Intersection of 2 surface plots in pgfplots

Question

My question is pretty much a duplicate of this one. I'd like pgfplots to display the intersection between tho surface plots properly. Unfortunately, my function is quite different from the one in the mentioned question, so I don't know, where the surfaces will meet. Would there be any kind of workaround for this kind of function?

\documentclass{scrartcl}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0.01:30]
\addplot3[surf] {0};
\addplot3[surf] {(1-0.3)*e^(-x*(y/100)*(1-0.3))-e^(-x*(y/100))};
\end{axis}
\end{tikzpicture}
\end{document}

Answer 1

Another hackish solution:

\documentclass{scrartcl}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0.01:30]
\addplot3[surf] {min(0.,(1-0.3)*e^(-x*(y/100)*(1-0.3))-e^(-x*(y/100))};
\addplot3[surf] {max(0.,(1-0.3)*e^(-x*(y/100)*(1-0.3))-e^(-x*(y/100)))};
\addplot3[domain=4:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({x},{118.89/x},{0.});
\addplot3[domain=0:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({x},{30.},{max(0.,(1-0.3)*e^(-x*(30./100)*(1-0.3))-e^(-x*(30./100)))});
\addplot3[domain=0:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({x},{0.},{max(0.,(1-0.3)*e^(-x*(0./100)*(1-0.3))-e^(-x*(0./100)))});
\addplot3[domain=0:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({x},{0.},{min(0.,(1-0.3)*e^(-x*(0./100)*(1-0.3))-e^(-x*(0./100)))});
\addplot3[domain=0:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({0.},{x},{max(0.,(1-0.3)*e^(-0.*(x/100)*(1-0.3))-e^(-0.*(x/100)))});
\addplot3[domain=0:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({30.},{x},{max(0.,(1-0.3)*e^(-30.*(x/100)*(1-0.3))-e^(-30.*(x/100)))});
\addplot3[domain=0:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({30.},{x},{min(0.,(1-0.3)*e^(-30.*(x/100)*(1-0.3))-e^(-30.*(x/100)))});
\end{axis}
\end{tikzpicture}
\end{document}

Note that this kind of solution is less flexible, because the correct hidden surface removal depends on the position of the camera, and also on special properties of shape of the functions. If one could know the point of view internally, one could generalize the max and min function to make it camera dependent and in this way simulate hidden surfaces.

Answer 2

A combination of surface colors, opacities and parametric plots can get you close to the desired result:

Code follows:

\documentclass{scrartcl}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0.01:30]
\addplot3[surf, opacity=0.25, blue, shader=flat] {0};
\addplot3[surf, opacity=0.25] {(1-0.3)*e^(-x*(y/100)*(1-0.3))-e^(-x*(y/100))};
\addplot3+[domain=4:30,samples=80,samples y=0,mark=none,black, opacity=0.5,thick]({x},{118.89/x},{0.});
\end{axis}
\end{tikzpicture}
\end{document}

Answer 3

This question has already tons of excellent answers, but as far as I can see none of them exploits that (1-0.3)*e^(-x*(y/100)*(1-0.3))-e^(-x*(y/100))=0 implies that y=-1000*ln(0.7)/(3*x)=:pft/x (pft is marmot language and means something like "convenient constant", but is hard to translate precisely;-), and that, in order to draw a plane, one really does not need to use \addplot3. All I am doing here is to draw the part x*y>pft first, than the curved surface, and then the x*y<pft part of the plane.

\documentclass[border=3.14mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}
\begin{axis}[domain=0.01:30,xlabel=$x$]
\pgfmathsetmacro{\pft}{-1000*ln(0.7)/3}
\fill[blue] plot[variable=\x,domain={\pft/30}:30] ({\x},{\pft/\x},0) -- 
(30,30,0) -- cycle;
\addplot3[surf] {(1-0.3)*e^(-x*(y/100)*(1-0.3))-e^(-x*(y/100))};
\fill[blue] plot[variable=\x,domain={\pft/30}:30] ({\x},{\pft/\x},0) -- 
(30,0,0) -- (0,0,0) -- (0,30,0) -- cycle;
\end{axis}
\end{tikzpicture}
\end{document}

Answer 4

Use asymptote, this is a more appropriate tool for plotting in 3D. With some additional work you can make it look like a pgfplots graph. Below there are three ways of doing this with asymptote.

Transparent surfaces

settings.outformat="png";
settings.render=16;

import graph3;
import palette;

size3(150,IgnoreAspect);

currentprojection=orthographic(50,-60,0.4);
currentlight=(5,-3,1);

real f(pair z){return (1.-0.3)*exp(-z.x*(z.y/100.)*(1.-0.3))-exp(-z.x*(z.y/100.));}
real cero(pair z){return 0.;}

limits((0,0,-0.3),(30,30,0.1));
xaxis3(Label("$x$",1),blue,arrow=Arrow3);
yaxis3(Label("$y$",1),blue,arrow=Arrow3);
zaxis3(Label("$z$",1),blue,arrow=Arrow3);

surface s=surface(f,(0,0),(30,30),30,30,Spline);
surface s2=surface(cero,(0,0),(30,30),30,30,Spline);

s.colors(palette(s.map(zpart),Rainbow()+opacity(0.5))); 
s2.colors(palette(s2.map(zpart),Rainbow()+opacity(0.5))); 

draw(s,meshpen=blue);
draw(s2,meshpen=red);

shipout(bbox(2mm,Fill(white)));

Solid surfaces

settings.outformat="png";
settings.render=16;

import graph3;
import palette;

size3(150,IgnoreAspect);

currentprojection=orthographic(50,-60,0.4);
//currentlight=(-5,3,20);

real f(pair z){return (1.-0.3)*exp(-z.x*(z.y/100.)*(1.-0.3))-exp(-z.x*(z.y/100.));}
real cero(pair z){return 0.;}

limits((0,0,-0.3),(30,30,0.1));
xaxis3(Label("$x$",1),blue,arrow=Arrow3);
yaxis3(Label("$y$",1),blue,arrow=Arrow3);
zaxis3(Label("$z$",1),blue,arrow=Arrow3);

surface s=surface(f,(0,0),(30,30),30,30,Spline);
surface s2=surface(cero,(0,0),(30,30),30,30,Spline);

//s.colors(palette(s.map(zpart),Rainbow+opacity(0.5))); 
//s2.colors(palette(s2.map(zpart),Rainbow()+opacity(0.5))); 

draw(s, blue,meshpen=blue);
draw(s2, red, meshpen=red);

shipout(bbox(2mm,Fill(white)));

PGFplots style (still with Asymptote)

This is the closest I was able to make it look like a pgfplots plot.

(Things to do, help is welcomed: 1) automatic point of view (to not depend on the scale of the plot, 2) thick border around each surface 3) vector graphics and hidden surface (e.g. via bsd module) 4) details, like small tics and tics spacing)

settings.outformat="png";
settings.render=8;
import grid3;
import graph3;
import palette;

size3(160,125,90,IgnoreAspect);
currentprojection=orthographic(30,-60,0.8);

real f(pair z){return (1.-0.3)*exp(-z.x*(z.y/100.)*(1.-0.3))-exp(-z.x*(z.y/100.));}
real cero(pair z){return 0.;}

limits((0,0,-0.3),(30,30,0.1));

xaxis3(Bounds(Min,Min), InTicks());
xaxis3(Bounds(Max,Max));
xaxis3(Bounds(Max,Min));
yaxis3(Bounds(Max,Min), InTicks(beginlabel=false) );
yaxis3(Bounds(Min,Max));
yaxis3(Bounds(Min,Min));
zaxis3(Bounds(Min,Min), InTicks() );
zaxis3(Bounds(Max,Max));
zaxis3(Bounds(Min,Max));

surface s=surface(f,(0,0),(30,30),24,24);
surface s2=surface(cero,(0,0),(30,30),24,24);

draw(s, mean(palette(s.map(zpart),Gradient(blue,yellow,red))),meshpen=0.2pt + black, nolight);
draw(s2, red, meshpen=0.2pt + black, nolight);
draw(s2, red, meshpen=0.2pt + black, nolight);

shipout(bbox(2mm,Fill(white)));

---

Notes

asymptote can output vector graphics, but the output is not as good as a raster (rendered) output and removal of hidden faces is very difficult (like with pgfplots.) So, this is big drawback of using asymptote, the plot becomes raster. However, it is common that 3D plots quickly become too complex to render with vector graphics.

Answer 5

Yet another hackish solution, based on this answer. I am adding it here in response to the comment by alfC:

\documentclass{beamer}
\usefonttheme[onlymath]{serif}
\setbeamersize{text margin left=10pt}
\setbeamersize{text margin right=10pt}

\usepackage{pgfplots}

\pgfplotsset{
    every axis/.append style={font=\scriptsize},
    plain/.style={every axis plot/.append style={mark=none},enlargelimits=false,grid=none},
    z-sort/.style={z buffer=sort,unbounded coords=jump},
}

\newcommand{\python}[1]{python -c "%
import math, sys; import numpy as np; import scipy.linalg as la;%
#1 np.savetxt(sys.stdout, data)%
"}

\newcommand<>{\pyplot}[3][]%
{\only#4{\addplot[#1] shell[prefix=fig/data/,id=#2,] {\python{#3}};}}

\newcommand<>{\pyplott}[3][]%
{\only#4{\addplot3[z-sort,#1] shell[prefix=fig/data/,id=#2,] {\python{#3}};}}

\newcommand<>{\pyload}[3][]%
{\only#4{\addplot[#1] table[x index=0,y index=#2] {fig/data/#3.out};}}

\newcommand<>{\pyloadt}[2][]%
{\only#3{\addplot3[z-sort,#1] table {fig/data/#2.out};}}

\newcommand{\pysave}[2]{
    \begin{tikzpicture}[overlay,opacity=0]
    \begin{axis} \pyplot{#1}{#2} \end{axis}
    \end{tikzpicture}
}

\begin{document}

\begin{frame}

\pysave{surf}{
    n = 31; x = np.linspace(0,30,n); y = x;
    X, Y = np.meshgrid(x, y);
    Z1 = np.zeros([n, n]);
    Z2 = (1-0.3)*np.exp(-X*(Y/100)*(1-0.3))-np.exp(-X*(Y/100));
    M1 = np.ones([n, n]);
    M2 = 2 * M1;
    N = np.ones([1, n]) * np.NaN;
    X = np.r_[X,  N, X ].reshape([-1, 1]);
    Y = np.r_[Y,  N, Y ].reshape([-1, 1]);
    Z = np.r_[Z1, N, Z2].reshape([-1, 1]);
    M = np.r_[M1, N, M2].reshape([-1, 1]);
    data = np.c_[X, Y, Z, M];
}

\begin{center}
\begin{tikzpicture}
\begin{axis}[
    plain,width=\textwidth,height=.8\textwidth,
    colormap={summap}{color=(green);color=(red);color=(yellow);},
]
    \pyloadt[surf,opacity=.7,mesh/cols=31,point meta=\thisrowno{3}]{surf};
\end{axis}
\end{tikzpicture}
\end{center}

\end{frame}

\end{document}

Here is the result:

So, we only need to provide one expression for each surface, and nothing manual for the intersection(s). And it works for any viewpoint. Such automated solution is needed when surfaces are very complex.

However, the intersection looks ugly because each patch is either drawn or not - while two intersecting patches themselves should be partially visible. I guess a single self-intersecting surface would have the same problem.