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 predicates, as derived from the 9-intersection model, have been widely recognized in GIS, spatial database systems, and many other geo-related disciplines. They are based on the evaluation of nine Boolean predicates checking the intersections of the boundary, interior, and exterior of a spatial object with the respective parts of another spatial object for inequality to the empty set. In this paper, we replace each Boolean predicate, which is a topological invariant, by another topological invariant. This new invariant is given as a function yielding the dimension of the respective intersection in the 9-intersection matrix, resulting in a dimension matrix. The goal of this paper is to determine the definition and semantics of all predicates that can be derived from this matrix for all combinations of spatial data types. It turns out that these dimension-based predicates are special refinements of the aforementioned topological predicates; hence, we call them dimension-refined topological predicates. We show that these predicates allow us to pose a class of more fine-grained topological queries.

/pdf/dimension-refined-topological-predicates-uaql6rr664.pdf

Dimension-refined topological predicates

Topological relationships between objects in space are of great importance in many disciplines. Recently, topological relationships have been defined for complex spatial objects. However, this definition only expresses topological relationships between complex spatial objects as a whole (global view); therefore,topological information between the individual components (local view)that compose the objects is lost. In this paper we propose a novel, hybrid model of topological relationships for composite regions that provides access to the global topological relationships as well as the local topological relationships that exist between the simple regions that are the components of the composite regions involved.

/pdf/preserving-local-topological-relationships-3dacfdxkin.pdf

Preserving local topological relationships

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Reasey Praing

Author Tools

Chat about Author

Papers

Dimension-refined topological predicates

Preserving local topological relationships