pgfplots quadrics

Question

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

Answer 1

Yes, it is certainly possible

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

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

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}

Answer 2

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}

or with viewpoint=50 60 10 rtp2xyz: