Following on from rmhleo's fantastic answer here, where he states that the deformation of an ideally elastic circle is a problem of the calculus of variations which may be solved with an ODE of the form
\begin{equation} \frac{dy}{dx} = \frac{a^2 - c^2 + x^2}{\sqrt{(c^2 - x^2)(2a^2 - c^2 + x^2)}} \end{equation}
where
\begin{equation} \int_{p_1}^{p_2} (\frac{1}{R} - k_0)^2ds \end{equation}
is the proposed integral solution.
For curve $p_1\approx\{0,-0.45\},\ p_2\approx\{0,1.957\},\ \theta_1\approx 17.227^{\circ},\ \theta_2\approx 43.627^{\circ},$ and arc length $\approx 4.251$:
with curvature plot:
approximate calculations show with $k_0\approx 1.214$, $\int_{p_1}^{p_2} (\frac{1}{R} - k_0)^2ds\approx 4.476$, whereas measurement of the arc length of the original curve is $\approx 4.251$. Does this discrepancy imply that the curve shown is not of elastica (minimal energy) type? (Data can be provided if this helps.)