In this paper, we propose an efficient algorithm

to compute the centroidal Voronoi tessellation in 2D periodic

space. We first present a simple algorithm for constructing the

periodic Voronoi diagram (PVD) from a Euclidean Voronoi

diagram. The presented PVD algorithm considers only a small

set of periodic copies of the input sites, which is more efficient

than previous approaches requiring full copies of the sites

(9 in 2D and 27 in 3D). The presented PVD algorithm is

applied in a fast Newton-based framework for computing

the centroidal Voronoi tessellation (CVT). We observe that

full-hexagonal patterns can be obtained via periodic CVT

optimization attributed to the convergence of the Newton-based

CVT computation.