intersections package does not find second point of intersection of a circle and an ellipse

Question

I want to find the intersection of two paths, a circle and an ellipse. Only one intersection is found. Why? Thanks.

Here is the MWE:

\documentclass[12pt]{article}
\usepackage[pdftex]{graphicx}
\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{calc,3d,shapes, intersections}
\usepackage{xcolor}
\usepackage{tikz-3dplot}

\begin{document}
\begin{tikzpicture}
  \coordinate (O) at (0,0);

  \def\R{3cm}
  \def\c1{(O) circle (\R)}
  \fill[ball color=white!10, opacity=0.3, name path=c1] \c1;
  \draw[rotate=42, name path=c3, yscale=0.5, color=red] \c1;
  \path [name intersections={of=c1 and c3, by={c131, c132}}];
  \node[] at (c131) {c131};
  \node[] at (c132) {c132};
\end{tikzpicture}
\end{document}

Here the graph:

To expand a bit. It finds intersections in many cases but not in this. Here is my complete code so far:

\documentclass[12pt]{article}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{calc,3d,shapes, pgfplots.external, intersections}


\begin{document}

\begin{tikzpicture}[]
  \coordinate (O) at (0,0);

  \def\R{3cm}

  %outside sphere
  \def\c1{(O) circle (\R)}
  \fill[ball color=white!10, opacity=0.3, name path=c1] \c1;


  %one lune side
  \draw[rotate=96, name path=c2, yscale=0.5] \c1;
  %the other lune side
  \draw[rotate=42, name path=c3, yscale=0.5, color=red] \c1;

  % find intersections of each lune side with outside circle
  \path [name intersections={of=c1 and c2, 
           by={c121, c122}}];

  % perhaps a bug in intersections but I need to reverse the sign of c123
  \path [name intersections={of=c1 and c3, 
           by={c131, c132}}];

  % find intersections between c2 and c3
  \path [name intersections={of=c2 and c3, 
           by={c231, c232, c233, c234}}];


  % Locate points (a preview) uncomment the following lines
  % to better understand the figure
   \node[] at (c121) {c121};
   \node[] at (c122) {c122};
   \node[] at (c131) {c131};
   \node[] at (c132) {c132};
   \node[] at (c231) {c231};
   \node[] at (c232) {c232};
   \node[] at (c233) {c233};
   \node[] at (c234) {c234};


  \path[name path=c4, yscale=0.25 , rotate=42] \c1;
\end{tikzpicture}


 \end{document}

and here my figure:

Why does it find all the intersections in the other couple of ellipses and not in the points c131, c132?

Answer 1

The problem was that the intersection of two ellipses (one of them could be a circle) has 4 points (no two). For some reason three points are up and one is down (I expected two up and two down, but I do not care about this at this point). Here is the code:

\documentclass[12pt]{article}
\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{calc,3d,shapes, pgfplots.external, intersections}


\begin{document}

\begin{tikzpicture}[]
  \coordinate (O) at (0,0);

  \def\R{3cm}

  %outside sphere
  \def\c1{(O) circle (\R)}
  \fill[ball color=white!10, opacity=0.3, name path=c1] \c1;


  %one lune side
  \draw[rotate=96, name path=c2, yscale=0.5] \c1;
  %the other lune side
  \draw[rotate=42, name path=c3, yscale=0.5, color=red] \c1;

  % find intersections of each lune side with outside circle
  \path [name intersections={of=c1 and c2, 
           by={c121, c122}}];

  % perhaps a bug in intersections but I need to reverse the sign of c123
  \path [name intersections={of=c1 and c3, 
           by={c131, c132,c133,c134}}];

  % find intersections between c2 and c3
  \path [name intersections={of=c2 and c3, 
           by={c231, c232, c233, c234}}];


  % Locate points (a preview) uncomment the following lines
  % to better understand the figure
   \node[] at (c121) {c121};
   \node[] at (c122) {c122};
   \node[] at (c131) {c131};
   \node[] at (c132) {c132};
   \node[] at (c133) {c133};
   \node[] at (c134) {c134};
   \node[] at (c231) {c231};
   \node[] at (c232) {c232};
   \node[] at (c233) {c233};
   \node[] at (c234) {c234};


  \path[name path=c4, yscale=0.25 , rotate=42] \c1;
\end{tikzpicture}


 \end{document}

Here is the figure: