= Tensile elastica loop
{title2=$\theta=4\arctan e^{s\sqrt{T/(EI)}}$}
An ideal zero-thickness planar loop under tension $T$ is a separatrix of the elastica equation. Its tangent turns from zero to $2\pi$, the same physical orientation at both ends. With $\alpha=\sqrt{T/(EI)}$, integration gives $x=s-2\tanh(\alpha s)/\alpha$ and $y=2\operatorname{sech}(\alpha s)/\alpha$. Its height is $2\sqrt{EI/T}$. Self-contact and finite beam thickness are absent from this ideal crossing solution.
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