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Suppose we are working in the three-dimensional real vector space V. A cone is a subset C of V such that if an element v of V lies in C, then all of its nonnegative multiples also lie in C. I would like to draw, using TikZ, a cone that simulates infinitely many extremal rays, all accumulating toward a fixed one. A section of the cone should therefore look like a “polygon with infinitely many sides”.

For example, the cone here Tikz: cones with a wide base is a circular one. I would like mine to be "polyhedral" away from the accumulation ray.

How would you do that?

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7
  • Do you have a reference picture? I’m having a little trouble visualizing this. Commented yesterday
  • 1
    The best approximation I have is here (picture on page 78): math.ens.psl.eu/~debarre/M2.pdf Commented yesterday
  • So you want a more lightly tessellated cone, so that you can see the polygonal shape? Commented yesterday
  • What do you mean by "more lightly tessellated"? Sorry, I don't think I've got it... Commented yesterday
  • 1
    Ahh, you want a sampling which is not an arithmetic sequence. Commented yesterday

3 Answers 3

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2

I did it using powers of 0.5, with help from copilot.

I had to fiddle with it a bit, so the code is a bit clunky, but this should be it.

\documentclass[tikz,border=1cm]{standalone}
\usepackage{lua-tikz3dtools} % https://github.com/Pseudonym321/TikZ-Animations/tree/master1/TikZ/lua-tikz3dtools
\begin{document}
\begin{luatikztdtoolspicture}[
    C1 = { {{-10,-10,-10,1}} }
    ,C2 = { {{10,10,10,1}} }
    ,light = { {{0,0,1,1}} }
]
    % Parameters
    \def\N{6} % number of faces
    \def\decay{0.5} % shrink factor

    \pgfmathsetmacro{\uA}{0}
    \pgfmathsetmacro{\uB}{pi}
    \pgfmathsetmacro{\udiff}{\uB-\uA}
    \pgfmathsetmacro{\ustart}{\uA}

    \foreach \p in {-1,1} {
    \foreach \k in {0,...,\N} {
        \pgfmathsetmacro{\udelta}{\udiff*pow(\decay,\k)}
        \pgfmathsetmacro{\uend}{\ustart+\udelta}
        \pgfmathparse{\k!=0 && \k!=1}
        \ifnum\pgfmathresult=1
        \appendsurface[
            ustart = {\ustart}
            ,ustop = {\uend}
            ,usamples = {2}
            ,vstart = {0}
            ,vstop = {tau}
            ,vsamples = {2}
            ,x = {-v}
            ,y = {v*cos(u)/5}
            ,z = {\p*v*sin(u)/5}
            ,transformation = {euler(pi/2,pi/3,pi/6)}
            ,fill options = {
                preaction = {fill = white, fill opacity = 1}
                ,postaction = {
                    draw = black
                    ,ultra thin
                    ,line join = round
                }
            }
        ]
        \fi
        \pgfmathparse{\k==1}
        \ifnum\pgfmathresult=1
        \appendsurface[
            ustart = {pi}
            ,ustop = {\uend}
            ,usamples = {2}
            ,vstart = {0}
            ,vstop = {tau}
            ,vsamples = {2}
            ,x = {-v}
            ,y = {v*cos(u)/5}
            ,z = {\p*v*sin(u)/5}
            ,transformation = {euler(pi/2,pi/3,pi/6)}
            ,fill options = {
                preaction = {fill = white, fill opacity = 1}
                ,postaction = {
                    draw = black
                    ,ultra thin
                    ,line join = round
                }
            }
        ]
        \fi
        \global\let\ustart\uend
    }
    }
\end{luatikztdtoolspicture}
\end{document}

output

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  • Dear Jasper, that’s already amazing, thanks a lot. If you could draw a single cone that is symmetric with respect to the plane on which the current one "sits on", it would be perfect. I’m not sure if I explained that properly. Commented yesterday
  • 1
    @Fradns It was an interesting diagram. If you want it to approach closer, just increase N. Commented yesterday
  • Dear Jasper, thanks a lot, and thanks for your help. I'd have one last request: could you please make the two accumulation sides coincide (now they are on opposite sides) and delete the horizontal line in the interior of the cone? Commented yesterday
  • 1
    Thank you very much Jasper! Commented yesterday
  • 1
    @Fradns A word of caution: This package is Experimental, so it's syntax will change in the future. Commented yesterday
5

We can construct a pyramid whose base is a polygon with an accumulation point; here is an example using luaDraw.

\documentclass[border=5pt]{standalone}% compile with lualatex only
\usepackage[svgnames]{xcolor}
\usepackage[3d]{luadraw}%https://github.com/pfradin/luadraw
\usepackage{fourier-otf}
%https://tex.stackexchange.com/questions/755843/how-to-draw-cone-with-infinitely-many-extremal-rays
\begin{document}
\begin{luadraw}{name=quasi-cone}
local pi = math.pi
local g = graph3d:new{adjust2d=true, bbox=false, bg="lightgray"}
local S, A = -4*vecJ, M(-4,5,-1) -- apex of the pyramid and a point on the base
local nbdot, base = 25, {A}
local angle, ratio, theta = 180, 1/3, 0
for k = 1, nbdot do
    theta = theta + ratio*angle
    table.insert(base, rotate3d(A,theta,{Origin,-vecJ}) )
    table.insert(base,1, rotate3d(A,theta,{Origin,vecJ}) )
    angle = (1-ratio)*angle
end
local P = pyramid(base,S,true)
table.remove(P.facets) -- remove the last facet
g:Dpoly(P, {color="Crimson", opacity=0.7,edgecolor="Gold",edgewidth=6})
g:Show()
\end{luadraw}
\end{document}

enter image description here

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3

I tried with luadraw

\documentclass[12pt]{standalone}
\usepackage[svgnames]{xcolor}
\usepackage[3d]{luadraw}
\begin{document}
%https://tex.stackexchange.com/questions/755843/how-to-draw-cone-with-infinitely-many-extremal-rays
\begin{luadraw}{name=regular_pyramid_07_12_2025}
local g = graph3d:new{window={-10,10,-5,11},viewdir={30,70},size={12,12}}
g:Linejoin("round"); g:Linewidth(6); Hiddenlines = true; Hiddenlinestyle = "dashed"
local a, n, h = 3, 20, 10
local O = Origin
local P = regular_pyramid(n,a,h,O)
g:Dpoly(regular_pyramid(n,a,h,O), {mode=4,color="Crimson"})
g:Show()
\end{luadraw}
\end{document}

enter image description here

you can use the function cvx_hull3d(L) to make a section.

\documentclass[12pt]{standalone}
\usepackage[svgnames]{xcolor}
\usepackage[3d]{luadraw}
\begin{document}
\begin{luadraw}{name=section_cone}
local g = graph3d:new{window={-10,10,-5,21},viewdir={50,70},size={12,12}}
 Hiddenlines = true; Hiddenlinestyle = "dashed"
local a, n, h = 5, 10, 18
local O = Origin
local P = regular_pyramid(n,a,h,O)
local A,B,C,D,E,F,G,H,K,L,S = table.unpack(P.vertices)
local L = {A,B,C,D,E,F,G,H,S}
g:Dscene3d(
g:addFacet(cvx_hull3d(L), {color="Orange",edge=true})
)
g:Show()
\end{luadraw}
\end{document}

enter image description here

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2
  • Thank you for your contribution! I think the author specifically wanted a quasi-"cone", in which the triangular faces are not evenly sized. In particular, they should decrease in size sequentially, approaching some finite limit. Commented 18 hours ago
  • Thank you very much. Commented 18 hours ago

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