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

Abstract

Given points $P_1,P_2,\ldots,P_m$ in the complex plane, we are concerned with the problem of finding an interpolating curve with minimal bending energy (i.e., an optimal interpolating curve). It was shown previously that existence is assured if one requires that the pieces of the interpolating curve be s-curves. In the present article we also impose the restriction that these s-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 optimal interpolating curves. This sufficient condition relates to the stencil angles $\{\psi_j\}$, where $\psi_j$ is defined as the angular change in direction from segment $[P_{j-1},P_j]$ to segment $[P_j,P_{j+1}]$. A distinguished angle $\Psi$ ($\approx 37^\circ$) is identified, and we show that if the stencil angles satisfy $|\psi_j|<\Psi$, then optimal interpolating curves are globally $G^2$. As with the previous article, most of our effort is concerned with the geometric Hermite interpolation problem of finding an optimal s-curve which connects $P_1$ to $P_2$ with prescribed chord angles $(\alpha,\beta)$. Whereas existence was previously shown, and sometimes uniqueness, the present article begins by establishing uniqueness when $|\alpha |,|\beta |\leq\pi/2$ and $|\alpha -\beta|<\pi$.