A QTM-based Algorithm for Generation of the Voronoi Diagram on a Sphere

To efficiently store and analyse spatial data at a global scale, the digital expression of the Earth’s data must be global, continuous and conjugate, i.e., a spherical dynamic data model is needed. The Voronoi data structure is the only published attempt and only solution (which is currently available) for dynamic GIS. The complexity of the Voronoi algorithms for line and area data sets in a vector-based context limits its application in dynamic GISs. As yet, there is no raster-based Voronoi algorithm for objects (including points, arcs and regions). To overcome this deficiency, an algorithm for generating a spherical Voronoi diagram, that is a Voronoi diagram on a spherical surface, is presented based on O-QTM (Octahedral Quaternary Triangular Mesh). The basic idea is to apply the dilation operation developed in mathematical morphology to objects on the sphere in an effort to produce the effect of distance transformation. The distance contours of objects will form the Voronoi boundaries of the spherical objects. The algorithm presented in this paper can handle point, line and area objects. Additionally, it has been tested and concluded that the processing time required for this algorithm with point, arc and region data is proportional to the levels of complexity of the spherical surface tessellation. The difference (error) between the great circle distance and the QTM cells distance is related to the spherical distance.