Topological Data Structures for Surfaces


Book Description

In Geography and GIS, surfaces can be analysed and visualised through various data structures, and topological data structures describe surfaces in the form of a relationship between certain surface-specific features. Drawn from many disciplines with a strong applied aspect, this is a research-led, interdisciplinary approach to the creation, analysis and visualisation of surfaces, focussing on topological data structures. Topological Data Structures for Surfaces: an introduction for Geographical Information Science describes the concepts and applications of these data structures. The book focuses on how these data structures can be used to analyse and visualise surface datasets from a range of disciplines such as human geography, computer graphics, metrology, and physical geography. Divided into two Parts, Part I defines the topological surface data structures and explains the various automated methods used for their generation. Part II demonstrates a number of applications of surface networks in diverse fields, ranging from sub-atomic particle collision visualisation to the study of population density patterns. To ensure that the material is accessible, each Part is prefaced by an overview of the techniques and application. Provides GI scientists and geographers with an accessible overview of current surface topology research. Algorithms are presented and explained with practical examples of their usage. Features an accompanying website developed by the Editor - http://geog.le.ac.uk/sanjayrana/surface-networks/ This book is invaluable for researchers and postgraduate students working in departments of GI Science, Geography and Computer Science. It also constitutes key reference material for Masters students working on surface analysis projects as part of a GI Science or Computer Science programme.




Topological, Differential and Conformal Geometry of Surfaces


Book Description

This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes’ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss–Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow’s Theorem on compact holomorphic submanifolds in complex projective spaces. Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.




Geographic Information Science


Book Description

This book constitutes the refereed proceedings of the 4th International Conference on Geographic Information Science, GIScience 2006. The book presents 26 revised full papers. Among traditional topics addressed are spatial representations and data structures, spatial and temporal reasoning, computational geometry, spatial analysis, and databases. Many papers deal with navigation, interoperability, dynamic modeling, ontology, and semantics. Geosensors, location privacy, social issues and GI research networks rank among the new directions covered.




Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms


Book Description

Implicit objects have gained increasing importance in geometric modeling, visualisation, animation, and computer graphics, because their geometric properties provide a good alternative to traditional parametric objects. This book presents the mathematics, computational methods and data structures, as well as the algorithms needed to render implicit curves and surfaces, and shows how implicit objects can easily describe smooth, intricate, and articulatable shapes, and hence why they are being increasingly used in graphical applications. Divided into two parts, the first introduces the mathematics of implicit curves and surfaces, as well as the data structures suited to store their sampled or discrete approximations, and the second deals with different computational methods for sampling implicit curves and surfaces, with particular reference to how these are applied to functions in 2D and 3D spaces.




Morphological Modeling of Terrains and Volume Data


Book Description

This book describes the mathematical background behind discrete approaches to morphological analysis of scalar fields, with a focus on Morse theory and on the discrete theories due to Banchoff and Forman. The algorithms and data structures presented are used for terrain modeling and analysis, molecular shape analysis, and for analysis or visualization of sensor and simulation 3D data sets. It covers a variety of application domains including geography, geology, environmental sciences, medicine and biology. The authors classify the different approaches to morphological analysis which are all based on the construction of Morse or Morse-Smale decompositions. They describe algorithms for computing such decompositions for both 2D and 3D scalar fields, including those based on the discrete watershed transform. Also addressed are recent developments in the research on morphological shape analysis, such as simplification operators for Morse and Morse-Smale complexes and their multi-resolution representation. Designed for professionals and researchers involved with modeling and algorithm analysis, Morphological Modeling of Terrains and Volume Data is a valuable resource. Advanced-level students of computer science, mathematics and geography will also find the content very helpful.










Scalable topological data{structures for 2 and 3 manifolds


Book Description

Pesquisas na área de estrutura de dados são fundamentais para aumentar a generalidade e eficiência computacional da representacão de modelos geometricos. Neste trabalho, apresentamos duas estruturas de dados topológicas escalonáveis, uma para superfícies triânguladas, chamada CHE (Compact Half--Edge), e outra para malhas de tetraedros, chamada CHF (Compact Half--Face). Tais estruturas são compostas de diferentes níveis, que nos possibilitam alterar a quantidade de dados armazenados com objetivo de melhorar sua eficiência computacional. O uso de APIs baseadas no conceito de objeto, e de haran"ca de classes, possibilitam uma interface única para cada função em todos os níveis das estruturas. A CHE e a CHF requerem pouca memória e são simples de implementar já que substituem o uso de ponteiros pelo de contêineres genéricos e regras aritméticas.




Computational Topology for Data Analysis


Book Description

Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions – like zigzag persistence and multiparameter persistence – and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.




Applications of Computational Mechanics in Geotechnical Engineering


Book Description

The development of constitutive relations for geotechnical materials, with the help of numerical models, have increased notably the ability to predict and to interpret mechanical behaviour of geotechnical works. These proceedings cover the applications of computational mechanics in this area.