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Aug 5, 2012 at 19:01 answer added Alexandre Eremenko timeline score: 4
Dec 24, 2010 at 14:33 comment added Bill Thurston Quadratic differentials can give good pictures, but these are different. Quadratic differentials genericaly have vertices with 3-way branching, they give whole families of line (much like the foliation $f^{-1}$ of lines parallel to the real or imaginary axis), and the measurements of distances are immportant.
Dec 24, 2010 at 14:31 answer added Bill Thurston timeline score: 13
Dec 24, 2010 at 7:42 comment added j.c. Have you looked at the literature on quadratic differentials? Not having engaged it better, I can't write an answer, but the x-ray seems to be similar to the notion of "positive" and "negative" trajectories of a quadratic differential. See e.g. eom.springer.de/q/q076040.htm
Dec 24, 2010 at 5:49 history asked David Feldman CC BY-SA 2.5