Date of Award


Level of Access

Campus-Only Thesis

Degree Name

Master of Science (MS)


Spatial Information Science and Engineering


Max J. Egenhofer

Second Committee Member

Kate Beard-Tisdale

Third Committee Member

Torsten Hahmann


The spatial reality that people perceive is three-dimensional. Models that capture spatial phenomena that occur in R3 are essential to reason about them in a way that is close to the human reality. The spatial relations that three-dimensional simple volumes expose in R3 have been studied in the past and it has been established that they are homologous to the behavior of simple regions in R2. However, knowledge of the behavior of simple regions in R3 is missing. One crucial step towards bolstering the understanding is to identify the conceptual neighborhood framework for the 43 relations simple regions realized in R3 that is capable of capturing transformations to the relations under different types of topological deformations. This thesis develops a conceptual neighborhood framework for topological region-region relations in R3 and validates its effectiveness in representing transformations to such relations during topological deformations to objects in the relations through comparison with the conceptual neighborhood graph of scaling a region by fixing a point in its interior (iso-scaling).

Connecting the topologically least different—difference between the intersections in the relations’ corresponding 9-intersection matrices—relations among the 43 relations gives rise to a neighborhood framework. On the other hand, the conceptual neighborhood graph for iso-scaling is a union between the conceptual neighborhood graphs for isotropically reducing a region by fixing a point in its interior (iso-reduce) and isotropically expanding a region by fixing a point in its interior (iso-expand). By identifying 14 constraints that hold during isotropically reducing a region in a binary topological relation, a candidate set of 211 transformations is filtered from an initial set of 1,849 transformations. By visual analysis, the final set of 169 transformations is confirmed. The iso-reduce neighborhoods lead to the identification of iso-expand neighborhoods and eventually to the construction of the iso-scale conceptual neighborhood graph.

The framework is evaluated based on two metrics: (1) the edge count ratio between the framework and the iso-scale conceptual neighborhood graph and (2) the framework distance for transformations in iso-scale neighborhood graph. The assessment produced an edge count ratio of 8:23. Further, the average framework distance for a transformation in the iso-scale conceptual neighborhood graph is two. Compared to region-region relations in R2 that report an edge count ratio of 8:20 and a lower framework distance, it is concluded that the framework is a competent model that captures essential characteristics of deformations to regions in R3.