三维空间多边形
Polygons in three-dimensional space
摘要：
令$P=A_1\ldots A_n$是三维空间中的通用多边形，并且令$v_1,v_2,\ldots,v_n$分别是向量$\overline{A_1A_2},\overline{A_2A_3},\ldots,\overline{A_nA_1}$。如果存在向量$u_1,\ldots,u_n$使得叉积$[u_1,u_2],[u_2,u_3],\ldots,[u_n,u_1]$分别等于向量$v_2,v_3,\ldots,v_1$，则$P$将被称为\emph{regular}。在这种情况下，定义为矢量$u_2-u_1,u_3-u_2,\ldots,u_1-u_n$的多边形$P'$将被称为多边形$P$的\emph{derived polygon}或\emph{derivative}。在这项工作中，我们制定的规则性条件，并讨论衍生多边形$n=4,5,6$的几何性质。
英文摘要：
Let $P=A_1\ldots A_n$ be a generic polygon in three-dimensional space and let $v_1,v_2,\ldots,v_n$ be vectors $\overline{A_1A_2},\overline{A_2A_3},\ldots,\overline{A_nA_1}$, respectively. $P$ will be called \emph{regular}, if there exist vectors $u_1,\ldots,u_n$ such that cross products $[u_1,u_2],[u_2,u_3],\ldots,[u_n,u_1]$ are equal to vectors $v_2,v_3,\ldots,v_1$, respectively. In this case the polygon $P'$, defined be vectors $u_2-u_1,u_3-u_2,\ldots,u_1-u_n$ will be called the \emph{derived polygon} or the \emph{derivative} of the polygon $P$. In this work we formulate conditions for regularity and discuss geometric properties of derived polygons for $n=4,5,6$.
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