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 $n=4,5,6$.
Categories: math.MG math.CO