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I wonder whether it is possible to draw the three following quadric surfaces (cone, hyperboloids)

x^2+y^2-z^2=0

x^2+y^2-z^2=1

x^2+y^2-z^2=-1

(and the axis)

using pgfplots 3d or TikZ as it is done in the following image

enter image description here

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2 Answers 2

Yes, it is certainly possible

screenshot

Explanation

When working with such surfaces, you have at least two choices:

  • work with the surfaces explicitly
  • work with the surfaces parametrically

If you work with the surfaces explicitly, then for your first surface

x^2+y^2-z^2=0

you end up using something like

enter image description here

which is pretty ugly. Instead, if you parameterize it using polar coordinates, then you get

enter image description here

which is much more elegant. You can apply this same approach for each of your other surfaces.

Code

The only thing to note about the code below is that instead of using r and theta, I've used x and y; it might be possible to change this- see the documentation for details.

\documentclass{article}

\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}
    \begin{axis}[
            xmin=-3,xmax=3,
            ymin=-3,ymax=3,
            zmin=-2,zmax=2,
            xlabel={$x$},
            ylabel={$y$},
            zlabel={$z$},
            zlabel style={rotate=90},
            view={60}{40}]
         % x^2+y^2-z^2=0
        \addplot3[mesh,black,domain=0:2,y domain=0:2*pi]({x*cos(deg(y))},{x*sin(deg(y))},{x});    
        \addplot3[mesh,black,domain=0:2,y domain=0:2*pi]({x*cos(deg(y))},{x*sin(deg(y))},{-x});    
         % x^2+y^2-z^2=-1
        \addplot3[mesh,blue,domain=0:2,y domain=0:2*pi]({x*cos(deg(y))},{x*sin(deg(y))},{sqrt(x^2+1)});    
        \addplot3[mesh,blue,domain=0:2,y domain=0:2*pi]({x*cos(deg(y))},{x*sin(deg(y))},{-sqrt(x^2+1)});    
        % x^2+y^2-z^2=1
        \addplot3[mesh,green,domain=1:2,y domain=0:2*pi,samples=15]({x*cos(deg(y))},{x*sin(deg(y))},{sqrt(x^2-1)});    
        \addplot3[mesh,green,domain=1:2,y domain=0:2*pi,samples=15]({x*cos(deg(y))},{x*sin(deg(y))},{-sqrt(x^2-1)});    
    \end{axis}
\end{tikzpicture}
\end{document}
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run it with xelatex

\documentclass{article}    
\usepackage{pst-solides3d}
\begin{document}

\psset{unit=0.8,viewpoint=50 60 30 rtp2xyz,Decran=50,lightsrc=viewpoint,
  ngrid=20 20, incolor=red!30, fillcolor=blue!30}
\begin{pspicture}(-4,-4)(4,4)
\defFunction[algebraic]{f1}(u,v){u*cos(v)}{u*sin(v)}{u}
\defFunction[algebraic]{f3}(u,v){u*cos(v)}{u*sin(v)}{sqrt(u^2+1)}
\defFunction[algebraic]{f4}(u,v){u*cos(v)}{u*sin(v)}{-sqrt(u^2+1)}
\defFunction[algebraic]{f5}(u,v){u*cos(v)}{u*sin(v)}{sqrt(u^2-1)}
\defFunction[algebraic]{f6}(u,v){u*cos(v)}{u*sin(v)}{-sqrt(u^2-1)}
\psSolid[object=surfaceparametree,base=-4 4 pi pi neg,function=f1]
\psSolid[object=surfaceparametree,function=f4,base=-1 4 pi pi neg,opacity=0.4]
\psSolid[object=surfaceparametree,function=f3,opacity=0.4]
\psSolid[object=surfaceparametree,function=f6,base=1 4 pi pi neg,
          fillcolor=red!30,incolor=blue!30,opacity=0.2]
\psSolid[object=surfaceparametree,function=f5,base=1 4 pi pi neg,opacity=0.2]
\gridIIID[Zmin=-4,Zmax=4](-4,4)(-4,4)
\end{pspicture}

\end{document}

enter image description here

or with viewpoint=50 60 10 rtp2xyz:

enter image description here

share|improve this answer
    
Although the OP asked specifically about PGF, I want to thank you for an amazing image! BTW: Where does the need for XeLaTeX arise? –  Dror Aug 10 '12 at 6:41
    
@Dror: Maybe it is the simplest compilation method among others such as pdflatex --shell-escape with \usepackage[pdf]{pstricks} or latex->dvips->ps2pdf combo. –  stalking is prohibited Aug 10 '12 at 7:22

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