How can I get correct the point A and B automatically in this picture?

Question

I am trying to find a general method to find the points A and B automatically of this picture. I can not get it. I must calculate by hand.

\documentclass[tikz,border=3.14mm]{standalone}
    \usepackage{fouriernc}
    \usepackage{tikz}
    \usepackage{tkz-euclide}
    \usetkzobj{all}
    \usepackage{tikz-3dplot}
    \usetikzlibrary{calc,backgrounds}
    \begin{document}
    \tdplotsetmaincoords{60}{110}
    \begin{tikzpicture}
    [scale=1,tdplot_main_coords]
    \path
    coordinate (O) at (0,0,0)
    coordinate (T) at  (0,0,3)
    coordinate (B) at  (3,{-sqrt(7)},0)
    coordinate (A) at  (0,4,0);
    \draw[thick, dashed] (O) -- (A) node[midway,sloped,below] {$r$};
    \draw[thick, dashed] (T) -- (A) node[midway,right] {$R$};
    \draw[thick, dashed] (T) -- (O) node[midway,left] {$d$};
    \foreach \v/\position in {T/above,O/below,A/below,B/below} {
        \draw[draw =black, fill=black] (\v) circle (1.2pt) node [\position=0.2mm] {$\v$};
    }
    \begin{scope}[tdplot_screen_coords, on background layer]
    \pgfmathsetmacro{\R}{5}%
    \pgfmathsetmacro{\r}{4}%
    \fill[ball color=cyan!50, opacity=1.0] (T) circle (\R);
    \end{scope}
    \tkzMarkRightAngle[size = 0.3](T,O,A);
    \draw [thick] (B) arc (318.5:450:4);
    \draw [thick, dashed] (A) arc (90:318.5:4);
    \end{tikzpicture}
    \end{document}

Is there a way to find the point A and B automatically?

Answer 1

AFAIK the first post working out the critical values that distinguish the stretches on the foreside from those on the backside was this. (To all who are interested: this is what I mean with "giving credit". Even though the following will use different formulae and macros, it was nice for me to see a post in which the derivation is explained and which I can use to compare my results. I personally feel one should always add these "references" because then Alain Matthes answer has many links and users have simple way of looking for related posts. Personally I just feel that Alain deserves credit for that.) These things were done in different coordinates than those tikz-3dplot, but the analogous expressions in the tikz-3dplot conventions were put down in this answer. With those, you can switch to an xy plane and all you need to say is

 \draw[thick] ({alpha1(\tdplotmaintheta,\tdplotmainphi,{atan(4/5)})}:4)
      coordinate (A)
      arc({alpha1(\tdplotmaintheta,\tdplotmainphi,{atan(4/5)})}:
      {alpha2(\tdplotmaintheta,\tdplotmainphi,{atan(4/5)})}:4) coordinate(B);

Here, alpha1 and alpha2 are the critical phi values that determine where the path makes transitions from the fore- to the backside and thus fix the locations of A and B analytically.

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{fouriernc}
\usepackage{tikz-3dplot}
\usetikzlibrary{backgrounds}
\begin{document}
\tdplotsetmaincoords{60}{110}
\begin{tikzpicture}[scale=1,tdplot_main_coords,declare function={R=5;
 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
 coordinate (O) at (0,0,0)
 coordinate (T) at  (0,0,3);
 \begin{scope}[tdplot_screen_coords, on background layer]
 \fill[ball color=cyan!50, opacity=1.0] (T) circle (R);
 \end{scope}
 \begin{scope}[canvas is xy plane at z={3-sqrt(5^2-4^2)}]
  \draw[thick,dashed] circle[radius=4cm];
  \draw[thick] ({alpha1(\tdplotmaintheta,\tdplotmainphi,{atan(4/5)})}:4)
  coordinate (A)
  arc({alpha1(\tdplotmaintheta,\tdplotmainphi,{atan(4/5)})}:
  {alpha2(\tdplotmaintheta,\tdplotmainphi,{atan(4/5)})}:4) coordinate(B);
 \end{scope}
 \begin{scope}[on background layer]
  \draw[thick, dashed] (O) -- (A) node[midway,sloped,below] {$r$};
 \draw[thick, dashed] (T) -- (A) node[midway,right] {$R$};
 \draw[thick, dashed] (T) -- (O) node[midway,left] {$d$};
 \foreach \v/\position in {T/above,O/below,A/below,B/below} {
     \draw[draw =black, fill=black] (\v) circle (1.2pt) node [\position=0.2mm] {$\v$};
 }
 \end{scope}
\end{tikzpicture}
\end{document}

If you compare this very carefully to your output, then you see that there is a tiny discrepancy in the location of A. This could be a numerical issue, or one of use made a mistake.