Random Polygon to Ellipse: A Generalization

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 $\mathcal{P} (\vec{x}^{(0)}, \vec{y}^{(0)})$.