Random Polygon to Ellipse: A Generalization

TL;DR

This paper extends the transformation from polygons to ellipses by considering arbitrary division points, analyzing the iterative process using complex analysis and linear algebra, and exploring properties like limiting shapes, oscillations, and periodicity.

Contribution

It introduces a generalized transformation for polygons based on weighted barycenters and derives a closed-form solution for the iterative process, expanding previous results.

Findings

The limiting shape is an ellipse with oscillating semi-axes.

Special optimality occurs at the midpoint division case.

Periodic behavior of the resulting ellipse is characterized.

Abstract

This paper generalizes the result of Elmachtoub et al to any weighted barycenter, where a transformation is considered which takes an arbitrary point of division $\xi \in (0,1)$ of the segments of a polygon with $n$ vertices. We then consider connecting these new points to form another polygon, and iterate this process. After considering properties of our generalized transformation matrix, a surprisingly elegant interplay of elementary complex analysis and linear algebra is used to find a closed form for our iterative process. We then specify the new limiting ellipse, $\mathcal{E}$, which has oscillating semi-axes. Along the way we find that the case for $\xi = 1/2$ enjoys some special optimality conditions, and periodicity of the ellipse $\mathcal{E}$ is analyzed as well. To conclude, an even more generalized case is considered: taking a different point of division for every segment of our polygon $\mathcal{P} (\vec{x}^{(0)}, \vec{y}^{(0)})$.