[Tex/LaTex] How to draw a square on cylinder

Question

Let be a cylinder with radius of the base circle is equal to 3 and the altitude of the cylinder is equal to 2. How can I draw square ABCD so that A, B and C, D lie on two base circles. I tried

\documentclass[border=2mm,12pt,tikz]{standalone}
\usepackage{tikz-3dplot} 
\usetikzlibrary{3d,calc,backgrounds,patterns}
\begin{document}
\pgfmathsetmacro{\myr}{3}
\pgfmathsetmacro{\h}{2}
\def\angB{0}
\def\phi{120}
\def\angA{{\angB + \phi}}
\def\angC{\phi - 180}
\tdplotsetmaincoords{65}{100}
\begin{tikzpicture}[tdplot_main_coords,scale=1,line cap=butt,line join=round]
\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);
\draw[thick] (\tdplotmainphi:\myr) coordinate(BR) arc(\tdplotmainphi:\tdplotmainphi-180:\myr)
coordinate(BL);
\end{scope}
\begin{scope}[canvas is xy plane at z=\h]
\coordinate (O') at (0,0);
\coordinate (C) at (\angC:\myr);
\coordinate (D) at ($ (A) + (C) -(B) $);
\draw[thick]  (O') circle[radius=\myr];
\draw [thick](BR) -- (\tdplotmainphi:\myr) (BL) -- (\tdplotmainphi-180:\myr); 
\end{scope}
\fill (A) circle[radius=1pt] node[above] {$A$};
\fill (B) circle[radius=1pt] node[above] {$B$};
\fill (C) circle[radius=1pt] node[above] {$C$};
\fill (D) circle[radius=1pt] node[above] {$D$};
\draw[dashed]  (C) -- (B) --   (A)  -- (D) ;
\draw[] (C) -- (D);
\end{tikzpicture}
\end{document} 

This code don't true when I change \def\angB{0} into \def\angB{30}. How to get a general way?

Answer 1

Consider three points on the square, say B, C and D. Without loss of generality we can assume that they have the coordinates

 C = (r \cos \phi,r \sin\phi,h)
 D = (r \cos (180-\phi),r \sin(180-\phi),h)
 B = (r \cos \phi,-r \sin\phi,0)

The conditions that they form a square mean that

 CD = BC

where

 CD = 2r \cos \phi
 BC = \sqrt{(2r \sin\phi)^2+h^2}

This yields

 \phi = \acos(h^2/(4r^2))/2

This confirms hpechristiansens numerical result:

 CD = 2 r \cos \left(\frac{1}{2} \acos\left(\frac{h^2}{4 r^2}\right)\right)

MWE

\documentclass[border=2mm,12pt,tikz]{standalone}
\usepackage{tikz-3dplot} 
\usetikzlibrary{3d,calc,backgrounds,patterns}
\begin{document}
\pgfmathsetmacro{\myr}{3}
\pgfmathsetmacro{\h}{2}
\pgfmathsetmacro{\angA}{acos(\h*\h/(4*\myr*\myr))/2}
\tdplotsetmaincoords{65}{100}
\begin{tikzpicture}[tdplot_main_coords,scale=1,line cap=butt,line join=round]
\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 (180-\angA:\myr);
\draw[thick] (\tdplotmainphi:\myr) coordinate(BR) arc(\tdplotmainphi:\tdplotmainphi-180:\myr)
coordinate(BL);
\end{scope}
\begin{scope}[canvas is xy plane at z=\h]
\coordinate (O') at (0,0);
\coordinate (C) at (-180+\angA:\myr);
\coordinate (D) at (-\angA:\myr);
\draw[thick]  (O') circle[radius=\myr];
\draw [thick](BR) -- (\tdplotmainphi:\myr) (BL) -- (\tdplotmainphi-180:\myr); 
\end{scope}
\fill (A) circle[radius=1pt] node[above] {$A$};
\fill (B) circle[radius=1pt] node[above] {$B$};
\fill (C) circle[radius=1pt] node[above] {$C$};
\fill (D) circle[radius=1pt] node[above] {$D$};
\draw[dashed]  (C) -- (B) --   (A);
\draw[thick] (A) -- (D) -- (C);
\end{tikzpicture}
\end{document}