How to draw a double napped cone with TikZ

Question

I would like to reproduce something like this in black and white using TikZ (just the figure not the writing) :

Here is the code I have so far, but I don't know how to draw the three planes properly :

\documentclass[tikz, border=3pt]{standalone}
\usepackage{tikz,tikz-3dplot}
\tdplotsetmaincoords{80}{45}
\tdplotsetrotatedcoords{-90}{180}{-90}

%% style for surfaces
\tikzset{surface/.style={draw=black, fill=white, fill opacity=.6}}

%% macros to draw back and front of cones
%% optional first argument is styling; others are z, radius, side offset (in degrees)
\newcommand{\coneback}[4][]{
  %% start at the correct point on the circle, draw the arc, then draw to the origin of the diagram, then close the path
  \draw[canvas is xy plane at z=#2, #1] (45-#4:#3) arc (45-#4:225+#4:#3) -- (O) --cycle;
  }
\newcommand{\conefront}[4][]{
  \draw[canvas is xy plane at z=#2, #1] (45-#4:#3) arc (45-#4:-135+#4:#3) -- (O) --cycle;
  }

\begin{document}
\begin{tikzpicture}[tdplot_main_coords]
  \coordinate (O) at (0,0,0);

  %% make sure to draw everything from back to front
  \coneback[surface]{-3}{2}{-10}
  \draw (0,0,-5) -- (O);
  \conefront[surface]{-3}{2}{-10}
  \draw[->] (-6,0,0) -- (6,0,0) node[right] {$x$};
  \draw[->] (0,-6,0) -- (0,6,0) node[right] {$y$};
  \coneback[surface]{3}{2}{10}
  \draw[->] (O) -- (0,0,5) node[above] {$z$};
  \conefront[surface]{3}{2}{10}
\end{tikzpicture}
\end{document}

Update : Here is the result thanks to these two posts (@Schrödinger's cat's answer / The fix from another question). The code contains a fix to cope with outdated tikz 3d library.

\documentclass[tikz, border=3pt]{standalone}
\usepackage{tikz,tikz-3dplot}
\tdplotsetmaincoords{80}{110}

%This piece of code fixes the bug%
\usetikzlibrary{3d}
\makeatletter
\tikzoption{canvas is xy plane at z}[]{%
  \def\tikz@plane@origin{\pgfpointxyz{0}{0}{#1}}%
  \def\tikz@plane@x{\pgfpointxyz{1}{0}{#1}}%
  \def\tikz@plane@y{\pgfpointxyz{0}{1}{#1}}%
  \tikz@canvas@is@plane
}
\makeatother
%%%%%%%%%%%%

%% style for surfaces
\tikzset{surface/.style={draw=black, left color=orange,right color=orange,middle
color=orange!60!#1, fill opacity=1},surface/.default=white}

%% macros to draw back and front of cones
%% optional first argument is styling; others are z, radius, side offset (in degrees)
\newcommand{\coneback}[4][]{
  %% start at the correct point on the circle, draw the arc, then draw to the origin of the diagram, then close the path
  \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) 
  arc(\tdplotmainphi-#4:\tdplotmainphi+180+#4:#3) -- (O) --cycle;
  }
\newcommand{\conefront}[4][]{
  \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) arc
  (\tdplotmainphi-#4:\tdplotmainphi-180+#4:#3) -- (O) --cycle;
  }

\newcommand{\conetruncback}[6][]{
  \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi+180+#4] 
  ({#3*cos(\t)},{#3*sin(\t)},#2)
  -- plot[variable=\t,domain=\tdplotmainphi+180-#4:\tdplotmainphi+#4] 
  ({#6*cos(\t)},{#6*sin(\t)},#5)
  --cycle;
  }

\newcommand{\conetruncfront}[6][]{
  \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi-180+#4] 
  ({#3*cos(\t)},{#3*sin(\t)},#2)
  -- plot[variable=\t,domain=\tdplotmainphi-180-#4:\tdplotmainphi+#4] 
  ({#6*cos(\t)},{#6*sin(\t)},#5)
  --cycle;
  }

\begin{document}
\begin{tikzpicture}[tdplot_main_coords]
  \coordinate (O) at (0,0,0);
  \conetruncback[surface=black]{-2}{4/3}{0}{-3}{2}
  \draw (0,0,-5) -- (0,0,-2);
  \conetruncfront[surface]{-2}{4/3}{0}{-3}{2}
  \draw[canvas is xy plane at z=-2,fill=green!40!black,fill opacity=1] (-3,-3) rectangle (3,3);
  \coneback[surface=black]{-2}{4/3}{-10}
  \draw (0,0,-2) -- (O);
  \conefront[surface]{-2}{4/3}{-10}
  \draw[canvas is xy plane at z=0,fill=teal,fill opacity=1] (-3,-3) rectangle (3,3);
  \draw[->] (-6,0,0) -- (7,0,0) node[below] {$x$};
  \draw[->] (0,-6,0) -- (0,6,0) node[right] {$y$};
  \coneback[surface=white]{2}{4/3}{10}
  \draw[-] (O) -- (0,0,2);
  \conefront[surface=black]{2}{4/3}{10}
  \draw[canvas is xy plane at z=2,fill=pink,fill opacity=1] (-3,-3) rectangle (3,3);
  \conetruncback[surface=white]{2}{4/3}{0}{3}{2}
  \draw[->] (0,0,2) -- (0,0,5) node[above] {$z$};
  \conetruncfront[surface=black]{2}{4/3}{0}{3}{2}
\end{tikzpicture}
\end{document}

And the result :

The cross hatching on the cone on the original book cover is nice. I wish we had a good way to do that in TikZ or asymptote. — Benjamin McKay, Nov 08 '19 at 12:52

score 16 · Accepted Answer · edited Oct 08 '21 at 23:24

You were almost there. Drawing a plane is as simple as saying

\draw[canvas is xy plane at z=2,fill=blue,fill opacity=0.6] (-4,-4) rectangle (4,4);

Other than that you need to draw the parts of the cone below and above the planes separately, which is why I added a macro for the truncated cone, \conetruncfront. I also replaced the hardcoded 45 with \tdplotmainphi.

\documentclass[tikz, border=3pt]{standalone}
\usepackage{tikz,tikz-3dplot}
\tdplotsetmaincoords{80}{45}
%% style for surfaces
\tikzset{surface/.style={draw=black, fill=white, fill opacity=.6}}
%% macros to draw back and front of cones
%% optional first argument is styling; others are z, radius, side offset (in degrees)
\newcommand{\coneback}[4][]{
  %% start at the correct point on the circle, draw the arc, then draw to the origin of the diagram, then close the path
  \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) 
  arc(\tdplotmainphi-#4:\tdplotmainphi+180+#4:#3) -- (O) --cycle;
  }
\newcommand{\conefront}[4][]{
  \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) arc
  (\tdplotmainphi-#4:\tdplotmainphi-180+#4:#3) -- (O) --cycle;
  }
\newcommand{\conetruncfront}[6][]{
  \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi-180+#4] 
  ({#3cos(\t)},{#3sin(\t)},#2)
  -- plot[variable=\t,domain=\tdplotmainphi-180-#4:\tdplotmainphi+#4] 
  ({#6cos(\t)},{#6sin(\t)},#5)
  --cycle;
  }
\begin{document}
\begin{tikzpicture}[tdplot_main_coords]
  \coordinate (O) at (0,0,0);
%% make sure to draw everything from back to front
  \coneback[surface]{-3}{2}{-10}
  \draw (0,0,-5) -- (O);
  \conefront[surface]{-3}{2}{-10}
  \draw[canvas is xy plane at z=-2,fill=green!60!black,fill opacity=0.6] (-4,-4) rectangle (4,4);
  \coneback[surface]{-2}{4/3}{-10}
  \draw (0,0,-2) -- (O);
  \conefront[surface]{-2}{4/3}{-10}
  \draw[canvas is xy plane at z=0,fill=blue,fill opacity=0.6] (-4,-4) rectangle (4,4);
  \draw[->] (-6,0,0) -- (6,0,0) node[right] {$x$};
  \draw[->] (0,-6,0) -- (0,6,0) node[right] {$y$};
  \coneback[surface]{3}{2}{10}
  \draw[->] (O) -- (0,0,5) node[above] {$z$};
  \conefront[surface]{3}{2}{10}
  \draw[canvas is xy plane at z=2,fill=purple,fill opacity=0.6] (-4,-4) rectangle (4,4);
  \draw[->] (0,0,2) -- (0,0,5) node[above] {$z$};
  \conetruncfront[surface]{2}{4/3}{0}{3}{2}
\end{tikzpicture}
\end{document}

However, I'd slightly change things to get

 \documentclass[tikz, border=3pt]{standalone}
    \usepackage{tikz,tikz-3dplot}
    \tdplotsetmaincoords{80}{45}
%% style for surfaces
\tikzset{surface/.style={draw=black, left color=yellow,right color=yellow,middle
color=yellow!60!#1, fill opacity=.6},surface/.default=white}

%% macros to draw back and front of cones
%% optional first argument is styling; others are z, radius, side offset (in degrees)
\newcommand{\coneback}[4][]{
  %% start at the correct point on the circle, draw the arc, then draw to the origin of the diagram, then close the path
  \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) 
  arc(\tdplotmainphi-#4:\tdplotmainphi+180+#4:#3) -- (O) --cycle;
  }
\newcommand{\conefront}[4][]{
  \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) arc
  (\tdplotmainphi-#4:\tdplotmainphi-180+#4:#3) -- (O) --cycle;
  }

\newcommand{\conetruncback}[6][]{
  \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi+180+#4] 
  ({#3*cos(\t)},{#3*sin(\t)},#2)
  -- plot[variable=\t,domain=\tdplotmainphi+180-#4:\tdplotmainphi+#4] 
  ({#6*cos(\t)},{#6*sin(\t)},#5)
  --cycle;
  }

\newcommand{\conetruncfront}[6][]{
  \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi-180+#4] 
  ({#3*cos(\t)},{#3*sin(\t)},#2)
  -- plot[variable=\t,domain=\tdplotmainphi-180-#4:\tdplotmainphi+#4] 
  ({#6*cos(\t)},{#6*sin(\t)},#5)
  --cycle;
  }

\begin{document}
\begin{tikzpicture}[tdplot_main_coords]
  \coordinate (O) at (0,0,0);
  \conetruncback[surface=black]{-2}{4/3}{0}{-3}{2}
  \draw (0,0,-5) -- (0,0,-2);
  \conetruncfront[surface]{-2}{4/3}{0}{-3}{2}
  \draw[canvas is xy plane at z=-2,fill=green!60!black,fill opacity=0.6] (-4,-4) rectangle (4,4);
  \coneback[surface=black]{-2}{4/3}{-10}
  \draw (0,0,-2) -- (O);
  \conefront[surface]{-2}{4/3}{-10}
  \draw[canvas is xy plane at z=0,fill=blue,fill opacity=0.6] (-4,-4) rectangle (4,4);
  \draw[-&gt;] (-6,0,0) -- (6,0,0) node[right] {$x$};
  \draw[-&gt;] (0,-6,0) -- (0,6,0) node[right] {$y$};
  \coneback[surface=white]{2}{4/3}{10}
  \draw[-] (O) -- (0,0,2);
  \conefront[surface=black]{2}{4/3}{10}
  \draw[canvas is xy plane at z=2,fill=purple,fill opacity=0.6] (-4,-4) rectangle (4,4);
  \conetruncback[surface=white]{2}{4/3}{0}{3}{2}
  \draw[-&gt;] (0,0,2) -- (0,0,5) node[above] {$z$};
  \conetruncfront[surface=black]{2}{4/3}{0}{3}{2}
\end{tikzpicture}
\end{document}

Or with slightly different view angles and opacity set to 1, and adjustments suggested by minhthien_2016.

\documentclass[tikz, border=3pt]{standalone}
\usepackage{tikz,tikz-3dplot}
\tdplotsetmaincoords{80}{60}
%% style for surfaces
\tikzset{surface/.style={draw=black, left color=yellow,right color=yellow,middle
        color=yellow!60!#1, fill opacity=1},surface/.default=white}
%% macros to draw back and front of cones
%% optional first argument is styling; others are z, radius, side offset (in degrees)
\newcommand{\coneback}[4][]{
    %% start at the correct point on the circle, draw the arc, then draw to the origin of the diagram, then close the path
    \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) 
    arc(\tdplotmainphi-#4:\tdplotmainphi+180+#4:#3) -- (O) --cycle;
}
\newcommand{\conefront}[4][]{
    \draw[canvas is xy plane at z=#2, #1] (\tdplotmainphi-#4:#3) arc
    (\tdplotmainphi-#4:\tdplotmainphi-180+#4:#3) -- (O) --cycle;
}
\newcommand{\conetruncback}[7][]{
    \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi+180+#4] 
    ({#3cos(\t)},{#3sin(\t)},#2)
    -- plot[variable=\t,domain=\tdplotmainphi+180-#7:\tdplotmainphi+#7] 
    ({#6cos(\t)},{#6sin(\t)},#5)
    --cycle;
}
\newcommand{\conetruncfront}[7][]{
    \draw[line join=round,#1] plot[variable=\t,domain=\tdplotmainphi-#4:\tdplotmainphi-180+#4] 
    ({#3cos(\t)},{#3sin(\t)},#2)
    -- plot[variable=\t,domain=\tdplotmainphi-180-#7:\tdplotmainphi+#7] 
    ({#6cos(\t)},{#6sin(\t)},#5)
    --cycle;
}
\begin{document}
    \begin{tikzpicture}[tdplot_main_coords]
    \coordinate (O) at (0,0,0);
    \conetruncback[surface=black]{-2}{4/3}{-5}{-3}{2}{5}
    \draw (0,0,-5) -- (0,0,-2);
    \conetruncfront[surface]{-2}{4/3}{-5}{-3}{2}{5}
    \draw[canvas is xy plane at z=-2,fill=green!60!black,fill opacity=1] (-4,-4) rectangle (4,4);
    \coneback[surface=black]{-2}{4/3}{-10}
    \draw (0,0,-2) -- (O);
    \conefront[surface]{-2}{4/3}{-10}
    \draw[canvas is xy plane at z=0,fill=blue,fill opacity=1] (-4,-4) rectangle (4,4);
    \draw[->] (-6,0,0) -- (6,0,0) node[right] {$x$};
    \draw[->] (0,-6,0) -- (0,6,0) node[right] {$y$};
    \coneback[surface=white]{2}{4/3}{10}
    \draw[-] (O) -- (0,0,2);
    \conefront[surface=black]{2}{4/3}{10}
    \draw[canvas is xy plane at z=2,fill=purple,fill opacity=1] (-4,-4) rectangle (4,4);
    \conetruncback[surface=white]{2}{4/3}{5}{3}{2}{-5}
    \draw[->] (0,0,2) -- (0,0,5) node[above] {$z$};
    \conetruncfront[surface=black]{2}{4/3}{5}{3}{2}{-5}
    \end{tikzpicture}
\end{document}

Thanks for your answer, however I tested the second version of your code and even after trying different values for \tdplotsetmaincoords{80}{60}, I can't get the same output as you. I've updated my post with a screenshot — Emperor_Udan, Nov 03 '19 at 20:25
@Emperor_Udan You might have an old version of the 3d library on your machine, in which the xy plane had a bug. The cleanest solution will be to update your installation. Alternatively you could use yx plane for the planes. Please add the very precise code that you run, and indicate the compiler you are using. — , Nov 03 '19 at 20:28
I think you're right, I'm using TexLive from the Ubuntu 18.04.3 LTS repo so it might be quite old. — Emperor_Udan, Nov 03 '19 at 20:31
This fixed the problem, thanks a lot. Should I update my post, to include the code with the patch ? — Emperor_Udan, Nov 03 '19 at 20:36
@Emperor_Udan This is up to you. Anyway, I am glad it worked. ;-) — , Nov 03 '19 at 20:37
@Schrödinger'scat I tried your code and zoom 500/100, how about the base of above cone? Not smooth. — minhthien_2016, Nov 04 '19 at 15:15
@minhthien_2016 Thanks! I tried to fix it. (To be honest, is this the answer that has problems with not everything being smooth? ;-) — , Nov 04 '19 at 16:34
@Schrödinger'scat This you very much. Please see at here https://chat.stackexchange.com/rooms/100707/room-for-minhthien-2016-and-marmot https://chat.stackexchange.com/rooms/100707/room-for-minhthien-2016-and-marmot — minhthien_2016, Nov 05 '19 at 15:52

user187802 · Answer 2 · 2019-11-04T14:19:57.690

Run with xelatex:

\documentclass[border=10pt,pstricks]{standalone}
\usepackage{pst-solides3d}
\begin{document}

\psset{unit=0.5}
\begin{pspicture}[linewidth=0.1pt](-7,-7)(7,7)
\psset[pst-solides3d]{viewpoint=20 10 20 rtp2xyz,Decran=50,lightsrc=20 10 5,solidmemory}
\psSolid[object=grille,base=-2 2 -2 2,ngrid=30,name=A](0,0,-1.5)
\psSolid[object=grille,base=-2 2 -2 2,ngrid=30,name=B](0,0,0)
\psSolid[object=grille,base=-2 2 -2 2,ngrid=30,name=C](0,0,1.5)
\defFunction{cone}(u,v){u v Cos mul}{u v Sin mul}{u}
\psSolid[object=surfaceparametree,base=-2 2 0 2 pi mul,
  inhue=0.8 0.2,hue=0.8 0.2,function=cone,linewidth=0.1pt,ngrid=25 40,name=D]
\psSolid[object=fusion,base=A B C D,linecolor=black!10]
\axesIIID(0,0,1.5)(2,2,3)
\end{pspicture} 

\end{document}

That was not exactly what I was looking for, but it looks good. I could use it in the future thanks — Emperor_Udan, Nov 04 '19 at 18:47

score 4 · Answer 3 · answered Oct 02 '22 at 05:08

An Asymptote solution for fun! I have not cleaned my code yet.

Without opacity (opacity(1))

With opacity(.5)

// http://asymptote.ualberta.ca/
import graph3;
currentprojection=orthographic(2,2,.5,zoom=.9);
unitsize(1cm);
pen plane1=red, plane2=cyan, plane3=blue,pcone=yellow;
real a=1.8,h=3; 
surface c=scale(a,a,h)*shift(0,0,-1)*unitcone;
draw(zscale3(-1)*c,pcone+opacity(.5));
draw(c,pcone+opacity(.3));
real b=2.2;
path3 g=scale3(b/a)unitcircle3;
draw(shift(0,0,b)g,plane1+1pt);
draw(shift(0,0,-b)*g,plane3+1pt);
surface pl=shift(-3,-4,0)scale(6,8,0)unitplane;
draw(shift(0,0,b)pl,plane1+opacity(.3));
draw(pl,plane2+opacity(.3));
draw(shift(0,0,-b)pl,plane3+opacity(.3));
draw(Label("$\eta_1$",EndPoint),O--4X,Arrow3);
draw(Label("$\eta_2$",EndPoint),O--5Y,Arrow3);
draw(Label("$\eta_3$",EndPoint,align=E),-3Z--4*Z,Arrow3);

How to draw a double napped cone with TikZ

3 Answers3