How can I draw a cylinder inscribed a sphere? - TeX

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marmot

The computation of the critical angles that deterimine the points at which the solid line turns dashed, and vice versa, is identical to the one for [the base of the inscribed cone](https://topanswers.xyz/tex?q=1239#a1472). I also added a more general discussion to the [3dtools manual](https://github.com/marmotghost/tikz-3dtools/blob/master/3DToolsManual.pdf). of The vertical boundaries are correct, I think.

```
\documentclass[border=3mm,tikz,fleqn]{standalone}
\usepackage{tikz-3dplot}
\begin{document}
\tdplotsetmaincoords{60}{60}
\begin{tikzpicture}[tdplot_main_coords, axis/.style={blue,thick},
		declare function={R=4;a=8;h=4;r=sqrt(R*R-h*h/4);alpha=\tdplotmainphi-60;},
		dot/.style={circle,inner sep=0pt,minimum size=2pt,fill}]
	\draw[dashed] (0,0,0) coordinate[dot,label=left:{$I$}] (I)
		(0,0,-h/2) coordinate[dot,label=below:{$O$}] (O) --
		(0,0,h/2) coordinate[dot,label=above:{$O'$}] (O')
		(O) --
		({r*cos(alpha)},{r*sin(alpha)},-h/2) coordinate[dot,label=above:{$A$}] (A)
		(I) -- (A);
	% top circle
	\pgfmathsetmacro{\rdiscT}{h/r/2*cot(\tdplotmaintheta)}
	\pgfmathtruncatemacro{\itestT}{\rdiscT<-1?2:(\rdiscT>1?1:0)}
  	\pgfmathsetmacro{\betacritT}{asin(min(1,max(\rdiscT,-1)))}
	\typeout{\betacritT,\itestT}
	\begin{scope}[canvas is xy plane at z=h/2]
	  \ifcase\itestT 
		\draw[dashed]  (\tdplotmainphi+180-\betacritT:r) 
	  	  arc[start angle=\tdplotmainphi+180-\betacritT,
		  end angle=\tdplotmainphi+\betacritT,radius=r];
		\draw[thick]  (\tdplotmainphi+180-\betacritT:r) 
	  	  arc[start angle=\tdplotmainphi+180-\betacritT,
		  end angle=\tdplotmainphi+\betacritT+360,radius=r];
	  \or
	     \draw[thick] (O') circle[radius=r];
	  \or
	     \draw[dashed] (O') circle[radius=r];
	  \fi	
	  \path (\tdplotmainphi:r) coordinate (TR)
	   (\tdplotmainphi+180:r) coordinate (TL);
	\end{scope}
	% bottom circle
	\pgfmathsetmacro{\rdiscB}{-h/r/2*cot(\tdplotmaintheta)}
	\pgfmathtruncatemacro{\itestB}{\rdiscB<-1?2:(\rdiscT>1?1:0)}
  	\pgfmathsetmacro{\betacritB}{asin(min(1,max(\rdiscB,-1)))}
	\typeout{\betacritB,\itestB}
	\begin{scope}[canvas is xy plane at z=-h/2]
	  \ifcase\itestT 
		\draw[dashed]  (\tdplotmainphi+180-\betacritB:r) 
	  	  arc[start angle=\tdplotmainphi+180-\betacritB,
		  end angle=\tdplotmainphi+\betacritB,radius=r];
		\draw[thick]  (\tdplotmainphi+180-\betacritB:r) 
	  	  arc[start angle=\tdplotmainphi+180-\betacritB,
		  end angle=\tdplotmainphi+\betacritB+360,radius=r];
	  \or
	     \draw[thick] (O) circle[radius=r];
	  \or
	     \draw[dashed] (O) circle[radius=r];
	  \fi	
	  \path (\tdplotmainphi:r) coordinate (BR)
	   (\tdplotmainphi+180:r) coordinate (BL);
	\end{scope}
	% vertical lines
	\draw[dashed] (TL) -- (BL);
	\draw[dashed] (TR) -- (BR);
	% sphere
	\begin{scope}[tdplot_screen_coords] 
	  \draw[thick] (I) circle[radius=R]; 
    \end{scope}
\end{tikzpicture}
\end{document}
```

![Screen Shot 2020-08-18 at 9.34.07 AM.png](/image?hash=35e2b523d8ed18f2bdc56af2ebab87711006fcde7462d49936934b9808e42dcf)

Yes, this is very good! Here is a variation thereof. ``` \documentclass[tikz,border=3mm]{standalone} \usetikzlibrary{3dtools}% https://github.com/marmotghost/tikz-3dtools \begin{document} \begin{tikzpicture}[3d/install view={phi=60,theta=70},scale=1,line cap=butt,line join=round,c/.style={circle,fill,inner sep=1pt}, declare function={R=4;h=4;r=sqrt(R*R-h*h/4);}] \path (0,0,0) coordinate (O) (0,0,h) coordinate (O') (0,0,h/2) coordinate (M); \draw[3d/screen coords] (-r,0|-O) edge[3d/hidden] (-r,0|-O') (r,0|-O) edge[3d/hidden] (r,0|-O') (O) edge[3d/hidden] (O') (M) circle[radius=R]; \path pic{3d/circle on sphere={R=R,C={(M)},P={(O)}}} pic{3d/circle on sphere={R=R,C={(M)},P={(O')}}}; \path foreach \p/\g in {O/-90,O'/90} {(\p)node[c]{}+(\g:3mm) node{$\p$}}; \end{tikzpicture} \end{document} ``` ![Screen Shot 2021-03-31 at 9.58.28 AM.png](/image?hash=64b319c0c20bd31e2b472cf4f298ddd5dc28b60dbff50cae55bbaa9bb616fe13)

I tried ``` \documentclass[tikz,border=3mm]{standalone} \usepackage{tikz-3dplot-circleofsphere} \usetikzlibrary{3dtools,calc}% https://github.com/marmotghost/tikz-3dtools \begin{document} \tdplotsetmaincoords{70}{70} \begin{tikzpicture}[tdplot_main_coords,scale=1,line cap=butt,line join=round,c/.style={circle,fill,inner sep=1pt}, declare function={R=4;h=4;r=sqrt(R*R-h*h/4);Angle=acos(r/R);}] \path (0,0,0) coordinate (O) (0,0,h) coordinate (O'); %(0,0,h/2) coordinate (M); \path[3d coordinate ={(M)=0.5*(O) + 0.5*(O')}]; \path[3d/visible/.style={draw,dashed},3d/hidden/.style={draw=none}] pic{3d/frustum={R=r,r=r,h=h}}; \draw[3d/screen coords] (M) circle[radius=R]; \draw[3d/hidden] (O) -- (O'); \path foreach \p/\g in {O/-90,O'/90} {(\p)node[c]{}+(\g:3mm) node{$\p$}}; \begin{scope}[shift={(M)}] \tdplotCsDrawLatCircle[tdplotCsFront/.style={}]{R}{{Angle}} \tdplotCsDrawLatCircle[tdplotCsFront/.style={}]{R}{{-Angle}} \end{scope} \end{tikzpicture} \end{document} ```

This code ``` \documentclass[tikz,border=3mm]{standalone} \usetikzlibrary{3dtools,calc}% https://github.com/marmotghost/tikz-3dtools \begin{document} \begin{tikzpicture}[3d/install view={phi=60,theta=70},scale=1,line cap=butt,line join=round,c/.style={circle,fill,inner sep=1pt}, declare function={R=4;h=4;r=sqrt(R*R-h*h/4);}] \path (0,0,0) coordinate (O) (0,0,h) coordinate (O'); \path[3d coordinate ={(M)=0.5*(O) + 0.5*(O')}]; \path[3d/visible/.style={draw,very thin,cheating dash}] pic{3d/frustum={R=r,r=r,h=h}}; \draw[3d/screen coords] (M) circle[radius=R]; \draw[3d/hidden] (O) -- (O'); \path foreach \p/\g in {O/-90,O'/90} {(\p)node[c]{}+(\g:3mm) node{$\p$}}; \end{tikzpicture} \end{document} ```

You need the newest version of `3dtools`, in which the `frustum` can have the two radii equal, which yields a cylinder. ``` \documentclass[tikz,border=3mm]{standalone} \usetikzlibrary{calc,3dtools}%https://github.com/marmotghost/tikz-3dtools \begin{document} \begin{tikzpicture}[declare function={R=2;h=2*R;}] \begin{scope}[3d/install view={phi=0,psi=0,theta=70}] \path (0,0,h/2) coordinate (M); \draw[3d/screen coords,3d/hidden] (M) circle[radius=R]; \pic{3d/frustum={r=R,R=R,h=h}}; \end{scope} \end{tikzpicture} \end{document} ```

@marmot, re: [your answer](#a1474), I tried to draw a sphere in cylinder like this ``` \documentclass[border=2mm,12pt,tikz]{standalone} \usepackage{tikz-3dplot} \usetikzlibrary{3d,calc,backgrounds,patterns} \usepackage{fouriernc} \begin{document} \def\myr{2} \def\h{4} \def\angA{0} \def\angB{110} \tdplotsetmaincoords{70}{70} \begin{tikzpicture}[tdplot_main_coords] \begin{scope}[canvas is xy plane at z=0] \draw[dashed] (\tdplotmainphi:\myr) arc(\tdplotmainphi:\tdplotmainphi+180:\myr); \coordinate (O) at (0,0); \coordinate (A) at (\angA:\myr); \coordinate (B) at (\angB:\myr); \coordinate (M) at ($(A)!0.5!(B)$); \draw[dashed] (\tdplotmainphi:\myr) arc(\tdplotmainphi:\tdplotmainphi+180:\myr); \draw[thick] (\tdplotmainphi:\myr) coordinate(BR) arc(\tdplotmainphi:\tdplotmainphi-180:\myr) coordinate(BL); %\draw[dashed] (BL) -- (BR); \end{scope} % \begin{scope}[canvas is xy plane at z=\h] \coordinate (O') at (0,0); \coordinate (A') at (\angA:\myr); \coordinate (B') at (\angB:\myr); \coordinate (M') at ($(A')!0.5!(B')$); \draw[thick] (O') circle[radius=\myr]; \draw[dashed] (O) -- (O'); \draw [thick](BR) -- (\tdplotmainphi:\myr) (BL) -- (\tdplotmainphi-180:\myr); % vẽ hai đường sinh %\draw[thick] (\tdplotmainphi:\myr) -- (\tdplotmainphi-180:\myr); %\draw (O')-- (\tdplotmainphi:\myr) node[midway, above]{$R$} ; \end{scope} \coordinate (T) at ($ (O) !0.5! (O') $); \foreach \v/\position in {O/below,O'/above,T/left} {\draw[draw =black, fill=black] (\v) circle (1pt) node [\position=0.2mm] {$\v$}; } \begin{scope}[tdplot_screen_coords, on background layer] %\fill[ball color=cyan!90, opacity=1.0] (T) circle (\myr); % 3D lighting effect \draw[dashed] (T) circle (\myr); \end{scope} \end{tikzpicture} \end{document} ``` How can I use `3dtools` to draw this?

There is always a huge trade-off when defining these pics, and I have not yet found a solution to the problem. What I want to say is that if you look e.g. at `tikz-3dplot-circleofsphere` or at `tkz-euclide`. These are surely great packages and super useful as long as you do precisely what the package authors had in mind. But as we both know, things are less glorious if we want to do something that was not foreseen by the author, e.g. knowing what `\alphacrit` is explicitly. So from this perspective it is arguably better to provide the user with some general tools, and this is the main purpose of `3dtools`. Nonetheless there is definitely room for extensions. One extension I personally would be very much interested in would be tools to solve numerically solve equations to, say, determine `\alphacrit`.

Not sure if this is true. `3dtools` is not even an official package, and likely will never be one. I am not too optimistic with regards to the future of Ti*k*Z or LaTeX as a whole. Let's hope that I am wrong.

I tried to collect some basic facts in the [3dtools manual](https://github.com/marmotghost/tikz-3dtools/blob/master/3DToolsManual.pdf). (Maybe this is the first manual that links to topanswers.xyz.;-) )

1 Answer