37,843 research outputs found
Beyond two dimensions: architecture through three dimensional visibility graph analysis
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 âaccessibleâ and âinaccessibleâ space in two and three dimensions are computed. This leads to a graph representation, which uses a mix of âdirectedâ and âundirectedâ 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 Terragniâs 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
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
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 and
edges are unobstructed lines of sight parallel to the - or -axis. We
prove that: Every complete bipartite graph admits a 2.5D-BR; The
complete graph admits a 2.5D-BR if and only if ; Every
graph with pathwidth at most 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 -vertex graph that admits a
2.5D-GBR has at most edges and this bound is tight. Finally,
we prove that deciding whether a given graph admits a 2.5D-GBR with a given
footprint is NP-complete. The footprint of a 2.5D-BR is the set of
bottom faces of the boxes in .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
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
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
Representation and generation of plans using graph spectra
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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