If you want to allow for the `M` line being in the back:
```
\documentclass[border=2mm,tikz]{standalone}
\usetikzlibrary{3dtools}% https://github.com/marmotghost/tikz-3dtools
\begin{document}
\foreach \Angle in {10,20,...,300}
{\begin{tikzpicture}[line cap=round,line join=round,
3d/install view={phi=\Angle,theta=70},same bounding box=A,declare function={h=3;r=1.6;a=2.5;alpha=-90;},c/.style={circle,fill,inner sep=1pt}]
\path (0,0,0) coordinate[label=below:{$O'$}] (O')
(0,0,h) coordinate[label=above:{$O$}] (O)
({r*cos(alpha)},{r*sin(alpha)},0) coordinate[label=below:{$M'$}] (M')
({r*cos(alpha)},{r*sin(alpha)},h) coordinate[label=above:{$M$}] (M);
\draw[3d/hidden] (O) -- (O');
\path[save named path=M] (M) -- (M');
\begin{scope}[canvas is xy plane at z=0,radius=r]
\path[shift={(O')},save named path=bplane] (-a,-a) rectangle (a,a);
\path[shift={(O)},save named path=tplane] (-a,-a) rectangle (a,a);
\end{scope}
\pgfmathsetmacro{\myscreenex}{TDunit("(0,0,1)x(nscreenx,nscreeny,nscreenz)")}
\path[3d coordinate={(L)=(O)-r*(\myscreenex)},3d coordinate={(R)=2*(O) -(L)},3d coordinate={(L')=(O')- (O) + (L)}, 3d coordinate={(R')=2*(O') -(L')} ];
\path pic{3d/circle on sphere={R=r, P={(O')}}};
\draw (O) circle [radius = r];
\begin{scope}[3d/screen coords]
\path[save named path=L] (L) -- (L');
\path[save named path=R] (R) -- (R');
\path[save named path=cyl] (L) -- (L') -- (R') -- (R) -- cycle;
\end{scope}
\pgfmathsetmacro{\ntest}{TDunit("(M)-(O)")}
\pgfmathtruncatemacro{\itest}{screendepth(\ntest)>0?1:0}
\ifnum\itest=0
\tikzset{3d/ordered paths/.cd,cyl/.style={draw=none},M/.style={3d/hidden}}
\tikzset{3d/draw ordered paths={bplane,cyl,L,R,M,tplane}}
\else
\tikzset{3d/ordered paths/.cd,cyl/.style={draw=none}}
\tikzset{3d/draw ordered paths={bplane,cyl,L,R,M,tplane}}
\fi
%\path foreach \p/\g in {L/0,L'/0,R/0,R'/0}{(\p)node[c]{}+(\g:2.5mm) node{$\p$}};
\end{tikzpicture}}
\end{document}
```
![ani.gif](/image?hash=dd83c6cb02681045dedbdc672d9a307912374e8ec7652740166383f617178487)