Reformulating Space Syntax: The Automatic Definition and Generation of Axial Lines and Axial Maps

Space syntax is a technique for measuring the relative accessibility of different locations in a spatial system which has been loosely partitioned into convex spaces.These spaces are approximated by straight lines, called axial lines, and the topological graph associated with their intersection is used to generate indices of distance, called integration, which are then used as proxies for accessibility. The most controversial problem in applying the technique involves the definition of these lines. There is no unique method for their generation, hence different users generate different sets of lines for the same application. In this paper, we explore this problem, arguing that to make progress, there need to be unambiguous, agreed procedures for generating such maps. The methods we suggest for generating such lines depend on defining viewsheds, called isovists, which can be approximated by their maximum diameters,these lengths being used to form axial maps similar to those used in space syntax. We propose a generic algorithm for sorting isovists according to various measures,approximating them by their diameters and using the axial map as a summary of the extent to which isovists overlap (intersect) and are accessible to one another. We examine the fields created by these viewsheds and the statistical properties of the maps created. We demonstrate our techniques for the small French town of Gassin used originally by Hillier and Hanson (1984) to illustrate the theory, exploring different criteria for sorting isovists, and different axial maps generated by changing the scale of resolution. This paper throws up as many problems as it solves but we believe it points the way to firmer foundations for space syntax

The State of the Art in Cartograms

Cartograms combine statistical and geographical information in thematic maps, where areas of geographical regions (e.g., countries, states) are scaled in proportion to some statistic (e.g., population, income). Cartograms make it possible to gain insight into patterns and trends in the world around us and have been very popular visualizations for geo-referenced data for over a century. This work surveys cartogram research in visualization, cartography and geometry, covering a broad spectrum of different cartogram types: from the traditional rectangular and table cartograms, to Dorling and diffusion cartograms. A particular focus is the study of the major cartogram dimensions: statistical accuracy, geographical accuracy, and topological accuracy. We review the history of cartograms, describe the algorithms for generating them, and consider task taxonomies. We also review quantitative and qualitative evaluations, and we use these to arrive at design guidelines and research challenges

Drawing graphs for cartographic applications

Graph Drawing is a relatively young area that combines elements of graph theory, algorithms, (computational) geometry and (computational) topology. Research in this field concentrates on developing algorithms for drawing graphs while satisfying certain aesthetic criteria. These criteria are often expressed in properties like edge complexity, number of edge crossings, angular resolutions, shapes of faces or graph symmetries and in general aim at creating a drawing of a graph that conveys the information to the reader in the best possible way. Graph drawing has applications in a wide variety of areas which include cartography, VLSI design and information visualization. In this thesis we consider several graph drawing problems. The first problem we address is rectilinear cartogram construction. A cartogram, also known as value-by-area map, is a technique used by cartographers to visualize statistical data over a set of geographical regions like countries, states or counties. The regions of a cartogram are deformed such that the area of a region corresponds to a particular geographic variable. The shapes of the regions depend on the type of cartogram. We consider rectilinear cartograms of constant complexity, that is cartograms where each region is a rectilinear polygon with a constant number of vertices. Whether a cartogram is good is determined by how closely the cartogram resembles the original map and how precisely the area of its regions describe the associated values. The cartographic error is defined for each region as jAc¡Asj=As, where Ac is the area of the region in the cartogram and As is the specified area of that region, given by the geographic variable to be shown. In this thesis we consider the construction of rectilinear cartograms that have correct adjacencies of the regions and zero cartographic error. We show that any plane triangulated graph admits a rectilinear cartogram where every region has at most 40 vertices which can be constructed in O(nlogn) time. We also present experimental results that show that in practice the algorithm works significantly better than suggested by the complexity bounds. In our experiments on real-world data we were always able to construct a cartogram where the average number of vertices per region does not exceed five. Since a rectangle has four vertices, this means that most of the regions of our rectilinear car tograms are in fact rectangles. Moreover, the maximum number vertices of each region in these cartograms never exceeded ten. The second problem we address in this thesis concerns cased drawings of graphs. The vertices of a drawing are commonly marked with a disk, but differentiating between vertices and edge crossings in a dense graph can still be difficult. Edge casing is a wellknown method—used, for example, in electrical drawings, when depicting knots, and, more generally, in information visualization—to alleviate this problem and to improve the readability of a drawing. A cased drawing orders the edges of each crossing and interrupts the lower edge in an appropriate neighborhood of the crossing. One can also envision that every edge is encased in a strip of the background color and that the casing of the upper edge covers the lower edge at the crossing. If there are no application-specific restrictions that dictate the order of the edges at each crossing, then we can in principle choose freely how to arrange them. However, certain orders will lead to a more readable drawing than others. In this thesis we formulate aesthetic criteria for a cased drawing as optimization problems and solve these problems. For most of the problems we present either a polynomial time algorithm or demonstrate that the problem is NP-hard. Finally we consider a combinatorial question in computational topology concerning three types of objects: closed curves in the plane, surfaces immersed in the plane, and surfaces embedded in space. In particular, we study casings of closed curves in the plane to decide whether these curves can be embedded as the boundaries of certain special surfaces. We show that it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we can determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2n=2 combinatorially distinct spatial embeddings and we discuss the existence of fixed-parameter tractable algorithms for related problems

Spatial arrangements in architecture and mechanical engineering: some aspects of their representation and construction

Spatial arrangements in architecture and mechanical engineering are represented by incidence structures and classified according to properties of these incidence structures. The relationships between classes are given by ornamentation operations and the construction of elements in fundamental classes by substructure replacement operations. Thus representations of the spatial arrangements for possible designs are generated. Planar maps represent spatial arrangements in architecutral plans. The edges correspond to walls and vertices to incidence between walls. Plans represented by 3-vertex connected maps are ornamented by rooting and extension operations. Further ornamentation specifies access between regions. Plans with all regions adjacent to the exterior correspond to outerplane maps. Trivalent maps represent an important class of plans. Fundamental plans with r internal regions and s regions adjacent to the exterior are represented by [r,s] triangulations. Ornamentations of simple [r,s] triangulations are specified which represent plans with rectangular regions. Plans with walls aligned along two directions are represented by rectangular shapes whose maximal lines correspond to contiguous aligned walls. Rules of construction for various classes are given and the incidence structures of maximal lines and regions are characterized. Spatial arrangements in machines are represented by systems whose blocks correspond to links and vertices to joints. The dual systems are also used. Coplanar kinematic chains with revolute pairs are classified according to mobility and connectedness. Two fundamental classes are considered. First, the chains with binary joints, represented by simple graphs and constructed by two new methods: (i) suspended chain and cycle addition and (ii) subgraph replacement. Second, the chains with binary links which are constructed by subgraph replacement

A Data Structure for Spatio-Temporal Databases

The advantages and applications of spatial mechanisms are well documented; however, there are very few being designed. The principal hinderance to the design of spatial mechanisms is the great difficulty involved in specifying spatial problems and in interpreting spatial solutions. Similarly, the development of spatial codes to implement these techniques is held back by the lack of means to easily visualize and verify solutions, particularly in the realm of relational databases. If spatial mechanisms are to be successful, the designer must be able to synthesize, analyse and evaluate, as well as load and extract information, using a single code representing a spatial structure. This entails the implementation of spatial relationships involving spatial data structures. It is with this in mind that the Canadian Hydrographic Service database group embarked on the development of a new type of spatial database structure based on the quadtree concept

Improving predictive mapping of deep-water habitats

In the deep sea, biological data are often sparse; hence models capturing relationships between observed fauna and environmental variables (acquired via acoustic mapping techniques) are often used to produce full coverage species assemblage maps. Many statistical modelling techniques are being developed, but there remains a need to determine the most appropriate mapping techniques. Predictive habitat modelling approaches (redundancy analysis, maximum entropy and random forest) were applied to a heterogeneous section of seabed on Rockall Bank, NE Atlantic, for which landscape indices describing the spatial arrangement of habitat patches were calculated. The predictive maps were based on remotely operated vehicle (ROV) imagery transects, high-resolution autonomous underwater vehicle (AUV) sidescan backscatter maps and ship-based multibeam bathymetry. Area under the curve (AUC) and accuracy indicated similar performances for the three models tested, but performance varied by species assemblage, with the transitional species assemblage showing the weakest predictive performances. Spatial predictions of habitat suitability differed between statistical approaches, but niche similarity metrics showed redundancy analysis and random forest predictions to be most similar. As one statistical technique could not be found to outperform the others when all assemblages were considered, ensemble mapping techniques, where the outputs of many models are combined, were applied. They showed higher accuracy than any single model. Different statistical approaches for predictive habitat modelling possess varied strengths and weaknesses and by examining the outputs of a range of modelling techniques and their differences, more robust predictions, with better described variation and areas of uncertainties, can be achieved. As improvements to prediction outputs can be achieved without additional costly data collection, ensemble mapping approaches have clear value for spatial management