Many properties of houses are of topological nature. The problem of three-dimensional encoding is... more Many properties of houses are of topological nature. The problem of three-dimensional encoding is solved here by first giving an axiomatic description of a simplified concept of >househouse<) still much topolgical information is kept in these structures still making them a useful approach to encoding topological spaces. Finally, a lossless representation of observation structures in a relational database scheme which we call PLAV (Points, Lines, Areas, Volumes) is given. We expect PLAV to be useful for encoding higher dimensional (architectural) space-time complexes
Contemporary Strategies and Approaches in 3-D Information Modeling, 2018
A novel approach to higher dimensional spatial database design is introduced by replacing the can... more A novel approach to higher dimensional spatial database design is introduced by replacing the canonical solid–face–edge–vertex schema of topological data by a common type SpatialEntity, and the individual “bounded-by” relations between two consecutive classes by one separate binary relation BoundedBy on SpatialEntity defining an Alexandrov topology. This exposes mathematical principles of spatial data design. The first consequence is a mathematical definition of topological “dimension” for spatial data. Another is that every topology for spatial data is an Alexandrov topology. Also, version histories have a canonical Alexandrov topology, and generalizations can be consistently modeled by continuous foreign keys between LoDs. The result is a relational database schema for spatial data of dimension 6 and more, seamlessly integrating space-time, LoDs, and version history. Topological constructions enable queries across these different aspects. Giving points coordinates amounts can give...
Many properties of houses are of topological nature. The problem of three-dimensional encoding is... more Many properties of houses are of topological nature. The problem of three-dimensional encoding is solved here by first giving an axiomatic description of a simplified concept of >house< as a certain generalisation of a cw-complex and, secondly, by generalising local observation structures of embedded unconnected planar graphs to the three-dimensional case and proving that they allow retrieving all topological properties of these simplified houses. In the more general case of an architectural complex (a certain generalisation of a >house
This article introduces a novel approach to higher dimensional spatial database design. Instead o... more This article introduces a novel approach to higher dimensional spatial database design. Instead of extending the canonical Solid–Face–Edge–Vertex schema of topological data, these classes are replaced altogether by a common type SpatialEntity, and the individual “bounded-by” relations between two consecutive classes are replaced by one separate binary relation BoundedBy on SpatialEntity which defines a so-called Alexandrov topology on SpatialEntity and thus exposes mathematical principles of spatial data design. This has important consequences: First, a mathematical definition of topological “dimension” for spatial data can be given. Second, every topology for data of arbitrary dimension has such a simple representation. Also, version histories have a canonical Alexandrov topology, and generalizations can be consistently modeled by continuous foreign keys between LoDs. The result is a relational database schema for spatial data of dimension 6 and more which seamlessly integrates 4D ...
Many properties of houses are of topological nature. The problem of three-dimensional encoding is... more Many properties of houses are of topological nature. The problem of three-dimensional encoding is solved here by first giving an axiomatic description of a simplified concept of "house" as a certain generalisation of a cw-complex and, secondly, by generalising local observation structures of embedded unconnected planar graphs to the three-dimensional case and proving that they allow retrieving all topological properties of these simplified houses. In the more general case of an architectural complex (a certain generalisation of a "house") still much topolgical information is kept in these structures still making them a useful approach to encoding topological spaces. Finally, a lossless representation of observation structures in a relational database scheme which we call PLAV (Points, Lines, Areas, Volumes) is given. We expect PLAV to be useful for encoding higher dimensional (architectural) space-time complexes. Schlagworte / Keywords: topologisches Modell; Gebä...
ABSTRACT Das Relationale Modell der Datenhaltung beruht auf der Mengenlehre und steht damit auf d... more ABSTRACT Das Relationale Modell der Datenhaltung beruht auf der Mengenlehre und steht damit auf dem gleichen mathematischen Fundament wie die Topologie, eine wichtige mathematische Disziplin und gleichzeitig wesentliche Grundlage der räumlichen Datenmodellierung. Die enge Verwandtschaft von Topologie und Relationalem Modell kann genutzt werden, um topologische Konzepte in das Relationale Modell einzuführen: Jede Topologie für eine endliche Menge, etwa eine Datenstruktur oder eine Tabelle einer Datenbank, kann durch eine Relation dargestellt werden. Damit kann eine Tabelle zu einem topologischen Raum werden, und auf derartigen Räumen operieren die relationalen Anfrageoperatoren als topologische Fundamentalkonstruktionen, die wiederum Räume erzeugen. Der relationalen Abgeschlossenheit der Relationalen Algebra entspricht also eine Art „räumlicher Abgeschlossenheit“ in der Topologie. Die relationale Darstellung von Topologien ist nachweisbar effizient und hat für beliebige Topologien zu einer gegebenen Menge optimalen Speicherbedarf. Dieser ist auch im Wesentlichen unabhängig von der Dimension des modellierten Objekts. Eine erste prototypische Implementierung dieser topologisch-Relationalen Algebra illustriert, wie Relationen zu topologischen Räumen werden können und wie die entsprechend erweiterte Relationale Algebra auf diesen Räumen operiert. Zudem gibt es dediziert topologische Anfragen, wie Inneres, Rand oder Abschluss von Mengen in Räumen. An einem Beispiel aus der räumlichen Wissensverarbeitung, dem Region-Connection-Calculus (RCC-8), wird der Nutzen dieses generischen Ansatzes deutlich: Mit räumlichen Datenbankanfragen lassen sich die topologisch definierten RCC-8-Prädikate realisieren und deren Eigenschaften genauer untersuchen.
ABSTRACT This article compares two approaches to storing spatial information: On the one hand the... more ABSTRACT This article compares two approaches to storing spatial information: On the one hand there are topological datatypes where primitives and their connectivity are explicitly stored, on the other hand there is the G-maps-approach storing abstract “darts” and groups acting on these darts such that their orbits implicitly give the elements and topology of the stored space. First these concepts are mutually related from a categorial viewpoint and, second, their storage complexity is compared.
Many properties of houses are of topological nature. The problem of three-dimensional encoding is... more Many properties of houses are of topological nature. The problem of three-dimensional encoding is solved here by first giving an axiomatic description of a simplified concept of >househouse<) still much topolgical information is kept in these structures still making them a useful approach to encoding topological spaces. Finally, a lossless representation of observation structures in a relational database scheme which we call PLAV (Points, Lines, Areas, Volumes) is given. We expect PLAV to be useful for encoding higher dimensional (architectural) space-time complexes
Contemporary Strategies and Approaches in 3-D Information Modeling, 2018
A novel approach to higher dimensional spatial database design is introduced by replacing the can... more A novel approach to higher dimensional spatial database design is introduced by replacing the canonical solid–face–edge–vertex schema of topological data by a common type SpatialEntity, and the individual “bounded-by” relations between two consecutive classes by one separate binary relation BoundedBy on SpatialEntity defining an Alexandrov topology. This exposes mathematical principles of spatial data design. The first consequence is a mathematical definition of topological “dimension” for spatial data. Another is that every topology for spatial data is an Alexandrov topology. Also, version histories have a canonical Alexandrov topology, and generalizations can be consistently modeled by continuous foreign keys between LoDs. The result is a relational database schema for spatial data of dimension 6 and more, seamlessly integrating space-time, LoDs, and version history. Topological constructions enable queries across these different aspects. Giving points coordinates amounts can give...
Many properties of houses are of topological nature. The problem of three-dimensional encoding is... more Many properties of houses are of topological nature. The problem of three-dimensional encoding is solved here by first giving an axiomatic description of a simplified concept of >house< as a certain generalisation of a cw-complex and, secondly, by generalising local observation structures of embedded unconnected planar graphs to the three-dimensional case and proving that they allow retrieving all topological properties of these simplified houses. In the more general case of an architectural complex (a certain generalisation of a >house
This article introduces a novel approach to higher dimensional spatial database design. Instead o... more This article introduces a novel approach to higher dimensional spatial database design. Instead of extending the canonical Solid–Face–Edge–Vertex schema of topological data, these classes are replaced altogether by a common type SpatialEntity, and the individual “bounded-by” relations between two consecutive classes are replaced by one separate binary relation BoundedBy on SpatialEntity which defines a so-called Alexandrov topology on SpatialEntity and thus exposes mathematical principles of spatial data design. This has important consequences: First, a mathematical definition of topological “dimension” for spatial data can be given. Second, every topology for data of arbitrary dimension has such a simple representation. Also, version histories have a canonical Alexandrov topology, and generalizations can be consistently modeled by continuous foreign keys between LoDs. The result is a relational database schema for spatial data of dimension 6 and more which seamlessly integrates 4D ...
Many properties of houses are of topological nature. The problem of three-dimensional encoding is... more Many properties of houses are of topological nature. The problem of three-dimensional encoding is solved here by first giving an axiomatic description of a simplified concept of "house" as a certain generalisation of a cw-complex and, secondly, by generalising local observation structures of embedded unconnected planar graphs to the three-dimensional case and proving that they allow retrieving all topological properties of these simplified houses. In the more general case of an architectural complex (a certain generalisation of a "house") still much topolgical information is kept in these structures still making them a useful approach to encoding topological spaces. Finally, a lossless representation of observation structures in a relational database scheme which we call PLAV (Points, Lines, Areas, Volumes) is given. We expect PLAV to be useful for encoding higher dimensional (architectural) space-time complexes. Schlagworte / Keywords: topologisches Modell; Gebä...
ABSTRACT Das Relationale Modell der Datenhaltung beruht auf der Mengenlehre und steht damit auf d... more ABSTRACT Das Relationale Modell der Datenhaltung beruht auf der Mengenlehre und steht damit auf dem gleichen mathematischen Fundament wie die Topologie, eine wichtige mathematische Disziplin und gleichzeitig wesentliche Grundlage der räumlichen Datenmodellierung. Die enge Verwandtschaft von Topologie und Relationalem Modell kann genutzt werden, um topologische Konzepte in das Relationale Modell einzuführen: Jede Topologie für eine endliche Menge, etwa eine Datenstruktur oder eine Tabelle einer Datenbank, kann durch eine Relation dargestellt werden. Damit kann eine Tabelle zu einem topologischen Raum werden, und auf derartigen Räumen operieren die relationalen Anfrageoperatoren als topologische Fundamentalkonstruktionen, die wiederum Räume erzeugen. Der relationalen Abgeschlossenheit der Relationalen Algebra entspricht also eine Art „räumlicher Abgeschlossenheit“ in der Topologie. Die relationale Darstellung von Topologien ist nachweisbar effizient und hat für beliebige Topologien zu einer gegebenen Menge optimalen Speicherbedarf. Dieser ist auch im Wesentlichen unabhängig von der Dimension des modellierten Objekts. Eine erste prototypische Implementierung dieser topologisch-Relationalen Algebra illustriert, wie Relationen zu topologischen Räumen werden können und wie die entsprechend erweiterte Relationale Algebra auf diesen Räumen operiert. Zudem gibt es dediziert topologische Anfragen, wie Inneres, Rand oder Abschluss von Mengen in Räumen. An einem Beispiel aus der räumlichen Wissensverarbeitung, dem Region-Connection-Calculus (RCC-8), wird der Nutzen dieses generischen Ansatzes deutlich: Mit räumlichen Datenbankanfragen lassen sich die topologisch definierten RCC-8-Prädikate realisieren und deren Eigenschaften genauer untersuchen.
ABSTRACT This article compares two approaches to storing spatial information: On the one hand the... more ABSTRACT This article compares two approaches to storing spatial information: On the one hand there are topological datatypes where primitives and their connectivity are explicitly stored, on the other hand there is the G-maps-approach storing abstract “darts” and groups acting on these darts such that their orbits implicitly give the elements and topology of the stored space. First these concepts are mutually related from a categorial viewpoint and, second, their storage complexity is compared.
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