Suppose $c_1$ and $c_2$ are segments of smooth plane curves. To be concrete, say $c_1$ and $c_2$ are graphs of smooth functions $f_i:[a_i,b_i]\to \mathbb R$, $i=1,2$. If the curves were lines, then any translate of $c_1$ intersects $c_2$ at most once. What kind of quantitative assumptions (say on derivatives, curvature, etc.) can be put on the $f_i$ to preserve this? i.e. how flat do $c_1$ and $c_2$ need to be so that no translate of $c_1$ intersects $c_2$ twice?
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$\begingroup$ The sets of slopes are sufficiently disjoint? As a motivating example, take c_1 to be some part of the exponential graph $y=e^x$ for positive x, and c_2 some part of the inverse, or graph of the logarithmic curve. Gerhard "Plays Around With Pretty Pictures" Paseman, 2016.05.16. $\endgroup$– Gerhard PasemanCommented May 16, 2016 at 15:41
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$\begingroup$ Perhaps, the images of the Gauss maps of $c_1$ and $c_2$ have disjoint convex hulls? (THis seems to be the same as Gerhard proposes, but may be generalized to higher dimensions.) $\endgroup$– Ilya BogdanovCommented May 17, 2016 at 10:01
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