We consider four problems connected by the common thread of geometry. The first three arise in applications that apriori do not involve geometry but this turns out to be the right language for visualizing and analyzing them. In the fourth, we generalize some well known results in geometry to the topological plane. The techniques we use come from probability and topology. First, we consider two algorithms that work well in practice but the theoretical mechanism behind whose success is not very well understood. Greedy routing is a routing mechanism that is commonly used in wireless sensor networks. While routing on the Internet uses standard established protocols, routing in ad-hoc networks with little structure (like sensor networks) is more difficult. Practitioners have devised algorithms that work well in practice, however there were no known theoretical guarantees. We provide the first such result in this area by showing that greedy routing can be made to work on Planar triangulations. Linear Programming is a technique for optimizing a linear function subject to linear constraints. Simplex Algorithms are a family of algorithms that have proven quite successful in solving Linear Programs in practice. However, examples of Linear Programs on which these algorithms are very inefficient have been obtained by researchers. In or-

Computational geometry.

Lösung: Man kann die Ebene mit Sechsecken überdecken. Nimmt man die Mittelpunkte der Sechsecke als Punkte für ein Voronoidiagramm, dann haben alle genau 6 Nachbarn, und die Voronoi Polygone sind gerade die Sechsecke. Gäbe es eine Punkteverteilung, bei der jeder Punkt mehr als sechs Nachbarn hätte, dann könnte man die Punkte so hinschieben, dass die Voronoi Polygone regelmässige n-Ecke mit n > 6 wären. Eine solche Überdeckung der Ebene gibt es aber nicht (wie man sich leicht aus einer Betrachtung der Winkel überlegen kann).

Computational geometry

Topological Spaces By Eduard Cech. Revised edition by Z. Frolik and M. Katetov. Pp. 893. (Prague: Publishing House of the Czechoslovak Academy of Sciences; London and New York: Interscience Publishers, a Division of John Wiley and Sons, 1966.) 145s.

Point Set Topology

This chapter focuses on the topological and mereological relations, contact, and parthood, between spatiotemporal regions as axiomatized in so-called mereotopologies. Despite, or because of, their simplicity, a variety of different first-order axiomatizations have been proposed. This chapter discusses their underlying ontological choices and different ways of systematically looking at them. The chapter further gives an overview of the algebraic, topological, and graph-theoretic representations of mereotopological models which help to better understand the model-theoretic consequences of the various ontological choices. While much work on mereotopologies has been primarily theoretical, the focus started shifting towards applications and domain-specific extensions of mereotopology. These aspects will most likely guide the future direction of the field: How can mereotopologies be extended or otherwise adjusted to better suit practical needs? Moreover, the integration of mereotopology into more comprehensive and maybe more pragmatic ontologies of space and time remains another challenge in the field of region-based space.

Qualitative Spatio-Temporal Representation and Reasoning: Trends and Future Directions

Moving objects databases should be able to manage trips that pass through several real world environments, e.g., road network, indoor. However, the current data models only deal with the movement in one situation and cannot represent comprehensive trips for humans who can move inside a building, walk on the pavement, drive on the road, take the public vehicles (bus or train), etc. As a result, existing queries are solely limited to one environment. In this paper, we design a data model that is able to represent moving objects in multiple environments in order to support novel queries on trips in different surroundings and various transportation modes (e.g., Car, Walk, Bus). A generic and precise location representation is proposed that can apply in all environments. The idea is to let the space for moving objects be covered by a set of so-called infrastructures each of which corresponds to an environment and defines the available places for moving objects. Then, the location is represented by referencing to the infrastructure. We formulate the concept of space and infrastructure and propose the methodology to represent moving objects in different environments with the integration of precise transportation modes. Due to different infrastructure characteristics, a set of novel data types is defined to represent infrastructure components. To efficiently support new queries, we design a group of operators to access the data. We present how such a data model is implemented in a database system and report the experimental results. The new model is designed with attention to the data models of previous work for free space and road networks to have a consistent type system and framework of operators. In this way, a powerful set of generic query operations is available for querying, together with those dealing with infrastructures and transportation modes. We demonstrate these capabilities by formulating a set of sophisticated queries across all infrastructures.

A generic data model for moving objects

Topological relationships between spatial objects are important for querying, reasoning, and indexing of data within spatial databases. These relationships are qualitative and respond to questions about the relative positions (e.g., disjointedness or containment) of spatial objects. Several models have been proposed that effectively define formal sets of topological relationships between simple spatial data types. The generalization of topological relationship models to complex spatial data types, which are roughly defined as multi-component versions of their simple counterparts, has raised awareness of the fact that these models only provide a global view of topological relationships whereas details of the topological relationships between individual components of the spatial objects involved are often ignored. In this paper, we introduce a fine-grained view on topological relationships between complex regions. Our model focuses on leveraging information about the local topological relationships that hold between the components of two spatial objects, thereby providing a localized view of the overall global topological relationship.

/pdf/local-topological-relationships-for-complex-regions-1r9jl37aax.pdf

Local topological relationships for complex regions

Moving object databases have recently found large interest in the data-base community, geographical information systems, and geosciences. So far, moving object models have focused exclusively either on past or future movements. The so-called Balloon Model is the first attempt to treat past and future movements of moving objects in a consistent and integrated manner. Based on a new set of spatio-temporal data types and operations as well as former research, in this paper, we propose a design of spatio-temporal predicates on Balloon objects. These predicates explore the topological relationships of Balloon objects over time and take into account both past and future movements. We present new kinds of queries to demonstrate the use of these predicates.

Modeling historical and future spatio-temporal relationships of moving objects in databases

Ensuring the semantic and topological correctness of spatial data is an important requirement in geographical information systems and spatial database systems in order to preserve spatial data quality and enable correct operation execution. Spatial objects like complex regions are usually represented as an ordered sequence of segments (sequence view) to support and ease the computation of spatial operations by means of plane sweep algorithms. The semantic correctness of such a sequence is usually simply assumed but is not easy to see. In this paper, we present a novel and efficient algorithm to discover the cyclic structure and check for the semantic correctness of the sequence representation of a complex region by determining its cyclic structures (component view) in terms of multiple faces possibly containing holes. The algorithm producing the component view is also interesting for object construction, manipulation, and visualization.

/pdf/ensuring-the-semantic-correctness-of-complex-regions-2lrhepuc7y.pdf

Ensuring the semantic correctness of complex regions

Spatiotemporal databases deal with geometries changing over time. Recently, moving objects like moving points and moving regions have been the focus of research. They represent time-dependent geometries that continuously change their location and/or extent and are interesting for many disciplines including the geosciences, geographical information science, moving objects databases, location-based services, robotics, and mobile computing. So far, a few moving object data models and query languages have been proposed. Each of them supports either exclusively historical movements relating to the past or exclusively predicted movements relating to the future. Thus, the query support for each model is limited by the type of supported movements. This presents a problem in modeling the dynamic nature of a moving object when both its known historical movement and its predicted future movement are desired to be simultaneously managed and made available for spatiotemporal operations and queries. Furthermore, current definitions of moving objects are too general and rather vague. It is unclear how a moving object is allowed to move or evolve through space and time. That is, the properties of movement (like its continuity) have not been precisely specified. It is also unclear how, in a database context, future predictions of a moving object can be modeled taking into account the inherent uncertainty of future evolution. Moreover, implementations of spatiotemporal data types and operations are rare and their integration into extensible database management systems has been so far nonexistent. In this research, we present a new type system and query language called Moving Balloon Algebra consisting of a moving object model that is able to represent the dynamic nature of moving objects while providing integrated and seamless support for both historical and predicted movements of moving objects. The goal is to go beyond existing moving object models by collectively integrating existing functionalities as well as introducing new ones. From a conceptual standpoint, this algebra provides a formal definition of novel spatiotemporal data types, operations, and predicates as well as introduces new types of spatiotemporal queries. Beside these conceptual contributions, an implementation of the algebra is provided in the form of a database-independent type system library, and its integration into a relational database management system is demonstrated.

Moving balloon algebra: design, implementation, and database integration of a spatiotemporal data model for historical and predictive moving objects

Topological predicates between spatial objects have for a long time been a focus of intensive research in a number of diverse disciplines. In the context of spatial databases and geographical information systems, they support the construction of suitable query languages for spatial data retrieval and analysis. Whereas to a large extent conceptual aspects of topological predicates have been emphasized, the development of efficient evaluation techniques for them has been rather neglected. Recently, the design of topological predicates for different combinations of complex spatial data types has led to a large increase of their numbers and accentuated the need for their efficient implementation. The goal of this paper is to develop efficient implementation techniques for them within the framework of the spatial algebra SPAL2D.

/pdf/efficient-evaluation-techniques-for-topological-predicates-2uauhibn0r.pdf

Efficient Evaluation Techniques for Topological Predicates on Complex Regions

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