Current practice in mechanical product description and computational processing is rooted in the theory of solid modeling whose principles were established almost four decades ago. Despite providing a concrete foundation for computerizing the product development process at the time, the limited capacity of the traditional models to answer the growing design and manufacturing needs is only recently being recognized in the scientific and engineering communities. I propose an alternative approach that employs spatial functions (i.e., 3D signals) instead of (or in addition to) topological pointsets in the Euclidean 3-space to abstract physical objects and their various geometric, physical, and material properties. The central benefit of this new approach is its great promise and potential extensibility in response to the emerging needs for more meaningful form-function correlations (e.g., for conceptual design) and more powerful tools for modeling new materials (e.g., knitted composites) and processes (e.g., additive manufacturing). This requires a paradigm shift from 'explicit' descriptions equipped with 'combinatorial' methods--e.g., combinatorial intersection test between objects modeled as r-sets, approximated by sphere trees, and tested using set-theoretic operations that exploit efficient tree traversal--to 'implicit' descriptions equipped with 'analytic' methods--e.g., analytic intersection test between objects modeled as density functions, approximated by frequency domain samples, and tested using measure-theoretic operations that are streamlined via fast Fourier transforms (FFT). The results obtained so far suggest that the proposed approach is a powerful unifying alternative to the conventional approaches to geometric computing and overcomes some of the key shortcomings of the traditional models. It opens up new promising theoretical and computational directions for future research in the nascent but emerging field of analytic solid geometry. I hope that the early-stage results in this thesis will inspire other researchers to develop more rigorous formal models and that it will eventually encourage broad industrial adoption of the analytic techniques into future generations of the product life-cycle management (PLM) software.

Presents the basic mathematical ideas and algorithms of the matrix analytic theory in a readable, up-to-date, and comprehensive manner.

This book gathers selected contributions presented at the INdAM Workshop "DREAMS", held in Rome, Italy on January 22−26, 2018. Addressing cutting-edge research topics and advances in computer aided geometric design and isogeometric analysis, it covers distinguishing curve/surface constructions and spline models, with a special focus on emerging adaptive spline constructions, fundamental spline theory and related algorithms, as well as various aspects of isogeometric methods, e.g. efficient quadrature rules and spectral analysis for isogeometric B-spline discretizations. Applications in finite element and boundary element methods are also discussed. Given its scope, the book will be of interest to both researchers and graduate students working in these areas.

Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing is the first book that applies advanced analytical and geometrical methods used in physics and mathematics to the financial field. It even obtains new results when only approximate and partial solutions were previously available.Through the problem of option pricing, th

Fundamentals of Matrix-Analytic Methods targets advanced-level students in mathematics, engineering and computer science. It focuses on the fundamental parts of Matrix-Analytic Methods, Phase-Type Distributions, Markovian arrival processes and Structured Markov chains and matrix geometric solutions. New materials and techniques are presented for the first time in research and engineering design. This book emphasizes stochastic modeling by offering probabilistic interpretation and constructive proofs for Matrix-Analytic Methods. Such an approach is especially useful for engineering analysis and design. Exercises and examples are provided throughout the book.

The first book on digital geometry by the leaders in the field.

The Blaubeuren Conference "Theory and Practice of Geometric Modeling" has become a meeting place for leading experts from industrial and academic research institutions, CAD system developers and experienced users to exchange new ideas and to discuss new concepts and future directions in geometric modeling. The relaxed and calm atmosphere of the Heinrich-Fabri-Institute in Blaubeuren provides the appropriate environment for profound and engaged discussions that are not equally possible on other occasions. Real problems from current industrial projects as well as theoretical issues are addressed on a high scientific level. This book is the result of the lectures and discussions during the conference which took place from October 14th to 18th, 1996. The contents is structured in 4 parts: Mathematical Tools Representations Systems Automated Assembly. The editors express their sincere appreciation to the contributing authors, and to the members of the program committee for their cooperation, the careful reviewing and their active participation that made the conference and this book a success.