How to draw correct arc of two circles?

Question

I saw this answer How to draw this arc (intersection of a plane and a sphere) automatically? and tried to draw this sphere. But, I can't obtain the correct result. How can I get the correct result?

\documentclass[tikz,border=1mm, 12 pt]{standalone}
\usepackage{tikz-3dplot}
\usepackage{fouriernc}
\usetikzlibrary{intersections,calc,backgrounds} 
\makeatletter
% retrieves the 3D coordinates
\def\RawCoord(#1){\csname tikz@dcl@coord@#1\endcsname}%
\def\scalprod#1=#2.#3;{%
\edef\coordA{\RawCoord#2}%
\edef\coordB{\RawCoord#3}%
\pgfmathsetmacro\pgfutil@tmpa{scalarproduct({\coordA},{\coordB})}
\edef#1{\pgfutil@tmpa}}%
\makeatother 
\newcommand{\spaux}[6]{(#1)*(#4)+(#2)*(#5)+(#3)*(#6)}  
\pgfmathdeclarefunction{scalarproduct}{2}{% scalar product of two 3-vectors
  \begingroup%
  \pgfmathparse{\spaux#1#2}%
  \pgfmathsmuggle\pgfmathresult\endgroup}  
\tikzset{reverseclip/.style={insert path={(current bounding box.south west) -- (current bounding box.north west) -- (current bounding box.north east) --(current bounding box.south east) -- cycle} }} 
\begin{document}
\tdplotsetmaincoords{70}{50}
    \begin{tikzpicture}[scale=1,tdplot_main_coords,declare function={R=5*sqrt(7)*(1/3);r=5*sqrt(3)*(1/3);
        alpha1(\th,\ph,\b)=\ph+asin(cot(\th)*tan(\b));%
        alpha2(\th,\ph,\b)=-180+\ph-asin(cot(\th)*tan(\b));%
        beta1(\th,\ph,\a)=90+atan(cot(\th)/sin(\a-\ph));%
        beta2(\th,\ph,\a)=270+atan(cot(\th)/sin(\a-\ph));%
    }]
    \path  
       (5/2, {5* sqrt(3)/6}, 0) coordinate (O)
      (0,0, 0) coordinate (A) 
      (5, 0, 0) coordinate (B) 
      (5/2, {5* sqrt(3)/2}, 0) coordinate (C) 
      (32/5,0, 24/5)  coordinate (S) 
     (5/2, 0, 10/3) coordinate (I)
      (5/2, {5* sqrt(3)/6}, 10/3) coordinate (T)
        ($ (A)!0.5!(B) $) coordinate (M)
        (0,0,1) coordinate(Z);
    \begin{scope}[tdplot_screen_coords, on background layer]
    \draw[thick,name path global=ball] (T) circle (R);
    \end{scope}

    \begin{scope}[canvas is xy plane at z={0}]
    \draw[dashed,cyan] (O) circle (r);
    \scalprod\myz=(T).(Z); % z component of T
    \pgfmathsetmacro{\myel}{atan(-1*\myz/r)}
    \draw[thick] ({alpha1(\tdplotmaintheta,\tdplotmainphi,{\myel})}:r) 
    arc({alpha1(\tdplotmaintheta,\tdplotmainphi,{\myel})}: 
    {alpha2(\tdplotmaintheta,\tdplotmainphi,{\myel})}:r) ;
    \end{scope}

    \begin{scope}[canvas is xz plane at y={0}]
     \draw[dashed,red,name path=c1] (I) circle (25/6);
    \clip[name intersections={of=c1 and ball,total=\t}] 

    let \p1=($(intersection-1)-(I)$),\n1={atan2(\y1,\x1)+180},\n2={2*veclen(\y1,\x1)} in 
    (intersection-1) -- ++ (\n1:\n2) -- (current bounding box.north west) 
    -- (current bounding box.south west) -- cycle; 
     \draw[thick] (I) circle (25/6); 
    \end{scope} 

    \begin{scope}[on background layer]
        \foreach \v/\position in {T/above,O/below,A/below,B/below,C/below,S/right,I/above,M/below} {
        \draw[draw =black, fill=black] (\v) circle (1.2pt) node [\position=0.2mm] {$\v$};
    }
    \end{scope}
    \foreach \X in {A,B,C} \draw[dashed] (\X) -- (S); 
    \draw[dashed] (A) -- (B) -- (C) -- cycle
    (T) -- (I) -- (M) -- (O) -- cycle
    ;
     \end{tikzpicture}
\end{document}

Answer 1

This is some attempt to answer your question following the strategy you devised. As for the first arc, everything works, you only forgot to shift the center of the arc to the center of the circle, (O). The second scope also works as you intended to do except that one needs to be somewhat careful since the clip path is subject to the transformations from canvas is xz plane at y=.... I used here the y coordinate of (I), but this is not really important since this is only a shift. I also followed minthien_2016's suggestion to shift (O) back to the origin. This is not crucial here but if you ever intend to produce animations, this is advantageous. In principle the second arc can be obtained analytically, too, but I did not attempt to do this here.

\documentclass[tikz,border=1mm, 12 pt]{standalone}
\usepackage{tikz-3dplot}
\usepackage{fouriernc}
\usetikzlibrary{intersections,backgrounds} % calc gets loaded by tikz-3dplot
\makeatletter
% retrieves the 3D coordinates
\def\RawCoord(#1){\csname tikz@dcl@coord@#1\endcsname}%
\def\scalprod#1=#2.#3;{%
\edef\coordA{\RawCoord#2}%
\edef\coordB{\RawCoord#3}%
\pgfmathsetmacro\pgfutil@tmpa{scalarproduct({\coordA},{\coordB})}
\edef#1{\pgfutil@tmpa}}%
\makeatother 
\newcommand{\spaux}[6]{(#1)*(#4)+(#2)*(#5)+(#3)*(#6)}  
\pgfmathdeclarefunction{scalarproduct}{2}{% scalar product of two 3-vectors
  \begingroup%
  \pgfmathparse{\spaux#1#2}%
  \pgfmathsmuggle\pgfmathresult\endgroup}  
\begin{document}
\tdplotsetmaincoords{70}{50}
\begin{tikzpicture}[scale=1,tdplot_main_coords,
declare function={R=5*sqrt(7)*(1/3);r=5*sqrt(3)*(1/3);
      alpha1(\th,\ph,\b)=\ph+asin(cot(\th)*tan(\b));%
      alpha2(\th,\ph,\b)=-180+\ph-asin(cot(\th)*tan(\b));%
      beta1(\th,\ph,\a)=90+atan(cot(\th)/sin(\a-\ph));%
      beta2(\th,\ph,\a)=270+atan(cot(\th)/sin(\a-\ph));%
  }]
  %\path[tdplot_screen_coords,use as bounding box] (-2*R,-0.6*R) rectangle (2*R,2.1*R);
  \path  % I shifted O to the origin
     (0, 0, 0) coordinate (O)
    (-5/2,{-5* sqrt(3)/6}, 0) coordinate (A) 
    (5/2, {-5* sqrt(3)/6}, 0) coordinate (B) 
    (0, {5* sqrt(3)/3}, 0) coordinate (C) 
    (32/5-5/2,{-5* sqrt(3)/6}, 24/5)  coordinate (S) 
   (0, {-5* sqrt(3)/6}, 10/3) coordinate (I)
    (0, 0, 10/3) coordinate (T)
      ($ (A)!0.5!(B) $) coordinate (M)
      (0,1,0) coordinate(Y)
      (0,0,1) coordinate(Z);
  \begin{scope}[tdplot_screen_coords, on background layer]
   \draw[thick,name path global=ball] (T) circle (R);
  \end{scope}
  \scalprod\myz=(T).(Z); % z component of T
  \begin{scope}[canvas is xy plane at z={0}]
  \draw[dashed,cyan] (O) circle (r);
  \pgfmathsetmacro{\myel}{atan(-1*\myz/r)}
  \draw[thick] ($(O)+({alpha1(\tdplotmaintheta,\tdplotmainphi,{\myel})}:r)$)
  arc({alpha1(\tdplotmaintheta,\tdplotmainphi,{\myel})}: 
  {alpha2(\tdplotmaintheta,\tdplotmainphi,{\myel})}:r) ;
  \end{scope}
  \scalprod\myy=(I).(Y); % y component of I
  \begin{scope}[canvas is xz plane at y={\myy}]
   \draw[dashed,red,name path=c1] (I) circle[radius=25/6];
   \clip[name intersections={of=c1 and ball,total=\t},overlay]
   %\pgfextra{\message{\t\space intersections found.^^J}}
    (intersection-1) -- (intersection-1|-current bounding box.north)
    -- (current bounding box.north east)
    -- (current bounding box.south east)
    -- (intersection-\t|-current bounding box.south west)  
    -- (intersection-\t) -- cycle;
   \draw[thick] (I) circle[radius=25/6];    
  \end{scope} 

  \begin{scope}[on background layer]
      \foreach \v/\position in {T/above,O/below,A/below,B/below,C/below,S/right,I/above,M/below} {
      \draw[draw =black, fill=black] (\v) circle (1.2pt) node [\position=0.2mm] {$\v$};
  }
  \end{scope}
  \foreach \X in {A,B,C} \draw[dashed] (\X) -- (S); 
  \draw[dashed] (A) -- (B) -- (C) -- cycle
  (T) -- (I) -- (M) -- (O) -- cycle;
\end{tikzpicture}
\end{document}