Use of rational numbers in the design of robust geometric primitives for three-dimensional spatial database systems

摘要

A necessary step in the implementation of three-dimensional spatial data types for spatial database systems and GIS is the development of robust geometric primitives. The authors have previously shown the need for 3D spatial data types and rigorously defined them. In this paper, we propose a set of 3D geometric primitives that can be used to implement them robustly. We provide for their robustness by specifying them using rational numbers. In the discretization of space, the developers of two-dimensional spatial data types have used simplicial complexes, realms or dual grids to produce robustness, but extending any of these to 3D is not adequate. Furthermore, rational number theory is sufficiently developed to apply to 3D implementation primitives. Efforts are lacking, however, in the field of spatial databases to show that spatial operations involving 3D spatial data types are closed under rational arithmetic. We therefore define four geometric primitives using rational numbers: point, segment, facet and solid which correspond to 0D, 1D, 2D and 3D spatial objects respectively. Also, we compare the rational specification of 3D primitives to the discretization methods used in 2D. Finally, we show that intersections involving these primitives have rational closure. We therefore conclude that use of rational numbers in the design of geometric primitives provides for a robust implementation of three-dimensional spatial data types.