Is it possible to trim a line when crossing another?

I'm trying to draw a cycle, where the lines A-B and C-D overlap the 1-2 and 3-4 lines.

1. Is it possible to trim or clip the numbered lines above and below when they cross the other ones?
2. Is there alternatively a way to calculate the cross points so I can simply put them into the coordinates?

I've read through the documentary for hours and couldn't find a way though.

    \begin{tikzpicture} [>=latex]

\draw [->, thick] (0,0) -- (10,0);
\draw [->, thick] (0,0) -- (0,8);

\coordinate (A) at (0.5,1.5);
\coordinate (B) at (8.5,7.5);
\coordinate (C) at (8.5,4);
\coordinate (D) at (0.5,0.5);

\coordinate [label=below:$1$] (1) at (1.5,0.5);
\coordinate [label=above:$2$] (2) at (1.5,2.5);
\coordinate [label=above:$3$] (3) at (7.5,7);
\coordinate [label=below:$4$] (4) at (7.5,3);

\draw [-] (A) to[out=25,in=230] (B);
\draw [-] (C) to[out=215,in=15] (D);
\draw [-] (1) to (2);
\draw [-] (3) to (4);

\end{tikzpicture}

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You could use clipping, but here's an answer using the intersections library:

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}[>=latex]
\draw [->, thick] (0,0) -- (10,0);
\draw [->, thick] (0,0) -- (0,8);

\coordinate (A) at (0.5,1.5);
\coordinate (B) at (8.5,7.5);
\coordinate (C) at (8.5,4);
\coordinate (D) at (0.5,0.5);

\coordinate [label=below:$1$] (1) at (1.5,0.5);
\coordinate [label=above:$2$] (2) at (1.5,2.5);
\coordinate [label=above:$3$] (3) at (7.5,7);
\coordinate [label=below:$4$] (4) at (7.5,3);

\draw [-, name path=A to B] (A) to[out=25,in=230] (B);
\draw [-, name path=C to D] (C) to[out=215,in=15] (D);
\path [-, name path=1 to 2] (1) to (2);
\path [-, name path=3 to 4] (3) to (4);

\tikzset{
name intersections={of=A to B and 1 to 2, by={X1}},
name intersections={of=C to D and 1 to 2, by={Y1}},
name intersections={of=A to B and 3 to 4, by={X2}},
name intersections={of=C to D and 3 to 4, by={Y2}}
}
\draw (X1) -- (Y1);
\draw (X2) -- (Y2);

\useasboundingbox (-0.5,-0.5) rectangle (10.5, 8.5);
\end{tikzpicture}
\end{document}


EDIT: Changed specification of intersections to a single \tikzset rather than multiple \path commands.

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Haven't thought of intersections yet. I will have a deeper look into it. Thank you very much. – Stephanus Sutor Feb 25 '13 at 21:09