2-point Centroidal Voronoi Tessellations

In this paper, we discuss important properties of the Centroidal Voronoi Tessellation, and the geometry of CVTs on ellipses. We discuss some definitions, algorithms, and applications for CVTs. We then proceed to prove the main result of this paper: that the only CVTs of ellipses with two generators are those where the boundary between the two Voronoi regions is a line of symmetry of the ellipse. We also generalize this result to a similar class of shapes -- particularly, convex shapes with rotational symmetry of order 2 whose boundaries mirrored across any line through their point of rotation intersect the original boundary in exactly four locations. To achieve both of these proofs, we first prove an important theorem which states that the Voronoi boundary of CVTs with two generators on convex shapes with order 2 rotational symmetry must intersect the origin.