How to draw a stack of cubes (or convex polyhedra) in such a way that hidden lines get distinguished? - TeX

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marmot

I hope that some techniques will allow us in the future to simplify such tasks. As often in this context, the challenge is to order lists, and to thread over lists. In Mathematica, these are simple `Sort` and `Map` commands, and I do hope that in some future we will have similar tools in LaTeX, yet at this point to the best of my knowledge we do not. This is something that works, I think, but takes a while to compile.
```
\documentclass[tikz,border=3mm]{standalone}
\usepackage{tikz-3dplot}
\usetikzlibrary{3dtools} % https://github.com/marmotghost/tikz-3dtools
\makeatletter
\newif\if@pgf@list@first@item
\pgfkeys{/handlers/.add item/.code=\pgfkeys{\pgfkeyscurrentpath/.append={,#1}},
/handlers/.remove item/.code={%
\edef\pgfutil@tmpa{\pgfkeysvalueof{\pgfkeyscurrentpath}}%
\edef\pgfutil@tmpb{#1}%
\edef\pgfutil@tmpc{\noexpand\@removeelement{\pgfutil@tmpb}{\pgfutil@tmpa}{\noexpand\pgfutil@tmpa}}%
\pgfutil@tmpc
\pgfkeyssetvalue{\pgfkeyscurrentpath}{\pgfutil@tmpa}},% maybe test if the list is now empty?
}
\tikzset{3d/draw behind/.code 2 args={% #1 object, #2 list of objects in front
   %\typeout{Draw #1 behind #2.}
   \tikzset{3d/current path=#1,
   	3d/draw clipped against/.list={#2},
	3d/draw avoiding={#2}}
  },3d/draw clipped against/.code={% #1=clip path
  %\typeout{Clip \pgfkeysvalueof{/tikz/3d/current path} against #1.}
  \begin{scope}
   \tikzset{/tikz/3d/polyhedron/before hidden/.code={%\typeout{clip!}%
   	\clip[use named path/.expanded=#1];},
	/tikz/3d/polyhedron/before visible/.code={%\typeout{clip!}%
   	\clip[use named path/.expanded=#1];},
   	all hidden,
   	/tikz/polyhedron \pgfkeysvalueof{/tikz/3d/current path}} 
  \end{scope}},
  3d/draw avoiding/.code={% #1=list of paths to be avoided
  %\typeout{Draw \pgfkeysvalueof{/tikz/3d/current path} avoiding #1.}
  \begin{scope}
   \tikzset{protect/.list={#1}}
   \tikzset{/tikz/3d/polyhedron/before hidden/.code={%\typeout{avoid!}%
   	\tikzset{protect/.list={#1}}},
	/tikz/3d/polyhedron/before visible/.code={%\typeout{avoid #1!}%
   	\tikzset{protect/.list={#1}}},
   	/tikz/polyhedron \pgfkeysvalueof{/tikz/3d/current path}} 
  \end{scope}},3d/current path/.initial={},
  3d/draw ordered objects/.code={%
  \tikzset{3d/temporary list 1/.initial={#1},
  	3d/draw ordered object/.list={#1}}%  
  },3d/draw ordered object/.code={%
  % test if remaining list has only one element
  \edef\pgf@util@tempa{\noexpand\pgfutil@in@{,}{\pgfkeysvalueof{/tikz/3d/temporary list 1}}}% 
  \pgf@util@tempa
  \tikzset{3d/temporary list 1/.remove item={#1}}
  \ifpgfutil@in@% 
	\tikzset{3d/draw behind/.expanded={#1}{\pgfkeysvalueof{/tikz/3d/temporary list 1}}} 
  \else
	\begin{scope}
    \tikzset{polyhedron #1};
	\end{scope}
  \fi  
  }}
\makeatother
\def\pgfmathsetmacroFPU#1#2{\begingroup%
\pgfkeys{/pgf/fpu,/pgf/fpu/output format=fixed}%
\pgfmathsetmacro#1{#2}%
\pgfmathsmuggle#1\endgroup}%  
\tikzset{
	protect/.code={%\typeout{protecting #1.}
		\clip[overlay,reset cm,even odd clip,use named path=#1] 
		(-16383.99999pt,-16383.99999pt) rectangle (16383.99999pt,16383.99999pt);
}}
  
\begin{document}
	\foreach \Angle in {5,15,...,85} %{55} % {5,15,...,355} %  {5,25,...,355} % {345} %   {155} % 
	{\tdplotsetmaincoords{60}{\Angle}
		\begin{tikzpicture}[same bounding box=A,%<- newly added to `3dtools`
			scale=1,tdplot_main_coords,line join = round, 
			line cap = round, declare function={a = 4;b = 4;h=4;}]
	  \tikzset{3d/polyhedron/.cd,
	  		%face edges/.style={},%
			back/.style={3d/polyhedron/complete dashes,fill=none},
			%
			fore/.style={3d/visible,solid,fill=none,
				3d/polyhedron/edges have complete dashes=false},
			complete dashes,
			/tikz/all hidden/.style={%
				3d/polyhedron/fore/.style={3d/polyhedron/complete dashes,fill=none},
				}}
      \def\zbottom{0}
	  \def\icube{0}
	  \tikzset{set up cube/.code={\edef\icube{\the\numexpr\icube+1}%
		\pgfmathsetmacroFPU{\acube}{pow(##1,1/3)/2}% note that you may need #1 instead of ##1
		\path (-\acube,-\acube,\zbottom) coordinate (v-\icube-1)
		 (\acube,-\acube,\zbottom) coordinate (v-\icube-2)
		 (\acube,\acube,\zbottom) coordinate (v-\icube-3)
		 (-\acube,\acube,\zbottom) coordinate (v-\icube-4);
	  	\pgfmathsetmacroFPU{\zbottom}{\zbottom-2*\acube}%
		\path (-\acube,-\acube,\zbottom) coordinate (v-\icube-5)
		 (\acube,-\acube,\zbottom) coordinate (v-\icube-6)
		 (\acube,\acube,\zbottom) coordinate (v-\icube-7)
		 (-\acube,\acube,\zbottom) coordinate (v-\icube-8);
		\path[save named path=cube-\icube,convex hull of={v-\icube-1,v-\icube-2,%
			v-\icube-3,v-\icube-4,v-\icube-5,v-\icube-6,v-\icube-7,v-\icube-8}];
	    \expanded{\noexpand\tikzset{polyhedron cube-\icube/.style={%
			/tikz/3d/polyhedron/.cd,
			O={(0,0,\zbottom+\acube)},
			draw face with corners={{(v-\icube-1)},{(v-\icube-2)},{(v-\icube-3)},{(v-\icube-4)}},
			draw face with corners={{(v-\icube-1)},{(v-\icube-2)},{(v-\icube-6)},{(v-\icube-5)}},
			draw face with corners={{(v-\icube-2)},{(v-\icube-3)},{(v-\icube-7)},{(v-\icube-6)}},
			draw face with corners={{(v-\icube-3)},{(v-\icube-4)},{(v-\icube-8)},{(v-\icube-7)}},
			draw face with corners={{(v-\icube-4)},{(v-\icube-1)},{(v-\icube-5)},{(v-\icube-8)}},
			draw face with corners={{(v-\icube-5)},{(v-\icube-6)},{(v-\icube-7)},{(v-\icube-8)}},
 		}}}
		\ifnum\icube=1
		 \edef\lstcubes{cube-1}
		\else
		 \edef\lstcubes{cube-\icube,\lstcubes}
		\fi
		}}
	  \tikzset{set up cube/.list={1,8,27,64,125,216,343}}
	  \tikzset{3d/draw ordered objects/.expanded={\lstcubes}}
   \end{tikzpicture}}
\end{document} 
```
![ani.gif](/image?hash=7257986f43cd243fea8820bd4dfe4ec42ec28d3068eb1ab1e3e3a1054543288c)

I think the code is fine if one does not want to change the view. If you want to change the view, then it is much more efforts. Here is a start. (No, one cannot add this to `3dtools` because, to the best of my knowledge, it is not generalizable.) ``` \documentclass[tikz,border=2mm]{standalone} \pagecolor{yellow!30} \usetikzlibrary{3dtools} \pgfmathdeclarefunction{symbolicscreendepth}{1}{\begingroup \pgfmathsetmacro{\myP}{TD("#1")}% \pgfmathparse{screendepth(\myP)}% \pgfmathsmuggle\pgfmathresult\endgroup} \newcounter{ipath} \tikzset{3d/create and measure paths/.code={ \tikzset{3d/create and measure single path={#1_1}{#1_5}{#1_3}} \tikzset{3d/create and measure single path={#1_2}{#1_6}{#1_3}} \tikzset{3d/create and measure single path={#1_1}{#1_7}{#1_4}} \tikzset{3d/create and measure single path={#1_2}{#1_8}{#1_4}} },3d/create and measure single path/.code n args={3}{ \stepcounter{ipath} \path[save named path=\number\value{ipath}] (O) -- (#1) -- (#2) -- (#3) -- cycle; \pgfmathsetmacro{\mysd}{max(symbolicscreendepth("(#1')"), symbolicscreendepth("(#2')"),symbolicscreendepth("(#3')"))} \ifnum\value{ipath}=1 \edef\lstsd{\mysd} \else \edef\lstsd{\lstsd,\mysd} \fi } } \begin{document} \foreach \Angle in {5,15,...,355} {\begin{tikzpicture}[3d/install view={phi=\Angle,theta=70}, same bounding box=A, line cap=round,line join=round,declare function={a=2;}, c/.style={circle,fill,inner sep=1pt}] \path (0,0,0) coordinate (O) (-a,0,0) coordinate (A_1) (a,0,0) coordinate (A_2) (0,2*a,0) coordinate (A_3) (0,-2*a,0) coordinate (A_4) (-a,2*a,0) coordinate (A_5) (a,2*a,0) coordinate (A_6) (-a,-2*a,0) coordinate (A_7) (a,-2*a,0) coordinate (A_8) (0,-a,0) coordinate (B_1) (0,a,0) coordinate (B_2) (0,0,2*a) coordinate (B_3) (0,0,-2*a) coordinate (B_4) (0,-a,2*a) coordinate (B_5) (0,a,2*a) coordinate (B_6) (0,-a,-2*a) coordinate (B_7) (0,a,-2*a) coordinate (B_8) (0,0,-a) coordinate (C_1) (0,0,a) coordinate (C_2) (2*a,0,0) coordinate (C_3) (-2*a,0,0) coordinate (C_4) (2*a,0,-a) coordinate (C_5) (2*a,0,a) coordinate (C_6) (-2*a,0,-a) coordinate (C_7) (-2*a,0,a) coordinate (C_8) (-a,0,0) coordinate (A_1') (a,0,0) coordinate (A_2') (0,a,0) coordinate (A_3') (0,-a,0) coordinate (A_4') (-a,a,0) coordinate (A_5') (a,a,0) coordinate (A_6') (-a,-a,0) coordinate (A_7') (a,-a,0) coordinate (A_8') (0,-a,0) coordinate (B_1') (0,a,0) coordinate (B_2') (0,0,a) coordinate (B_3') (0,0,-a) coordinate (B_4') (0,-a,a) coordinate (B_5') (0,a,a) coordinate (B_6') (0,-a,-a) coordinate (B_7') (0,a,-a) coordinate (B_8') (0,0,-a) coordinate (C_1') (0,0,a) coordinate (C_2') (a,0,0) coordinate (C_3') (-a,0,0) coordinate (C_4') (a,0,-a) coordinate (C_5') (a,0,a) coordinate (C_6') (-a,0,-a) coordinate (C_7') (-a,0,a) coordinate (C_8'); \setcounter{ipath}{0} \tikzset{3d/create and measure paths/.list={A,B,C}} \pgfkeys{/my lists/.cd,% my initial array/.is array=\lstsd,% my values/.initial=\pgfkeysvalueof{/my lists/my initial array/content},% my values/.sort numeric list={\temp}{\templ},% my sorted array/.is array/.expanded={\temp},% my index machinery/.is array/.expanded={\templ}}% \tikzset{1/.style={fill=blue!30},2/.style={fill=blue!30}, 3/.style={fill=blue!30},4/.style={fill=blue!30}, 5/.style={fill=red!30},6/.style={fill=red!30}, 7/.style={fill=red!30},8/.style={fill=red!30}, 9/.style={fill=orange!60},10/.style={fill=orange!60}, 11/.style={fill=orange!60},12/.style={fill=orange!60}} \foreach \X in \templ {\path[style/.expanded=\X,use named path/.expanded=\X];} \end{tikzpicture}} \end{document} ``` ![ani.gif](/image?hash=14b01b2fffbe2d465dda8e54055366274696a0894099135869d8f358f979a436)

@marmot, re: [your answer](#a1838), To draw it, my way is set all coordinates of vertices and midpoints of a cube [like this](https://tex.stackexchange.com/questions/599983/zscale-is-not-recognized/600141#600141). I think, this is not a nice way. How can draw the figure with `3dtools` without setting all coordinates of vertices and midpoints of a cube?

@marmot, re: [your answer](#a1838), I copy this code from group of latex on facebook ``` \documentclass[tikz,borde=5mm]{standalone} \pagecolor{yellow!30} \begin{document} \begin{tikzpicture} \def\rm{4} \pgfmathsetmacro{\R}{sqrt(1+(1+sqrt(5))^2/4+(1+sqrt(5))/2)*\rm} \pgfmathsetmacro{\rh}{sqrt(1+(1+sqrt(5))^2/4-(1+sqrt(5))/2)*\rm}; \pgfmathsetmacro{\b}{\rm*(1+sqrt(5))/2} \pgfmathsetmacro{\goc}{acos(((\rm)^2-(\b)^2+(\R)^2)/(2*(\rm)*(\R)))} \clip (0,0) circle(1.2*\R); \foreach \i in{0,1,2}{ \path (90+\i*120:\rm)coordinate(A\i); \draw (0,0)--(A\i); } \foreach \j in{0,1,2}{ \pgfmathsetmacro{\jh}{\j+1} \path (90+\goc+ \j*120:\rh)coordinate(B\j); } \foreach \k in{0,1,...,5}{ \pgfmathsetmacro{\kh}{\k+1} \draw[blue](90+\goc+60*\k:\R)coordinate(C\k)--(90+\goc+60*\kh:\R); } \foreach \y/\mau in{0/pink,120/yellow!65,240/blue!30}{ \draw[blue,rotate=\y](90+\goc:\R)--(90+300+\goc:\rh); \draw[blue,rotate=\y,fill=\mau](0,0)--(-30:\rm)--([turn]60:\b)--([turn]120:2*\rm)--(0,0); \draw [blue,rotate=\y](90+\goc+240:\rh)--(90+\goc+240:\R)--(390+\goc:\R)--cycle; } \foreach \n/\mau in{0/blue!30,120/pink,240/yellow!65}{ \draw[blue,rotate=\n,fill=\mau](90:\rm)--(0,0)--([turn]-60:\rm)--++(90:\b)--([turn]-60:2*\rm)--++(-90:\b)--cycle; } \draw [dash pattern ={on 2pt off 2pt},gray](0,0)circle(\R cm); \foreach \x in{0,1,2}{ \fill[ball color=red](B\x)circle(3pt);} \draw[blue](B0)--(B2)--(B1)--cycle; \foreach \x in{0,1,...,5}{ \fill[ball color=red](C\x)circle(3pt);} \end{tikzpicture} \end{document} ```

Me neither. `xscale` and `yscale` do not scale things in the actual `x` or `y` directions, but horizontally and vertically. One could thus argue that the `xscale` and `yscale` keys are not named very intuitively, but given that the projection is 2-dimensional, there is no well-defined `zscale` key.

If you take the preamble from the answer on the left, it is possible to make this work, too, with the following main documen: ``` \begin{document} \foreach \Angle in {55} % {5,15,...,85} % {5,15,...,355} % {5,25,...,355} % {345} % {155} % {\tdplotsetmaincoords{60}{\Angle} \begin{tikzpicture}[same bounding box=A,%<- newly added to `3dtools` scale=1,tdplot_main_coords,line join = round, line cap = round, declare function={a = 4;b = 4;h=4;}] \tikzset{3d/polyhedron/.cd, %face edges/.style={},% back/.style={3d/polyhedron/complete dashes,fill=none}, % fore/.style={3d/visible,solid,fill=none, 3d/polyhedron/edges have complete dashes=false}, complete dashes, /tikz/all hidden/.style={% 3d/polyhedron/fore/.style={3d/polyhedron/complete dashes,fill=none}, }} \def\zbottom{0} \def\icube{0} \tikzset{set up cube/.code={\edef\icube{\the\numexpr\icube+1}% \pgfmathsetmacroFPU{\acube}{##1}% note that you may need #1 instead of ##1 \path (0,0,\zbottom) coordinate (v-\icube-1) (\acube,0,\zbottom) coordinate (v-\icube-2) (\acube,\acube,\zbottom) coordinate (v-\icube-3) (0,\acube,\zbottom) coordinate (v-\icube-4); \pgfmathsetmacroFPU{\zbottom}{\zbottom-\acube}% \path (0,0,\zbottom) coordinate (v-\icube-5) (\acube,0,\zbottom) coordinate (v-\icube-6) (\acube,\acube,\zbottom) coordinate (v-\icube-7) (0,\acube,\zbottom) coordinate (v-\icube-8); \path[save named path=cube-\icube,convex hull of={v-\icube-1,v-\icube-2,% v-\icube-3,v-\icube-4,v-\icube-5,v-\icube-6,v-\icube-7,v-\icube-8}]; \expanded{\noexpand\tikzset{polyhedron cube-\icube/.style={% /tikz/3d/polyhedron/.cd, O={(\acube/2,\acube/2,\zbottom+\acube/2)}, draw face with corners={{(v-\icube-1)},{(v-\icube-2)},{(v-\icube-3)},{(v-\icube-4)}}, draw face with corners={{(v-\icube-1)},{(v-\icube-2)},{(v-\icube-6)},{(v-\icube-5)}}, draw face with corners={{(v-\icube-2)},{(v-\icube-3)},{(v-\icube-7)},{(v-\icube-6)}}, draw face with corners={{(v-\icube-3)},{(v-\icube-4)},{(v-\icube-8)},{(v-\icube-7)}}, draw face with corners={{(v-\icube-4)},{(v-\icube-1)},{(v-\icube-5)},{(v-\icube-8)}}, %draw face with corners={{(v-\icube-5)},{(v-\icube-6)},{(v-\icube-7)},{(v-\icube-8)}}, }}} \ifnum\icube=1 \edef\lstcubes{cube-1} \else \edef\lstcubes{cube-\icube,\lstcubes} \fi }} \tikzset{set up cube/.list={1,2,3,4}} \tikzset{3d/draw ordered objects/.expanded={\lstcubes}} \end{tikzpicture}} \end{document} ``` ![Screen Shot 2021-01-11 at 7.35.51 AM.png](/image?hash=da22aaa0c3b2e840e41eb9b7a7e7c1bcb4d63e69153c42978e83ce5c86a94273)

@marmot, re: [your answer](#a1838), Another stack of cubes is simpler at [here](https://artofproblemsolving.com/wiki/index.php/2014_AMC_10A_Problems/Problem_19) Four cubes with edge lengths $1$, $2$, $3$, and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3$? ![ScreenHunter 1072.png](/image?hash=69815fcf00025faca9c164dd679a2bf286ad8dddab8005b29513ea876d40d724)

I think we should ask the author of the question, but I agree that the title has some room for improvement. One possible title could be "How to draw a stack of cubes (or convex polyhedra) in such a way that hidden lines get distinguished".

@marmot, re: [your answer](#a1838), Can you edit the question title to make it reflect the functionality and beauty of your answer? Because `How to draw figure?` won't help any Googler or internet roamer looking for such drawing mechanism or this specific drawing.

You're welcome! Maybe some day there will be simpler and more general codes to do this. In principle things are rather easy: decompose the objects into convex "atoms" and order them according to their screen depth. In practice it is a bit harder, the user has to decompose the objects on their own (not a problem here, we are talking about cubes) and as of now there is no universally working prescription of ordering.

1 Answer