Let us take a rope 100m long. Tying it to two poles located at 100m from each other leaves only one possibility to lay it down flat: a straight line. We then reduce the distance between the two poles ==> there are now infinite ways to flat the rope out, tracing one of the infinite set of possible curves (including not simple curves). Just take one of said curves, and represent it in terms of its arc length function and unit tangent vector: $$ s_o(t) \,\,\,\, and \,\,\,\, T_o(s_o(t)) $$ Intuitively, if we now layout the rope following a curve featuring exactly the same $s(t)$ arc length function, but whose unit tangent vector is now rotated by an angle progressively increasing towards $\pi$ ($L$ is to make the 100m length more general, and $a>0$ is chosen sufficiently small so that the angle of rotation approaches $\pi$ as $s_o(t)$ approaches $s_o(t_{end}) =L$ )
$$ s(t) \, = \,s_o(t) \,\,\,\,\,\, and \,\,\,\,\,\, T(s(t)) \, = \,T_o(s(t)) \,\, e^{2 \, i \arctan{\frac{t}{a}}} $$ we could expect the rope never to be able to reach the end pole. So to speak, the rope is constantly "turned away" from the original curve up to its "near mirroring" when the unit tangent vector is turned around by almost $\pi$ (never exceeding it). Although it makes heuristically sense, I was unable to find any rigorous proof (if any exists at all).