Sketching conic sections as intersections of cone and plane?

Question

I would have thought this would be trivial, but I still haven't found an easy method. All I want to do is draw a cone, a plane, and their intersection (which will be a conic section). Something similar to, for example:

http://mathworld.wolfram.com/ConicSection.html

or the picture here:

http://www.ck12.org/book/CK-12-Algebra-II-with-Trigonometry-Concepts/section/10.0/

The same sort of standard pictures we've all seen a million times before.

I don't mind if I use a LaTeX drawing package, or something else, but I want to do it myself, so I can play around with the cone and plane to get the best possible diagram. I've tried a few online things such as 3dtin and shapesmith, but they're a bit flaky. It would be nice to use a vector drawing package and then export the result into LaTeX, but at this stage I'll settle for an image. It can also be a black and white wire diagram rather than a surface plot.

Has anybody created something like this before?

Answer 1

pst-solides3d seems a good option here. In the PS-Tricks gallery of 3D examples you have a nice starting point by Manuel Luque:

\documentclass{article}%    Manuel Luque
\usepackage{pst-solides3d}
\def\bicone{% generation du bicone
         /dZ 0.5 def
         /Rayon 3 def % rayon de la base
         /Hauteur 3 def % hauteur d'un cone
         /THETA Rayon Hauteur atan def
         /tanTHETA {Rayon Hauteur div} bind def
         /dPHI 10 def
         /tableau_des_sommets [
             Hauteur neg dZ Hauteur dZ sub{/Z exch def
             0 dPHI 360 {/PHI exch def
                  /r1 Z abs tanTHETA mul def
                  /r2 Z dZ add abs tanTHETA mul def
                  r1 PHI cos mul % x0
                  r1 PHI sin mul % y0
                  Z             % z0
                  r1 PHI dPHI add cos mul % x1
                  r1 PHI dPHI add sin mul % y1
                  Z                      % z1
                  r2 PHI dPHI add cos mul % x2
                  r2 PHI dPHI add sin mul % y2
                  Z dZ add               % z2
                  r2 PHI cos mul % x3
                  r2 PHI sin mul % y3
                  Z dZ add       % z3
             } for
             } for
                  ]
           def
         /Sommets {tableau_des_sommets aload pop} def
         /NbrePoints tableau_des_sommets length 3 div cvi def
        /Faces  {0 4 NbrePoints 4 sub {
                 /Ni exch def
               [ Ni Ni 1 add  Ni 2 add Ni 3 add]
                                     } for
              } def
        /Faces_internes {0 4 NbrePoints 4 sub {
                 /Ni exch def
               [Ni 3 add Ni 2 add Ni 1 add Ni]
                                     } for
             } def
             }
\begin{document}
\begin{center}
\psset{unit=0.7}
\psset{lightsrc=viewpoint}
\psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
\begin{pspicture}(-3,-5)(3,5)
\pstVerb{\bicone}%
\psSolid[object=new,sommets=Sommets,faces={Faces},
    hollow,
    incolor=yellow!25,
    intersectiontype=0,
    intersectionlinewidth=2,
    intersectioncolor=(rouge),
    intersectionplan={[0 0.5 1 1]}]
\axesIIID[linecolor=blue](0,0,3)(8.5,4.5,4)
\end{pspicture}
\begin{pspicture}(-5,-5)(5,5)
\pstVerb{\bicone}%
\psset{solidmemory}
\psSolid[object=new,sommets=Sommets,faces={Faces},
    hollow,
    plansepare={[0 0.5 1 1]},
    name=partbicone,
    action=none]
 \psSolid[object=load,
    hollow,rm=0,
    incolor=yellow!25,
    load=partbicone1,
    intersectiontype=0,
    intersectionlinewidth=2,
    intersectioncolor=(rouge),
    intersectionplan={[0 0.5 1 1.01]}]
\psSolid[object=plan,definition=equation,
    args={[0 0.5 1 1] 180},
    base=-3 3 -3 3,linewidth=0.01,
    planmarks,showBase,
    action=draw]
\psSolid[object=load,
    incolor=yellow!25,
    load=partbicone0](0,4,0)
\axesIIID[linecolor=blue](0,0,0)(8.5,4.5,4)
\end{pspicture}
\end{center}
\end{document}

The resulting image (taken directly from the PSTricks homepage):

Answer 2

The same a bit easier:

\documentclass[landscape]{article}
\usepackage{geometry}
\usepackage{pst-solides3d}
\begin{document}

\psset{unit=0.7,lightsrc=viewpoint,viewpoint=100 30 20 rtp2xyz,Decran=100}
\begin{pspicture}(-5,-5)(5,5)
\psSolid[object=cone,h=3,r=3,ngrid=5 30,intersectiontype=0,
   intersectionlinewidth=2,intersectioncolor=(red),
   intersectionplan={[0 0.5 1 1]}](0,0,-3)
\psSolid[object=conecreux,h=3,r=3,ngrid=5 30,RotX=180](0,0,3)
\axesIIID[linecolor=blue](0,0,3)(8.5,4.5,5)
\end{pspicture}
%
\begin{pspicture}[solidmemory](-4,-5)(7,5)
\psSolid[object=cone,h=3,r=3,ngrid=5 30,plansepare={[0 0.5 1 1]},
    name=partbicone,action=none](0,0,-3)
 \psSolid[object=load,hollow,rm=0,incolor=yellow!25,
    load=partbicone1,intersectiontype=0,intersectionlinewidth=2,
    intersectioncolor=(red),intersectionplan={[0 0.5 1 1.01]}]
\psSolid[object=plan,definition=equation,args={[0 0.5 1 1] 180},
    base=-3 3 -3 3,linewidth=0.01,planmarks,showBase,action=draw]
\psSolid[object=load,incolor=yellow!25,load=partbicone0](0,4,0)
\psSolid[object=conecreux,h=3,r=3,ngrid=5 30,RotX=180](0,4,3)
\axesIIID[linecolor=blue](0,0,3)(8.5,4.5,5)
\end{pspicture}

\end{document}