平面机器人手臂的临界位形
Critical configurations of planar robot arms
摘要：
已知闭多边形P是有向面积函数的临界点当且仅当P是循环多边形，即$P$可内接于圆.此外，还有一个简短的莫尔斯指数公式。进一步在这个方向上，我们扩展这些结果的情况下，开放的多边形链，或机器人手臂。引入了开多边形链的有向面积的概念，证明了临界点是端点对极的循环构形，并导出了临界构形Morse指数的计算公式.
英文摘要：
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.
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