3]=sph2cart(theta,phi,rho3);
[x4,y4,z4]=sph2cart(theta,phi,rho4);
[x5,y5,z5]=sph2cart(theta,phi,rho5);
[x6,y6,z6]=sph2cart(theta,phi,rho6);

surf(x1,y1,z1,'FaceColor','red','EdgeColor','none')
hold on
surf(x2,y2,z2,'FaceColor','red','EdgeColor','none')
surf(x3,y3,z3,'FaceColor','blue','EdgeColor','none')
surf(x4,y4,z4,'FaceColor','blue','EdgeColor','none')
surf(x5,y5,z5,'FaceColor','green','EdgeColor','none')
surf(x6,y6,z6,'FaceColor','green','EdgeColor','none')

alpha(0.6)
camlight left
lighting flat


Next I tried drawing the figure in tikz-3dplot and pst-solides3d. With both of these methods, I had the problem that the surfaces didn't intersect. Instead, 2-dimensional projections of the spheres were drawn separately on the canvas, one on top of the other, as shown in the figures below. The first figure is the output from tikz-3dplot, and the second figure is the output from pst-solides3d.

With tikz-3dplot, I also had an issue with certain axes appearing to be on top of the spheres, when they should rather appear to be inside them (ref. the negative half of the y axis).

Below are the codes for the above figures, in the same order.

\documentclass[11pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{tikz,tikz-3dplot}
\usetikzlibrary{arrows}

\begin{document}

\tdplotsetmaincoords{70}{135}

\begin{tikzpicture}[tdplot_main_coords,fill  opacity=.5,>=latex]
\pgfsetlinewidth{.1pt}

\tdplotsphericalsurfaceplot{72}{36}{4*abs(cos(\tdplotphi)*sin(\tdplottheta))}{black!85!white}{blue}
{\draw[color=black,thick,->]  (-6,0,0) --  (6,0,0)  node[anchor=north  east]{\textbf{x}};}
{\draw[color=black,thick,->]  (0,-6,0) --  (0,6,0)  node[anchor=north  west]{\textbf{y}};}
{\draw[color=black,thick,->]  (0,0,-6) --  (0,0,6)  node[anchor=south]{\textbf{z}};}

\tdplotsphericalsurfaceplot{72}{36}{4*abs(cos(\tdplottheta))}{black!85!white}{green}
{\draw[color=black,ultra thin,->]  (-6,0,0) --  (6,0,0)  node[anchor=north  east]{};}
{\draw[color=black,ultra thin,->]  (0,-6,0) --  (0,6,0)  node[anchor=north  west]{};}
{\draw[color=black,ultra thin,->]  (0,0,-6) --  (0,0,6)  node[anchor=south]{};}

\tdplotsphericalsurfaceplot{72}{36}{4*abs(sin(\tdplotphi)*sin(\tdplottheta))}{black!85!white}{red}
{\draw[color=black,ultra thin,->]  (-6,0,0) --  (6,0,0)  node[anchor=north  east]{};}
{\draw[color=black,ultra thin,->]  (0,-6,0) --  (0,6,0)  node[anchor=north  west]{};}
{\draw[color=black,ultra thin,->]  (0,0,-6) --  (0,0,6)  node[anchor=south]{};}

\end{tikzpicture}

\end{document}


\documentclass[11pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{pstricks,pst-solides3d}

\begin{document}

\begin{pspicture}(-6,-6)(6,6)

\psset{viewpoint=100  100  100,Decran=150}
\axesIIID[showOrigin=true,mathLabel=false,axisemph=\textbf,labelsep=8pt](-7,-7,-7)(7,7,7)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=green,linecolor=black!85!white,linewidth=0.5pt,opacity=0.6,ngrid=15 24,name=green2](0,0,-2)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=blue,linecolor=black!85!white,linewidth=0.5pt,opacity=0.6,ngrid=15 24,name=blue2](-2,0,0)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=red,linecolor=black!85!white,linewidth=0.5pt,opacity=0.6,ngrid=15 24,name=red2](0,-2,0)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=green,linecolor=black!85!white,linewidth=0.5pt,opacity=0.6,ngrid=15 24,name=green1](0,0,2)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=red,linecolor=black!85!white,linewidth=0.5pt,opacity=0.6,ngrid=15 24,name=red1](0,2,0)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=blue,linecolor=black!85!white,linewidth=0.5pt,opacity=0.6,ngrid=15 24,name=blue1](2,0,0)

\end{pspicture}

\end{document}


How can I obtain the intended result and make the spheres actually intersect as shown in the first figure? Is there anything I can add to my methods above to make the spheres intersect? If not, are there other methods I can use to obtain the intended result? The output must be vector graphics and use the same font as the main document.

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Have you tried using asymptote? It's the only graphics-maker that commonly shows up on this site and has genuine hidden surface removal. –  Charles Staats May 11 '14 at 12:46
@CharlesStaats: I looked into asymptote, but from the few tutorials and documentation I could find, it seems asymptote has similar limitations as the methods presented above. From what I could find, making surfaces intersect and display correctly (e.g. axes appearing inside the spheres rather than outside) appears to be very cumbersome in asymptote. –  eiterorm May 11 '14 at 21:37
Theoretically, it's automatic. In practice, there are still a few bugs to be worked out when you want to intersect transparent surfaces. –  Charles Staats May 12 '14 at 1:53

You can use object=fusion to merge all the spheres to a single object. For that you must also set the option solidmemory:

\documentclass[11pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{pstricks,pst-solides3d}

\begin{document}

\begin{pspicture}(-6,-6)(6,6)

\psset{viewpoint=100  100  100,Decran=150}
\axesIIID[showOrigin=true,mathLabel=false,axisemph=\textbf,labelsep=8pt](-7,-7,-7)(7,7,7)
\psset{linecolor=black!85!white,linewidth=0.5pt, opacity=0.6, strokeopacity=0.6,
object=sphere, r=2, ngrid=15 24, action=none, solidmemory, grid}
\psSolid[fillcolor=green, name=green2](0,0,-2)
\psSolid[fillcolor=blue, name=blue2](-2,0,0)
\psSolid[fillcolor=red, name=red2](0,-2,0)
\psSolid[fillcolor=green, name=green1](0,0,2)
\psSolid[fillcolor=red, name=red1](0,2,0)
\psSolid[fillcolor=blue, name=blue1](2,0,0)

\psSolid[object=fusion, base=green1 green2 red1 red2 blue1 blue2, action=draw**]

\end{pspicture}

\end{document}


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I actually tried that before I posted, but it didn't work. It turns out this is only possible with the draw** action. draw and draw* will not provide intersections. I see you have also added a strokeopacity option, but it doesn't seem to do anything? Could you explain what it's for? –  eiterorm May 11 '14 at 21:27
the action keyword has nothing to do with intersections. You have to collect all objects before they are plotted, that is the only important fact. –  Herbert May 12 '14 at 6:04
@eiterorm That was a remainder from some tests: I tried to remove the grid lines with grid. But that seems only to change the linecolor to the color of the faces. That works fine for opaque colors, but apparently not for transparent ones (also strokeopacity=0 doesn't hide the lines). –  Christoph May 12 '14 at 7:36
@Herbert: well, on my end the type of action actually does affect the intersections. If I use the exact same code, but change draw** to either draw or draw*, the spheres no longer intersect. –  eiterorm May 12 '14 at 9:53
@Herbert: Not sure what you meant by this, but like I said, running the exact same code and only changing the number of asterisks on the draw action, determines whether or not the spheres intersect. –  eiterorm May 12 '14 at 11:59
\documentclass[pstricks]{standalone}
\usepackage{pst-solides3d}
\begin{document}

\begin{pspicture}(-6,-6)(6,6)
\psset{viewpoint=100 100 100,Decran=150,solidmemory,object=sphere,r=2,
action=none,linewidth=0.5pt,ngrid=45 60,linecolor=black!85!white}
\axesIIID[mathLabel=false,axisemph=\textbf,labelsep=8pt](-7,-7,-7)(7,7,7)
\psSolid[fillcolor=green,name=green2](0,0,-2)
\psSolid[fillcolor=blue,name=blue2](-2,0,0)
\psSolid[fillcolor=red,name=red2](0,-2,0)
\psSolid[fillcolor=green,name=green1](0,0,2)
\psSolid[fillcolor=red,name=red1](0,2,0)
\psSolid[fillcolor=blue,name=blue1](2,0,0)
\psSolid[object=fusion,base=green2 blue2 red2 green1 red1 blue1,action=draw**,opacity=0.9]
\end{pspicture}

\end{document}


The same with a lightsource:

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

\begin{pspicture}(-6,-6)(6,6)
\psset{viewpoint=60 45 30 rtp2xyz,lightsrc=viewpoint,
Decran=60,solidmemory,object=sphere,r=2,
action=none,linewidth=0.5pt,
ngrid=45 60,linecolor=black!45!white}
\axesIIID[axisemph=\mathbf,labelsep=8pt](-5,-5,-5)(5,5,5)
\psSolid[fillcolor=green,name=green2](0,0,-2)
\psSolid[fillcolor=blue,name=blue2](-2,0,0)
\psSolid[fillcolor=red,name=red2](0,-2,0)
\psSolid[fillcolor=green,name=green1](0,0,2)
\psSolid[fillcolor=red,name=red1](0,2,0)
\psSolid[fillcolor=blue,name=blue1](2,0,0)
\psSolid[object=fusion,base=green2 blue2 red2 green1 red1 blue1,action=draw**,opacity=0.7,grid]
\end{pspicture}

\end{document}


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Here is an solution for pstricks, but it's not very nice.

Thomas

\documentclass[11pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{pstricks,pst-solides3d}

\begin{document}

\begin{pspicture}(-6,-6)(6,6)

\psset{viewpoint=100 100 100,Decran=150,solidmemory}
\axesIIID[showOrigin=true,mathLabel=false,axisemph=\textbf,labelsep=8pt](-7,-7,-7)(7,7,7)

\psSolid[object=sphere,r=2,action=draw*,fillcolor=green,linecolor=black!85!white,linewidth=0.5pt,ngrid=45 60,name=green2,action=none](0,0,-2)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=blue,linecolor=black!85!white,linewidth=0.5pt,ngrid=45 60,name=blue2,action=none](-2,0,0)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=red,linecolor=black!85!white,linewidth=0.5pt,ngrid=45 60,name=red2,action=none](0,-2,0)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=green,linecolor=black!85!white,linewidth=0.5pt,ngrid=45 60,name=green1,action=none](0,0,2)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=red,linecolor=black!85!white,linewidth=0.5pt,ngrid=45 60,name=red1,action=none](0,2,0)
\psSolid[object=sphere,r=2,action=draw*,fillcolor=blue,linecolor=black!85!white,linewidth=0.5pt,ngrid=45 60,name=blue1,action=none](2,0,0)

\psSolid[object=fusion,base=green2 blue2 red2 green1 red1 blue1,action=draw**,opacity=0.7,linewidth=0.01pt,grid]
\end{pspicture}

\end{document}


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After tinkering with the solutions provided here, I decided to give my own, summarized answer. Based on the previous answers, it seems the pst-solides3d package is the best option. (Charles Staats suggests asymptote as another possible method.) As pointed out by Christoph, one can make the spheres intersect through object=fusion (which also requires the solidmemory option). For me, this only works when the action of the fusion object is set to draw** (see the discussion below Christoph's answer).

Herbert's answer shows two other examples, one with grid lines, and one with lighting. The answer by Thomas S shows a similar example both without lighting and without grid lines.

Both Herbert's and Christoph's examples show how the code can be written more efficiently.

In all these examples, however, the axes are drawn beneath the spheres, as the axes are not a part of the fusion. My solution to this is to draw two sets of axes on top of each other. The first one is drawn beneath the spheres, and the second one is drawn on top of the spheres where appropriate.

In my example, I've increased the resolution of the spheres to the maximum (higher resolutions seem to cause the compiler to break down), to make the intersections as smooth as possible. I've also added lighting and reduced the intensity of the colors to make the result easier on the eyes.

Thank you all for your help.

EDIT: Language.

\documentclass[11pt]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{pstricks,pst-solides3d}

\begin{document}

\begin{pspicture}(-6,-6)(6,6)

\psset{lightsrc=100 50 130,viewpoint=100 100 100,Decran=150}
\axesIIID[showOrigin=true,axisnames={}](-6.5,-6.5,-6.5)(6.5,6.5,6.5)

\psset{linewidth=0.5pt,opacity=0.7,object=sphere,r=2,ngrid=39 90,action=none,solidmemory,grid}

\psSolid[fillcolor=green!70!gray,name=green2](0,0,-2)
\psSolid[fillcolor=blue!70!gray,name=blue2](-2,0,0)
\psSolid[fillcolor=red!70!gray,name=red2](0,-2,0)
\psSolid[fillcolor=green!70!gray,name=green1](0,0,2)
\psSolid[fillcolor=red!70!gray,name=red1](0,2,0)
\psSolid[fillcolor=blue!70!gray,name=blue1](2,0,0)

\psSolid[object=fusion,base=green1 green2 red1 red2 blue1 blue2,action=draw**]

\psset{linewidth=0.8pt,opacity=1}
\axesIIID[showOrigin=false,mathLabel=false,axisemph=\textbf,labelsep=8pt](4,4,4)(6.5,6.5,6.5)

\end{pspicture}

\end{document}


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