Polygons in three-dimensional space

Abstract

Let $P=A_1\dots A_n$ be a generic polygon in three-dimensional space and let $v_1,v_2,\dots ,v_n$ be vectors $\overline{A_1{A}_2},\overline{A_2{A}_3},\dots ,\overline{A_n{A}_1}$, respectively. $P$ will be called \emph{regular}, if there exist vectors $u_1,\dots ,u_n$ such that cross products $[u_1,u_2],[u_2,u_3],\dots ,[u_n,u_1]$ are equal to vectors $v_2,v_3,\dots ,v_1$, respectively. In this case the polygon $P^{\prime }$, defined be vectors $u_2-u_1,u_3-u_2,\dots ,u_1-u_n$ 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 $n=4,5,6$.

10 pages

References in corpus (1)

• Dual quadrangles in the plane