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Is it possible at every point $p=(x,y)$ on the unit circle, there is a continuous curve $C_p$ passing through it, a curve which is not only the single point $p$, and all these curves are pairwise disjoint?

And there is a constant $e$, such that all these curves have length longer then $e$.

Edit:

How to prove that they must be all the same curve locally around $p$, except translated and rotated?

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    Sure, take for example segments of radial lines.2012-05-17
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    Similar to @AndréNicolas, pick any point in the interior of the unit disk, and draw straight lines from this point outwards. Take segments of these lines.2012-05-17
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    @lbhezbgure: Of course they are continuous. Radial lines have end points. My suggestion is purely a deformation of Andre's suggestion.2012-05-17
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    "[Segments of lines] aren't continuous, because they have endpoints" -- Do you mean *closed* curves rather than *continuous* curves? Curves are necessarily continuous according to the usual definition.2012-05-17
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    oh ok, i get it. How to prove that they must be all the smae curve except translated and rotated?2012-05-17
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    You can't prove that because it's not true.2012-05-17
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    @TonyK Give a counterexample?2012-05-17
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    Take the segments of radial lines and wiggle them a bit.2012-05-17
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    Why dont you have to wiggle them all the same, so they coincide?2012-05-17
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    Why don't I have to bend them into the shape of a smiley face either?2012-05-17

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For $c\in {\mathbb R}$ consider the segments $$\sigma_c:\quad v(u):= c +\sin c\ u\qquad\bigl(-{1\over2} in the $(u,v)$-plane. The segment $\sigma_c$ intercepts the $v$-axis at the height $c$. As $|c-c'| > {1\over 2}|\sin c-\sin c'|$ when $c\ne c'$, these segments are disjoint; and when $c=c'$ ${\rm mod}\, 2\pi$ the segments $\sigma_c$ and $\sigma_{c'}$ are parallel.

The map $w\mapsto z:=e^w$, where $w=u+iv$, $z=x+iy$, maps the strip $-{1\over2} onto the annulus $e^{-1/2}<|z|. Thereby the segments $\sigma_c$ will be mapped onto arcs of logarithmic spirals which are disjoint when $c\ne c'$ ${\rm mod}\, 2\pi$ and noncongruent when $\sin c\ne \sin c'$.