The area as a natural pseudo-Hermitian structure on the spaces of plane polygons and curves

This paper is mainly concerned with the space of Euclidean polygons in the plane and the space of curves in the plane. We show that the area function determines a pseudo-Hermitian structure on the space of orientation-preserving similarity classes of Euclidean plane polygons. The area as a pseudo-Hermitian structure on the space of orientation-preserving similarity classes of plane closed curves up to reparametrization is also explored. Explicit geodesics of such spaces are calculated.