#### Date of Award

5-2009

#### Level of Access

Open-Access Thesis

#### Degree Name

Master of Science (MS)

#### Department

Spatial Information Science and Engineering

#### Advisor

Max J. Egenhofer

#### Second Committee Member

Robert D. Franzosa

#### Third Committee Member

Reinhard Moratz

#### Abstract

In the practice of mapmaking, it is commonplace to represent many spatial phenomena as if they were in a different dimension than they truly are. In this context, it is extremely important to understand the total set of relationships available in a two-dimensional setting. Currently, there exist many sets of relations in two dimensions that have been modeled by the *9-intersection matrix*. Many of these sets of relations have defined *conceptual neighborhood graphs* that show the inherent topological similarities between different types of configurations. Many of these sets of relations, however, do not have established conceptual neighborhood graphs. Furthermore, a conceptual neighborhood graph does not exist for the entire set of relations in a two-dimensional setting, a result of particular importance when considering relationships on maps which do not necessarily precisely represent reality. Through analyzing the 9-intersection matrix, a uniquely identifying labeling scheme is derived that takes away the semantic barriers of language. Its mapping function µ is found to be a *bijective* function. Through the values of µ, connection, negation, and *converseness* are defined and subsequently derived, culminating in a conceptual neighborhood graph of the power set of relations under μ;. μ is formed under the conditions of a single-direction Hamming distance. The definitions and theorems defining connection are then exploited to produce primitive conceptual neighborhood graphs for many sets of spatial relations. These primitive graphs serve as *bases* for conceptual neighborhood graphs, meaning that each of these connections derived must be found in any conceptual neighborhood graph modeling the topological deformations of *translation*, *rotation*, and *scaling*. This method, however, does not say that these are the only connections and shows that other connections are possible with restrictions upon the objects involved in the relations. In cases where the conceptual neighborhood graphs of sets of spatial relations with at least one object in codimension 0, the method is shown to produce identical results to those that currently exist in the literature, providing concrete evidence for the viability of the method. This method also shows a purpose for the perplexing *Lakes of Wada* topological example. These conceptual neighborhood graphs are the first step to defining an overarching conceptual neighborhood graph that encompasses all relations in a two-dimensional embedding space. It is shown that of 218 possible pairs of relations between two sets of relations differing by exactly one dimension in one object, 160 of these pairs exist. This result suggests that the 9-intersection and the method utilized in this thesis may provide the necessary utility to construct a conceptual neighborhood graph which relates spatial relations from different dimensional constraints.

#### Recommended Citation

Dube, Matthew P., "An Embedding Graph for 9-Intersection Topological Spatial Relations" (2009). *Electronic Theses and Dissertations*. 552.

https://digitalcommons.library.umaine.edu/etd/552