plot 3d graphs of functions defined implicitly

Question

How can I plot 3d graphs of functions defined implicitly (quadratic forms, for Linear Algebra course notes -- I'd like to include lots of examples)? As far as I can see it is not possible with pgfplots and not through gnuplot either. Is there any package that will help with that?

For example, parabolic and hyperbolic cylinders, hyperboloids, ellipsoids, etc. Concrete examples:

2xy + 2xz = 1 (hyperbolic cylinder)

(x^2)/(2^2) + (y^2)/(3^2) + (z^2)/(2^2) = 1 (ellipsoid)

I also see that Maxima can do this:

(%11) hc:2*x*y+2*x*z=2;
(%i2) draw3d(enhanced3d=true,implicit(hc,x,-5,5,y,-5,5,z,-5,5));

This will work fine (the hyperbolic cylinder is correctly plotted on the screen), but I don't know what backend Maxima uses for this, and I'd like to use a plain LaTeX method, or something that could be called from LaTeX, as I may have to send the document for others to compile themselves on different environments.

Answer 1

Asymptote contour3 package draws 3D surfaces described as the null space of real-valued functions of (x, y, z). Note that the images here are rendered into raster format (png).

% impsurf.tex :
%
\documentclass[10pt,a4paper]{article}
\usepackage{lmodern}
\usepackage{subcaption}
\usepackage[inline]{asymptote}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\begin{asydef}
settings.outformat="png";
settings.render=8;  
import graph3;
import contour3;
currentlight=light(gray(0.8),ambient=gray(0.1),specular=gray(0.7),
                     specularfactor=3,viewport=true,dir(42,48));
pen bpen=rgb(0.75, 0.7, 0.1);
material m=material(diffusepen=0.7bpen
,ambientpen=bpen,emissivepen=0.3*bpen,specularpen=0.999white,shininess=1.0);      
\end{asydef}

%
\begin{document}
%
\begin{figure}
\captionsetup[subfigure]{justification=centering}
    \centering
      \begin{subfigure}{0.49\textwidth}
\begin{asy}
size(200,0);
currentprojection=orthographic(camera=(9,10,4),up=Z,target=O,zoom=1);

// ellipsoid
real f(real x, real y, real z) {return (x^2)/(2^2) + (y^2)/(3^2) + (z^2)/(2^2)-1;}

draw(surface(contour3(f,(-3,-3,-3),(3,3,3),32)),m
     ,render(compression=Low,merge=true));

xaxis3(Label("$x$",1),-4,4,red);
yaxis3(Label("$y$",1),-4,4,red);
zaxis3(Label("$z$",1),-4,4,red);

\end{asy}
%
\caption{$(\frac{x}{2})^2+(\frac{y}{3})^2+(\frac{z}{2})^2= 1$ (ellipsoid)}
\label{fig:1a}
\end{subfigure}
%
\begin{subfigure}{0.49\textwidth}
\begin{asy}
size(200,0);
currentprojection=orthographic(camera=(9,4,4),up=Z,target=O,zoom=1);

// hyperbolic cylinder
real f(real x, real y, real z) {return 2*x*y + 2*x*z-1;}

draw(surface(contour3(f,(-3,-3,-3),(3,3,3),32)),m
     ,render(compression=Low,merge=true));

xaxis3(Label("$x$",1),-4,4,red);
yaxis3(Label("$y$",1),-4,4,red);
zaxis3(Label("$z$",1),-4,4,red);    
\end{asy}
%
\caption{$2xy + 2xz = 1$ (hyperbolic cylinder)}
\label{fig:1b}
\end{subfigure}
\caption{}
\label{fig:1}
\end{figure}
%
\end{document}
%
% Process:
%
% pdflatex impsurf.tex
% asy impsurf-*.asy
% pdflatex impsurf.tex

Answer 2

An example for the hyperbolic cylinder:

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

\begin{pspicture}(-4,-4)(4,4)% the main 2D area
\psset{lightsrc=viewpoint,viewpoint=50 -200 20 rtp2xyz,Decran=30}
\pstVerb{/constA 1 def /constB 1 def }
\defFunction[algebraic]{hcyl0}(u,v)
   { constA*SINH(u) }%                               x=f(u)
   { constB*COSH(u) }%                               y=f(u)
   { v }        %                               z=f(v)
\defFunction[algebraic]{hcyl1}(u,v)
   { constA*SINH(u) }%                               x=f(u)
   { -constB*COSH(u) }%                               y=f(u)
   { v }        %                               z=f(v)
\psSolid[object=surfaceparametree,base=-2 2 -3 3,
 fillcolor=red!40,function=hcyl0,linewidth=0.1\pslinewidth,ngrid=25]
\psSolid[object=surfaceparametree,base=-2 2 -3 3,
 fillcolor=red!40,function=hcyl1,linewidth=0.1\pslinewidth,ngrid=25]
\gridIIID[Zmin=-3,Zmax=3](-4,4)(-4,4)
\end{pspicture}

\end{document}

Answer 3

This answer is based on g.kov's excellent answer, but uses the relatively new smoothcontour3 module to produce a nicer-looking surface. The smoothcontour3 module has been incorporated into Asymptote version 2.33 (released 11 May 2015, just a little too late for the Tex Live 2015 cutoff), so if you have that version or later, you should not need to download the module. However, if you have an earlier version of Asymptote (for instance, the version incorporated into TeX Live 2015), you need to download the module from the line above or from the asymptote base code and put it in the same directory as the file you want to compile.

The code (again, heavily based on g.kov's code):

% impsurfsmooth.tex :
%
\documentclass[10pt,a4paper]{article}
\usepackage{lmodern}
\usepackage{subcaption}
\usepackage{asypictureB}
\usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
\begin{asyheader}
settings.outformat="png";
settings.render=8;
import graph3;
import smoothcontour3;

pen bpen = rgb(0.75, 0.7, 0.1);
material m = material(diffusepen=0.7bpen, ambientpen=bpen, emissivepen=0.3*bpen,
                      specularpen=0.999white, shininess=1.0);      
\end{asyheader}

%
\begin{document}
%
\begin{figure}
\captionsetup[subfigure]{justification=centering}
    \centering
      \begin{subfigure}[b]{0.49\textwidth}
\begin{asypicture}{name=ellipsoid}
size(200,0);
currentprojection = orthographic(9,10,4);

// ellipsoid
real f(real x, real y, real z) {
  return (x^2)/(2^2) + (y^2)/(3^2) + (z^2)/(2^2) - 1;
}

draw(implicitsurface(f, (-3,-3,-3), (3,3,3), overlapedges=true), surfacepen=m);

xaxis3(Label("$x$",1),-4,4,red);
yaxis3(Label("$y$",1),-4,4,red);
zaxis3(Label("$z$",1),-4,4,red);

\end{asypicture}
%
\caption{$(\frac{x}{2})^2+(\frac{y}{3})^2+(\frac{z}{2})^2= 1$ (ellipsoid)}
\label{fig:1a}
\end{subfigure}
%
\begin{subfigure}[b]{0.49\textwidth}
\begin{asypicture}{name=hyperboliccylinder}
size(200,0);
currentprojection=orthographic(9,4,4);

// hyperbolic cylinder
real f(real x, real y, real z) {
  return 2*x*y + 2*x*z - 1;
}

draw(implicitsurface(f, (-3,-3,-3), (3,3,3), overlapedges=true), surfacepen=m);

xaxis3(Label("$x$",1),-4,4,red);
yaxis3(Label("$y$",1),-4,4,red);
zaxis3(Label("$z$",1),-4,4,red);    
\end{asypicture}
%
\caption{$2xy + 2xz = 1$ (hyperbolic cylinder)}
\label{fig:1b}
\end{subfigure}
\caption{}
\label{fig:1}
\end{figure}
%
\end{document}
%
% Process:
%
% pdflatex --shell-escape impsurfsmooth.tex