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 ξ ∈ (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, E, which has oscillating semi-axes. Along the way we find that the case for ξ = 1/2 enjoys some special optimality conditions, and periodicity of the ellipse 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 P( x (0) , y (0) ).