弹性样条 I：存在性
Elastic Splines I: Existence
AI总结：
针对平面插值点，提出可容许插值曲线概念，并证明其中存在最小弯曲能量曲线，同时构造性地证明了连接两个单位切向量的最小能量 s-曲线的存在性。
摘要：
给定平面中的插值点 $P_1,P_2,\ldots,P_n$，已知除非这些点依次位于一条直线上，否则不存在具有最小弯曲能量的插值曲线。我们称一条插值曲线是{\it 可容许的}，如果每一段连接两个连续点 $P_i$ 和 $P_{i+1}$ 的曲线都是 s-曲线，其中{\it s-曲线}是一条平面曲线，它首先朝一个方向最多旋转 $180^\circ$，然后朝相反方向最多旋转 $180^\circ$。我们的主要结果是，在所有可容许的插值曲线中，存在一条具有最小弯曲能量的曲线。我们还以一种非常构造性的方式证明了存在一条连接两个给定单位切向量的具有最小弯曲能量的 s-曲线。
英文摘要：
Given interpolation points $P_1,P_2,\ldots,P_n$ in the plane, it is known that there does not exist an interpolating curve with minimal bending energy, unless the given points lie sequentially along a line. We say than an interpolating curve is {\it admissable} if each piece, connecting two consecutive points $P_i$ and $P_{i+1}$, is an s-curve, where an {\it s-curve} is a planar curve which first turns at most $180^\circ$ in one direction and then turns at most $180^\circ$ in the opposite direction. Our main result is that among all admissable interpolating curves there exists a curve with minimal bending energy. We also prove, in a very constructive manner, the existence of an s-curve, with minimal bending energy, which connects two given unit tangent vectors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
