Polygons in three-dimensional space

Yury Kochetkov

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arXiv · 2004.12106

Polygons in three-dimensional space

Yury Kochetkov

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Abstract

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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Yury Kochetkov. 2020-04-25. Polygons in three-dimensional space. https://arxiv.org/abs/2004.12106

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