论文标题
三维空间中的多边形
Polygons in three-dimensional space
论文作者
论文摘要
令$ p = a_1 \ ldots a_n $是三维空间中的通用多边形，让$ v_1，v_2，\ ldots，v_n $ be vectors $ \ overline {a_1a_2}，\ edimalline {a_2a_3}，a_2a_3}，\ ldots，\ ldots，\ ldots，\ line_na_na = $ p $将称为\ emph {常规}，如果存在vectors $ u_1，\ ldots，u_n $，使得交叉产品$ [u_1，u_2]，[u_2，u_3]，\ ldots，[u_n，u_1] $等于vectors $ v_2，v_2，v_2，v_3，v_3，\ ldots $ ldots，\ ldots，v_1 $ ldots，v。在这种情况下，polygon $ p'$，定义为向量$ u_2-u_1，u_3-u_2，\ ldots，u_1-u_n $将称为\ emph {derived polygon}或\ emph {derivative} polygon $ p $。在这项工作中，我们为规律性制定条件，并讨论$ n = 4,5,6 $的衍生多边形的几何特性。
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 $n=4,5,6$.
