How to make nice 3D "potatoes" with tikz (mathematical domain or structural mechanics)

Question

I am looking for an efficient way to create those 3D "potato" shapes frequently used in the description of mathematical domains or structural mechanics (an example is provided in Fig.1 of this link).

Because these shapes would be applied in the context of lecture notes, I would like to be able to manipulate the created shape with cut views and so on. I believe there are two suitable strategies:

tikz
asymptote

Aside from a few posts with respect to such shapes in 2D, I cannot find anything for 3D drawings...

Would you have any advice on this matter?

Ideally, I would like to reach this kind of result:

Thank you !

@Alain +1 for potato. However, aren't they simply 2D shapes drawn on a 3D axis? — Raaja_is_at_topanswers.xyz, May 30 '17 at 13:41
@RaajaG most of the time, I would say they are. But in the context of structural mechanics, you may want to "open" the potato to reveal internal components (typically stresses). In that case, having a 3D shape seems more appropriate. — Alain, May 30 '17 at 13:57
@RaajaG looking at old texts, I see a lot of 2-D potatoes in, e.g., Sokolnikov, Shames, & Timoshenko, though even they will use an occasional 3-D potato. But the Statics/Dynamics series by Beer & Johnson uses the 3-D potato in a prolific manner, particularly in its chapter on 3-D kinematics of rigid bodies. — Steven B. Segletes, May 30 '17 at 16:33
You say "potato", I say "well, I dunno what that thing is, but I've never seen a real potato that shape!" FWIW back in the early dates (1980s) of computational mechanics, those diagrams usually looked more like kidneys than potatoes, and the 3-D ones certainly weren't just solids of revolution. Maybe PETA has driven out the carnivores since then. — alephzero, May 30 '17 at 23:02
TikZ only does fake 3D. If you change perspective or rotate it or whatever, you start drawing again. — cfr, May 31 '17 at 01:56
@HenriMenke, well.. isn't that precisely the point of my question that I have no idea on how to make the mwe ? — Alain, May 31 '17 at 11:53
@alephzero I never heard of "kidneys" but it does make a lot of sense and certainly looks alike ! As a non anglophone native speaker, I can tell you that "potato" is widely used though in other languages :) — Alain, May 31 '17 at 11:55

score 11 · Accepted Answer · answered May 31 '17 at 21:23

Here is a brief Asymptote MWE:

// potato.asy

import graph3;
size(8cm,0);

currentprojection=orthographic(camera=(190,10,180),up=(0,-5,0),target=(0,0,0),zoom=1);

triple[] bot={( 34, 36,0),( 11, 64,0),(-23, 65,0),(-39, 32,0),(-30,  0,0),(-38,-38,0),(-22,-61,0),( 27,-61,0),( 45,-32,0),( 35, -5,0),};

guide3 gBot=graph(bot,operator..)..cycle;
guide3 gTop, gMid; 

gTop=shift(0,0, 50)*scale3(0.8)*gBot;
gMid=shift(0,0, 30)*scale3(1.1)*gBot;

triple f(pair uv){
    real u=uv.x, v=uv.y;
    guide3 g=relpoint(gBot,u)..relpoint(gMid,u)..relpoint(gTop,u);
    return relpoint(g,v);
}

surface s=surface(f,(0,0),(1,1),Spline);

draw(s,gray(0.85));
draw(surface(gTop),gray(0.87),nolight);

triple ref1=relpoint(gTop,0.6);
triple ref2=relpoint(gTop,0.9);
triple ref3=relpoint(gTop,0.4);

triple v0,dfn,dfs,df;

v0=0.5ref1+0.5ref2;
dfn=40*unit(cross(ref2-v0,ref3-v0));
dfs=50*unit(ref1-v0);
df =dfn+dfs;

real a=6, dz=0.01;

draw(
surface((v0+(-a,-a,dz))--(v0+(a,-a,dz))--(v0+(a,a,dz))--(v0+(-a,a,dz))--cycle )
,lightred,nolight
);

arrowbar3 ar=Arrow3(size=8);

draw(v0--(v0+dfn),red  +1.2bp, ar);
draw(v0--(v0+dfs),green+1.2bp, ar);
draw(v0--(v0+df ),blue +1.2bp, ar);

dot(v0,black+3bp);

label("$\Delta\mathbf{f_n}$",project(v0+dfn),plain.S);
label("$\mathbf{n}$",project(v0+dfn*0.5),plain.S);

label("$\Delta\mathbf{f_s}$",project(v0+dfs),plain.N);
label("$s$",project(v0+0.4dfs),plain.W);

label("$\Delta\mathbf{f}$",project(v0+df),plain.N);
label("$\Delta a$",project(v0+(0,a,dz)),plain.S);//rgb(0.97 ,0.6,0.6));
label("$M$",project(v0+(a,a,dz)),plain.W);//rgb(0.97    ,0.6,0.6));

triple fscar(real t){
    real u0=0.77, v0=0.6;
    real u1=0.81, v1=0.5;
    triple scarBot=f((0.81, 0.5));
    triple scarTop=f((0.77, 0.6));
    return f((u0*(1-t)+u1*t,v0*(1-t)+v1*t));
}

guide3 gscar=graph(fscar,0,1);

draw(gscar,red+1.4bp);

triple w0=relpoint(gscar,0);
triple w1=relpoint(gscar,0.5);
triple w2=relpoint(gscar,1);

triple dw=(-30,20,0);

draw((w0-dw)--(w0-0.07*dw),red+1.2bp,ar);
draw((w1-dw)--(w1-0.07*dw),red+1.2bp,ar);
draw((w2-dw)--(w2-0.07*dw),red+1.2bp,ar);

Compiled to potato.pdf with

asy -f pdf -noprc -render=0 potato.asy

The "potato" shape is based on a single contour

guide3 gBot=graph(bot,operator..)..cycle;

and two its scaled and shifted clones,

gTop=shift(0,0, 50)*scale3(0.8)*gBot;
gMid=shift(0,0, 30)*scale3(1.1)*gBot;

The point on the side surface is constructed as an u-v surface

triple f(pair uv){
    real u=uv.x, v=uv.y;
    guide3 g=relpoint(gBot,u)..relpoint(gMid,u)..relpoint(gTop,u);
    return relpoint(g,v);
}

for u,v=0..1:

surface s=surface(f,(0,0),(1,1),Spline);

Note that a "scar" is really a curve, located on the surface, using the same function f, scar points are found as

triple fscar(real t){
    real u0=0.77, v0=0.6;
    real u1=0.81, v1=0.5;
    triple scarBot=f((0.81, 0.5));
    triple scarTop=f((0.77, 0.6));
    return f((u0*(1-t)+u1*t,v0*(1-t)+v1*t));
}

for t=0..1:

guide3 gscar=graph(fscar,0,1);

score 1 · Answer 2 · answered May 31 '17 at 14:36

While I am not very happy of the result, here is a first attempt (still in 2D though) based on this post :

\documentclass{standalone}
\usepackage{tikz,pgfplots}
\pgfplotsset{compat=1.13}
\begin{document}

\begin{tikzpicture}[scale=0.5]
    % patate en arrière-plan
    \coordinate (K) at (-5,0);      
    \shade[ball color= gray!05!white, opacity=.3] (K) plot [smooth cycle,tension=0.7] coordinates {
    (-3+5,-0.95) (-3+5,-0.95) (-3+5,-0.95) (-2.9+5,-0.95) (0.75+5,-1) (1.75+5,2) (0.75+5,7) (-2.25+5,7) (-2.9+5,6.99) (-3+5,6.975) (-3+5,6.975) (-3+5,6.975) (-3.5+5,6) (-3.9+5,4.5) (-3.9+5,1.5) (-3.5+5,0)};  
    \coordinate (K) at (0,0);   
    \shade[ball color= gray!05!white, opacity=.3] (K) plot [smooth cycle,tension=0.7] coordinates {
    (-3.9,4.5) (-3.5,6)  (-3,6.975) (-3,6.975) (-3,6.975) (-3.1,6.95) (-8.25,6.5) (-8.55,1.5) (-7.25,-1) (-5.25,-0.8) (-3.1,-0.95) (-3,-0.95) (-3,-0.95) (-3,-0.95) (-3.5,0) (-3.9,1.5)};   
    \coordinate (K) at (-1.75,-4);      
    \shade[ball color= black!35, draw=black!35,very thick] (K) plot [smooth cycle,tension=0.7] coordinates {
    (-3,-0.95) (-2.5,1) (-2.5,5) (-3,6.975) (-3.5,6) (-3.9,4.5) (-3.9,1.5) (-3.5,0)};
    \node[black!35,above] at (-3,6.975) {$\mathcal{A}$};
    % centroïde de la section   
    \draw[green,very thick,-latex] (-3.25,3.15) -- (-1,6) node[right] {$\vec{F}_i$};
    \draw[green,very thick,-latex] (-3.25,3.15) -- (-1.5,1) node[right] {$\vec{M}_i$};
    \node[draw,shape=circle,fill=black!35,minimum size=1mm,line width=0mm,inner sep=0] at (-3.25,3.15) {};
    %
    \begin{scope}[x={(.7cm,.3cm)},z={(.8cm,-.4cm)}] 
        \begin{scope}[every path/.style={thick}]
            % repère
            \draw[-latex] (0,0,-7) -- (-2.5,0,-7) node[left] {$\vec{z}$};
            \draw[-latex] (0,0,-7) -- (0,3.5,-7) node[above] {$\vec{y}$};
            \draw[-latex] (0,0,-7) -- (0,0,-4.5) node[below] {$\vec{x}$};
            \node[above left] at(0,0,-7) {O};
            \node[draw,shape=circle,fill=blue,minimum size=1mm,line width=0mm,inner sep=0] at (0,0,-7) {};
        \end{scope}
    \end{scope}
    % efforts quelconques
    \draw[blue,ultra thick,latex-] (0.3+5,1) -- (2.7+5,0.8) node[right] {$\vec{F}_1$};
    \draw[blue,ultra thick,latex-] (0+5,5) -- (2+5,6) node[right] {$\vec{M}_2$}; 
    \draw[blue,ultra thick,latex-] (-6.9,5.2) -- (-9,6.5) node[left] {$\vec{F}_3$}; 
    \draw[blue,ultra thick,latex-] (-6,0.95) -- (-8,-1) node[left] {$\vec{M}_4$}; 
    % nom du solide: S
    \node[] at (-0.5+5,6) {$(\mathcal{S})$};
\end{tikzpicture}
\end{document}

How to make nice 3D "potatoes" with tikz (mathematical domain or structural mechanics)

2 Answers2