Modeling and Manipulating Cell Complexes in Two, Three and Higher Dimensions

Čomić, Lidija; Floriani, Leila De

doi:10.1007/978-94-007-4174-4_4

Cited by 10 publications

(4 citation statements)

References 22 publications

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“…Meanwhile, the secondary topological entities are partial entities called partial face (p-face), partial edge (p-edge) and partial vertex (p-vertex). The partial entity data structure considers each face to have one orientation geometrically defined based on its face normal, and the orientation of its boundary can thus be uniquely defined [48]. Basic query procedures are used to extract the adjacency relationship between the basic topological entities.…”

Section: Three-dimensional (3d) Spatial Modeling and Data Structurementioning

confidence: 99%

“…Basic query procedures are used to extract the adjacency relationship between the basic topological entities. The difference with the coupling entity data structure is that the partial entity data structure kept the relationship of the objects that were connected by a vertex and, as stated by [48], this data structure saves more storage cost than the Radial Edge data structure for non-manifold cell 2-complexes. Cell-tuple data structure [42] has a combination of nodes, edges, faces, and solids.…”

Section: Three-dimensional (3d) Spatial Modeling and Data Structurementioning

confidence: 99%

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Abstract Topological Data Structure for 3D Spatial Objects

Ujang

Castro

Azri

2019

IJGI

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In spatial science, the relationship between spatial objects is considered to be a vital element. Currently, 3D objects are often used for visual aids, improving human insight, spatial observations, and spatial planning. This scenario involves 3D geometrical data handling without the need for topological information. Nevertheless, in the near future, users will shift to more complex queries corresponding to the existing 2D spatial approaches. Therefore, having 3D spatial objects without having these relationships or topology is impractical for 3D spatial analysis queries. In this paper, we present a new method for creating topological information that we call the Compact Abstract Cell Complexes (CACC) data structure for 3D spatial objects. The idea is to express in the most compact way the topology of a model in 3D (or more generally in nD) without requiring the topological space to be discrete or geometric. This is achieved by storing all the atomic cycles through the models (null combinatorial homotopy classes). The main idea here is to store the atomic paths through the models as an ant experiences topology: each time the ant perceives a previous trace of pheromone, it knows it has completed a cycle. The main advantage of this combinatorial topological data structure over abstract simplicial complexes is that the storage size of the abstract cell cycles required to represent the geometric topology of a model is far lower than that for any of the existing topological data structures (including abstract simplicial cell cycles) required to represent the geometric decomposition of the same model into abstract simplicial cells. We provide a thorough comparative analysis of the storage sizes for the different topological data structures to sustain this.

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Section: Three-dimensional (3d) Spatial Modeling and Data Structurementioning

confidence: 99%

Section: Three-dimensional (3d) Spatial Modeling and Data Structurementioning

confidence: 99%

Abstract Topological Data Structure for 3D Spatial Objects

Ujang

Castro

Azri

2019

IJGI

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“…A linear cell complex is the special case of a CW complex, whose attaching maps are homeomorphisms and whose cells are polyhedra [16]. Cell complexes are used in applications such as computer graphics, Computer Aided Design (CAD) and geographic in order to discretise 3D geometric objects [17].…”

Section: Linear Cell Complexes and Combinatorial Mapsmentioning

confidence: 99%

Incorporating Topological Representation in 3D City Models

Vitalis

Ohori

Stoter

2019

IJGI

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3D city models are being extensively used in applications such as evacuation scenarios and energy consumption estimation. The main standard for 3D city models is the CityGML data model which can be encoded through the CityJSON data format. CityGML and CityJSON use polygonal modelling in order to represent geometries. True topological data structures have proven to be more computationally efficient for geometric analysis compared to polygonal modelling. In a previous study, we have introduced a method to topologically reconstruct CityGML models while maintaining the semantic information of the dataset, based solely on the combinatorial map (C-Map) data structure. As a result of the limitations of C-Map’s semantic representation mechanism, the resulting datasets could suffer either from semantic information loss or the redundant repetition of them. In this article, we propose a solution for a more efficient representation of geometry, topology and semantics by incorporating the C-Map data structure into the CityGML data model and implementing a CityJSON extension to encode the C-Map data. In addition, we provide an algorithm for the topological reconstruction of CityJSON datasets to append them according to this extension. Finally, we apply our methodology to three open datasets in order to validate our approach when applied to real-world data. Our results show that the proposed CityJSON extension can represent all geometric information of a city model in a lossless way, providing additional topological information for the objects of the model.

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“…An additional coordinate indicates a scale factor that is applied to all other coordinates. data structures and representation schemes Čomić & de Floriani (2012); Arroyo Ohori, Ledoux & Stoter (2015b). Possible approaches include incidence graphs (Rossignac & O'Connor, 1989;Masuda, 1993;Sohanpanah, 1989;Hansen & Christensen, 1993), Nef polyhedra Bieri & Nef (1988), and ordered topological models Brisson (1993); Lienhardt (1994).…”

Section: Higher-dimensional Modelling Of Space Time and Scalementioning

confidence: 99%

Visualising higher-dimensional space-time and space-scale objects as projections to ℝ³

Ohori

Ledoux

Stoter

2017

PeerJ Computer Science

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Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object’s shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from ℝ4to ℝ3. We present three projections that we believe are particularly intuitive for this purpose: (i) a simple ‘long axis’ projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (S3) followed by a stereographic projection to ℝ3, which results in an inwards-outwards fourth axis. Our focus is in using these projections from ℝ4to ℝ3, but they are formulated from ℝnto ℝn−1so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

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Modeling and Manipulating Cell Complexes in Two, Three and Higher Dimensions

Cited by 10 publications

References 22 publications

Abstract Topological Data Structure for 3D Spatial Objects

Abstract Topological Data Structure for 3D Spatial Objects

Incorporating Topological Representation in 3D City Models

Visualising higher-dimensional space-time and space-scale objects as projections to ℝ³

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Modeling and Manipulating Cell Complexes in Two, Three and Higher Dimensions

Cited by 10 publications

References 22 publications

Abstract Topological Data Structure for 3D Spatial Objects

Abstract Topological Data Structure for 3D Spatial Objects

Incorporating Topological Representation in 3D City Models

Visualising higher-dimensional space-time and space-scale objects as projections to ℝ3

Contact Info

Product

Resources

About

Visualising higher-dimensional space-time and space-scale objects as projections to ℝ³