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Given points $P_1,P_2,\ldots,P_m$ in the complex plane, we are concerned withthe problem of finding an interpolating curve with minimal bending energy(i.e., an optimal interpolating curve). It was shown previously that existenceis assured if one requires that the pieces of the interpolating curve bes-curves. In the present article we also impose the restriction that theses-curves have chord angles not exceeding $\pi/2$ in magnitude. With this setup,we have identified a sufficient condition for the $G^2$ regularity of optimalinterpolating curves. This sufficient condition relates to the stencil angles$\{\psi_j\}$, where $\psi_j$ is defined as the angular change in direction fromsegment $[P_{j-1},P_j]$ to segment $[P_j,P_{j+1}]$. A distinguished angle$\Psi$ ($\approx 37^\circ$) is identified, and we show that if the stencilangles satisfy $|\psi_j|<\Psi$, then optimal interpolating curves are globally$G^2$.  As with the previous article, most of our effort is concerned with thegeometric Hermite interpolation problem of finding an optimal s-curve whichconnects $P_1$ to $P_2$ with prescribed chord angles $(\alpha,\beta)$. Whereasexistence was previously shown, and sometimes uniqueness, the present articlebegins by establishing uniqueness when $|\alpha |,|\beta |\leq\pi/2$ and$|\alpha -\beta|<\pi$.

Elastic Splines II: unicity of optimal s-curves and $G^2$ regularity of splines