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A saddle connection on a translation surface $\omega$ is a geodesic in the flat metric determined by $\omega$ joining two zeros with no zeros in its interior.

Athreya, Jayadev S., and Howard Masur. Translation Surfaces. Vol. 242. American Mathematical Society, 2024. Quote from p.26.

Q. Why is it called a "saddle connection"? In particular, why saddle?

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    $\begingroup$ If you think of the zeros as cone points and imagine two cones pointing up with a saddle connection joining them, it really does look like a (horse) saddle. $\endgroup$
    – Andy Putman
    Commented Jul 11 at 1:08
  • $\begingroup$ @AndyPutman: That makes sense---Thanks! $\endgroup$
    – Joseph O'Rourke
    Commented Jul 12 at 14:07
  • $\begingroup$ mathcurve.com/surfaces.gb/selle/monkeysaddle.gif $\endgroup$
    – Alex Wright
    Commented Sep 17 at 18:10

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