Polygons in three-dimensional space
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 $n=4,5,6$.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.12106
- arXiv:
- arXiv:2004.12106
- Bibcode:
- 2020arXiv200412106K
- Keywords:
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- Mathematics - Metric Geometry;
- Mathematics - Combinatorics
- E-Print:
- 10 pages