Given points $P_1,P_2,\dots ,P_n$ in ${\mathbb{R}}^d$ ($d\ge 2$), we consider the problem of constructing a fair interpolating curve. For $d=2$, we proposed and analyzed, in Johnson and Johnson (2016), a method which first generates a family of $G^1$ interpolating curves, where each piece is a parametric cubic. An energy functional, that loosely approximates bending energy, is defined on this family and then one seeks a curve with minimal energy. Such optimal curves, called quasi-elastic cubic splines, always exist and are always $G^1$, but often they are both $G^2$ and unique. In the present article we extend the construction and analysis to $d\ge 2$ and prove sufficient a priori conditions (on the interpolation points) for $G^2$ regularity and uniqueness of the quasi-elastic cubic spline. These sufficient conditions constitute significant improvements over those obtained in Johnson and Johnson (2016). For example, we show that if the exterior angles of the data polygon do not exceed the threshold angle $\arccos (1/3)\approx {70.5}^{\circ }$, then the quasi-elastic cubic spline is $G^2$ and unique. In contrast, the threshold angle obtained for $d=2$ in Johnson and Johnson (2016) is only $\approx {30.5}^{\circ }$. As in Johnson and Johnson (2016), we first develop a framework and then apply it to the particular example of quasi-elastic cubic splines. This framework potentially applies to other minimal energy interpolation methods.

给定分数 $P_{\mathrm{1个}}，P_2，\dots ，P_{ñ}$ 在 ${\mathbb{[R}}^d$ （$d\ge 2$），我们考虑构造公平插值曲线的问题。对于$d=2$，我们在Johnson and Johnson（2016）中提出并分析了一种方法，该方法首先生成一个 $G^{\mathrm{1个}}$插值曲线，其中每段都是参数三次。在该族中定义了一种能量函数，大致近似于弯曲能量，然后人们寻求一条具有最小能量的曲线。这样的最优曲线（称为准弹性三次样条）始终存在并且始终$G^{\mathrm{1个}}$，但通常两者都是 $G^2$和独特。在本文中，我们将构造和分析扩展到$d\ge 2$ 并证明足够的先验条件（在插值点上） $G^2$准弹性三次样条的规则性和唯一性 这些条件相对于Johnson and Johnson（2016）获得的条件有了重大改进。例如，我们表明如果数据多边形的外角不超过阈值角 $\arccos （\mathrm{1个}/3）\approx {70.5}^{\circ }$，则准弹性三次样条为 $G^2$和独特。相反，获得的阈值角$d=2$ 仅在强生公司（2016）中 $\approx {30.5}^{\circ }$。与Johnson and Johnson（2016）中一样，我们首先开发一个框架，然后将其应用于准弹性三次样条的特定示例。该框架可能适用于其他最小能量插值方法。