In this paper, we give the definition for the voronoi diagram and its dual graph -- Delaunay triangulation for 3D cuboids. We prove properties of the 3D Delaunay triangulation, and provide algorithms to construct and update the Delaunay triangulation. The Delaunay triangulation data structure is used to perform proximity searches for both static and kinetic cases. We describe experimental results that show how the Delaunay triangulation is used on a mobile robot to model, understand and reason about the spatial information of the environment.