37,843 research outputs found

    Beyond two dimensions: architecture through three dimensional visibility graph analysis

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    Architecture consists of spatial relations that accommodate functions, afford social relations and create visual interest. Through openings and walls, architects manipulate continuities and discontinuities of visual fields in two and three dimensions. Analytical diagrams and models of these fields have been offered by space syntax, especially through visibility graph analysis (VGA), graphing visual relations in two dimensions. This paper introduces a new approach to VGA that departs from planar restrictions. We show how a graph can be generated of inter-visible locations on a planar surface that incorporates relations among elements in three dimensions. Using this method, we extend the current space syntax analysis of architectural space to a new methodology for diagramming and modelling three-dimensional visual relationships in architecture. The paper is structured in three parts. The first section provides an overview of the principles of visibility analysis using graphs, and explains the method by which visibility relations of ‘accessible’ and ‘inaccessible’ space in two and three dimensions are computed. This leads to a graph representation, which uses a mix of ‘directed’ and ‘undirected’ visibility connections, and a new multi-variant spatial categorisation analysis that informs the properties of multi-directional graphs. The second part of the paper tests the three-dimensional visibility model through the analysis of hypothetical and real spatial environments. The third part analyses Giuseppe Terragni’s Casa del Fascio, describing architectural characteristics that are not captured by two-dimensional analysis, and allowing a comparative understanding of spatial configuration in two and three dimensions. The paper concludes with a discussion about the significance of this new model as an analytical and architectural tool

    3D Visibility Representations of 1-planar Graphs

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    We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    Visibility Representations of Boxes in 2.5 Dimensions

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    We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane z=0z=0 and edges are unobstructed lines of sight parallel to the xx- or yy-axis. We prove that: (i)(i) Every complete bipartite graph admits a 2.5D-BR; (ii)(ii) The complete graph KnK_n admits a 2.5D-BR if and only if n≀19n \leq 19; (iii)(iii) Every graph with pathwidth at most 77 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an nn-vertex graph that admits a 2.5D-GBR has at most 4n−6n4n - 6 \sqrt{n} edges and this bound is tight. Finally, we prove that deciding whether a given graph GG admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR Γ\Gamma is the set of bottom faces of the boxes in Γ\Gamma.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

    Representation and generation of plans using graph spectra

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    Numerical comparison of spaces with one another is often achieved with set scalar measures such as global and local integration, connectivity, etc., which capture a particular quality of the space but therefore lose much of the detail of its overall structure. More detailed methods such as graph edit distance are difficult to calculate, particularly for large plans. This paper proposes the use of the graph spectrum, or the ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space as a whole. The result is a vector of high dimensionality that can be easily measured against others for detailed comparison. Several graph types are investigated, including boundary and axial representations, as are several methods for deriving the spectral vector. The effectiveness of these is evaluated using a genetic algorithm optimisation to generate plans to match a given spectrum, and evolution is seen to produce plans similar to the initial targets, even in very large search spaces. Results indicate that boundary graphs alone can capture the gross topological qualities of a space, but axial graphs are needed to indicate local relationships. Methods of scaling the spectra are investigated in relation to both global local changes to plan arrangement. For all graph types, the spectra were seen to capture local patterns of spatial arrangement even as global size is varied

    Transforming planar graph drawings while maintaining height

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    There are numerous styles of planar graph drawings, notably straight-line drawings, poly-line drawings, orthogonal graph drawings and visibility representations. In this note, we show that many of these drawings can be transformed from one style to another without changing the height of the drawing. We then give some applications of these transformations

    Representing Style by Feature Space Archetypes: Description and Emulation of Spatial Styles in an Architectural Context

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    Representation and generation of plans using graph spectra

    Get PDF
    Numerical comparison of spaces with one another is often achieved with set scalar measures such as global and local integration, connectivity, etc., which capture a particular quality of the space but therefore lose much of the detail of its overall structure. More detailed methods such as graph edit distance are difficult to calculate, particularly for large plans. This paper proposes the use of the graph spectrum, or the ordered eigenvalues of a graph adjacency matrix, as a means to characterise the space as a whole. The result is a vector of high dimensionality that can be easily measured against others for detailed comparison. Several graph types are investigated, including boundary and axial representations, as are several methods for deriving the spectral vector. The effectiveness of these is evaluated using a genetic algorithm optimisation to generate plans to match a given spectrum, and evolution is seen to produce plans similar to the initial targets, even in very large search spaces. Results indicate that boundary graphs alone can capture the gross topological qualities of a space, but axial graphs are needed to indicate local relationships. Methods of scaling the spectra are investigated in relation to both global local changes to plan arrangement. For all graph types, the spectra were seen to capture local patterns of spatial arrangement even as global size is varied
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