标题： 弹性样条 II：最优 s 曲线的唯一性以及$G^2$正则性 显示英文标题

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

摘要： 给定复平面上的点$P_1,P_2,\ldots,P_m$，我们关注的是寻找具有最小弯曲能量的插值曲线（即最优插值曲线）的问题。 之前已经证明，如果要求插值曲线的各段为s曲线，则存在性可以得到保证。 在本文中，我们还施加了限制条件，即这些s曲线的弦角幅度不超过$\pi/2$。 在这种设置下，我们确定了最优插值曲线$G^2$正则性的充分条件。 这个充分条件与模板角度$\{\psi_j\}$相关，其中$\psi_j$被定义为从线段$[P_{j-1},P_j]$到线段$[P_j,P_{j+1}]$的方向变化角度。 一个特殊的角$\Psi$($\approx 37^\circ$) 被确定，我们证明如果模板角满足$|\psi_j|<\Psi$，则最优插值曲线是全局的$G^2$。 与前一篇文章一样，我们的大部分努力集中在几何 Hermite 插值问题上，即找到一条最优的 s 曲线，该曲线连接$P_1$到$P_2$并具有规定的弦角$(\alpha,\beta)$。 尽管之前已经证明了存在性，有时也证明了唯一性，本文首先在 $|\alpha |,|\beta |\leq\pi/2$ 和 $|\alpha -\beta|<\pi$ 的情况下建立唯一性。 显示英文摘要

摘要： 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 $|\alpha -\beta|<\pi$.