The axial map of a town is one of the key components of the space syntax method – a tool for analysing urban layout. It is derived by placing the longest and fewest lines, called axial lines, to cross the adjacencies between convex polygons in a convex map of a town. Previous research has shown that placing axial lines to cross the adjacencies between a collection of convex polygons is NP-complete, even when the convex polygons are restricted to rectangles and the axial lines to have orthogonal orientation. In this document, we show that placing orthogonal axial lines in orthogonal rectangles where the adjacencies between the rectangles are restricted to be crossed only once (ALPSC- OLOR) is NP-complete. As a result, we infer the single adjacency crossing version of the general axial line placement problem is NP-complete. The transformation of NPcompleteness of ALP-SC-OLOR is from vertex cover for biconnected planar graphs. A heuristic is then presented that gives a reasonable approximate solution to ALP-SC-OLOR based on a greedy method.

Branches of complexity theory have been widely employed in geographic information systems (GIS) to explore phenomena that appear in urban environments. Among these, space syntax, as an urban morphological application of complexity theory, has attracted increasing attention in recent years. Accordingly, many computer-based tools have been developed to realize related analysis spatially, especially those that can be integrated as functions with GIS. In this thesis, a space syntax tool – Axwoman – is redeveloped and tested as an extension of ESRI ArcGIS Desktop in order to fulfill certain specific needs in urban morphological analysis. It is primarily used to calculate all space syntax measures for several urban systems and to explore the relationships between these measures. To meet the needs for this new version of Axwoman, several functions have been updated and changed, for drawing, coloring, and classifying axial lines as maps for visual thinking; ticking overpasses and excluding them from computing space syntax parameters; and integrating AxialGen and Axwoman. In accordance with this, several case studies have been performed on the urban street networks in large cities. In this paper, Stockholm was chosen as the study object at both the urban level and the building level. After the scaling analysis and time efficiency analysis, the results are also interpreted from a structural point of view and in terms of how the function of space is subject to its morphological structure. Finally, the connectivity of axial lines (a spatial measurement in space syntax theory) was found to follow a power-law distribution. Through this work, the new edition of Axwoman generating satisfactory outputs, the research have proved that the connectivity of axial lines follows a lognormal distribution or a power-law-like distribution, which is one of the heavy-tailed distributions. In addition, it was have found that axial lines better for capture the underlying urban morphologies showed in their study on redefining the generated axial lines from street center lines. Moreover, fewer longest axial lines will show up on the maps, just as coincidental as the shape of mental maps, which proved that the axial line representations can be a powerful tool for urban studies.

O objetivo deste trabalho é descrever a estrutura das linhas axiais principais e médias de imersões de superfícies em R4 na vizinhança de pontos axiumbílicos. Pontos axiumbílicos são aqueles nos quais a elipse de curvatura se degenera em um círculo. Estudamos a perturbação dos ciclos axiais principais, e obtemos resultados sobre a genericidade dos ciclos axiais principais hiperbólicos e semi-hiperbólicos. / The aim of this work is to describe the structure of principal and mean axial lines of immersions of surfaces into R4, close to axiumbilic points. Axiumbilic points are those which the ellipse of curvature denegerate in a circle. We study the perturbation of principal axial cycles, and we obtain results about genericity of hyperbolic and semi-hyperbolic principal axial cycles.

O objetivo deste trabalho é descrever a estrutura das linhas axiais principais e médias de imersões de superfícies em R4 na vizinhança de pontos axiumbílicos. Pontos axiumbílicos são aqueles nos quais a elipse de curvatura se degenera em um círculo. Estudamos a perturbação dos ciclos axiais principais, e obtemos resultados sobre a genericidade dos ciclos axiais principais hiperbólicos e semi-hiperbólicos. / The aim of this work is to describe the structure of principal and mean axial lines of immersions of surfaces into R4, close to axiumbilic points. Axiumbilic points are those which the ellipse of curvature denegerate in a circle. We study the perturbation of principal axial cycles, and we obtain results about genericity of hyperbolic and semi-hyperbolic principal axial cycles.