TikZ: Properly joining a curved line to the edge of a circle

Question

I'm trying to draw a diagram with a curved line inside of the circle, attaching to the edges. What I want is similar to that described here. Here is my code:

\documentclass[border=10pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}

\node[draw,circle,minimum size=40pt,red](circ){};
\draw[blue,looseness=1.5]
    (circ.west) to [out=70,in =110]
    (circ.center) to [out=-70,in=-110] (circ.east);

\end{tikzpicture}
\end{document}

The problem, is that the edges of the curve do not align in the way I would like with the edge of the circle. What I want is for the blue curve to extend only to the internal edge of the red circle. Now, I could simply draw the curve first, and use a invisible node, but the problem there is that some of it would still be visible since it extends past the outer edge of the circle. Also, it seems like a decidedly inelegant solution.

Answer 1

Something like this?

I set the circle outside sep=0pt. By this coordinates of circle lie on the circle and not outside of circle anymore. Also in the first case put the blue line behind circle:

\documentclass[border=10pt,tikz]{standalone}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}

\node[circle, draw=red,
      minimum size=40pt, outer sep=0pt](circ){};
\scoped[on background layer]
\draw[blue,looseness=1.5]
    (circ.west)   to [out =70,in =110]
    (circ.center) to [out=-70,in=-110] (circ.east);
\end{tikzpicture}
     \end{document}

Second example is with line in front, but line has rounded caps:

\documentclass[border=10pt,tikz]{standalone}
\usepackage{tikz}
\usetikzlibrary{backgrounds}
\begin{document}
\begin{tikzpicture}

\node[circle, draw=red,
      minimum size=40pt, outer sep=0pt](circ){};
 \draw[blue,cap=round,looseness=1.5]
    (circ.west)   to [out =70,in =110]
    (circ.center) to [out=-70,in=-110] (circ.east);
\end{tikzpicture}
     \end{document}

which gives:

Answer 2

Use the circle to clip everything outside it and draw it in the last place:

\documentclass[border=10pt]{standalone}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\node[circle,minimum size=40pt](circ){};
\clip (0,0) circle (20pt);
\draw[blue,looseness=1.5]
    (circ.west) to [out=70,in =110]
    (circ.center) to [out=-70,in=-110] (circ.east);
\node[draw,circle,minimum size=40pt,red,thick] {};
\end{tikzpicture}

\end{document}