On the Construction of a Generalized Voronoi Inverse of a Rectangular Tessellation

We introduce a new concept of constructing a generalized Voronoi inverse (GVI) of a given tessellation ${cal T}$ of the plane. Our objective is to place a set $S_i$ of one or more sites in each convex region (cell) $t_i in {cal T}$, such that all the edges of ${cal T}$ coincide with the edges of Voronoi diagram $V(S)$, where $S = bigcup_i S_i$, and $forall i, j, i ne j, S_ibigcap S_j = emptyset$. In this paper, we study the properties of GVI for the special case when $cal T$ is a rectangular tessellation and propose an algorithm that finds a minimal set of sites $S$. We also show that for a general tessellation, a solution of GVI always exists.