A topological model for a 3-dimensional spatial information system

This thesis proposes the topological theory necessary to extend the conventional topological models used in geographic information systems (GIS), computer-aided design (CAD) and computational geometry, to a 3-dirnensional spatial information system (SIS) which supports query and analysis of spatial relationships. To encompass a wide range of applications and minirniz.e fragmentation, we define a spatial object as a cell complex, where each k-cell is homeomorphic to a Euclidean kmanifold with one or more subdivided (k-1)-manifold boundary cycles. The sirnplicial and regular cell complexes currently used in topology and many spatial information systems, are restricted forms of these generaliz.ed regular cell complexes. Spatial relationships between the cells of the generalized regular cell complex are expressed 4i terms of their boundary and coboundary cells. To support query and traversal of the neighborhood of any cell via orderings of its cobounding cells, we embed the generalized regular k-cell complex in a Euclidean n-manifold which we represent as a 'world' n-cell. Spatial relationships between spatial objects can be expressed in terms of the boundary and co boundary relations between the cells of another complex formed from the union of the generalized regular cell complexes. If this complex is embedded in a Euclidean nmanifold, then co bounding cells may also be ordered. The cells of this complex have 'singular manifold' or 'pseudomanifold' boundary cycles, which we classify into three primitive types using identification spaces. The cell complex is known as the generaliz.ed singular cell complex - generalized regular, regular and simplicial complexes are restricted forms of this complex. To represent these cell complexes, we extend the implicit cell-tuple of Brisson (1990) since it encapsulates the boundary-coboundary relations and the ordering information. Topological operators are defined to construct spatial objects. Since the set of spatial objects has few restrictions, we define topological operators which consistently construct both subdivided manifolds and manifolds with boundary, from the strong deformation retract of a manifold with boundary. The theory underlying these operators is based on combinatorial homotopy. Generic versions of these topological construction operators can then be used to join these subdivided manifolds or manifolds with boundary, to form the generalized regular cell complex.