Dual quadrangles in the plane

Abstract

We consider quadrangles of perimeter $2$ in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^\bot$ defines a plane quadrangle $Q^\circ$ of perimeter $2$ and with marked directed edge. This quadrangle is defined uniquely (up to rotation and symmetry). Quadrangles $Q$ and $Q^\circ$ will be called dual to each other. The following properties of duality are proved: a) duality preserves convexity, non convexity and self-intersection; b) duality preserves the length of diagonals; c) the sum of lengths of corresponding edges in $Q$ and $Q^\circ$ is $1$.