GEOMETRIC COMPUTATIONS WITH INTERVAL AND NEW ROBUST METHODS
APPLICATIONS IN COMPUTER GRAPHICS, GIS AND COMPUTATIONAL GEOMETRY
"Geometric Computations with Interval and New Robust Methods: Applications in Computer Graphics, GIS, and Computational Geometry" is a comprehensive book that delves into the challenges posed by errors in numerical computations and explores robust techniques to address these issues in geometric computations. Written by esteemed experts in the field, the book serves as an invaluable resource for researchers, practitioners, and students interested in improving the accuracy and reliability of geometric calculations.
The book begins with an insightful introduction, highlighting the significance of errors in numerical computations and their impact on geometric computations, especially those reliant on floating-point computation. The authors discuss various approaches to controlling errors in geometric computations and introduce the Interval Analysis Approach as a potential solution. They also explore global interval aspects and present the Exact Sign of Sum Algorithm (ESSA) and arithmetic filters to further enhance accuracy.
The core of the book focuses on Interval Analysis, which is a key methodology used to perform reliable geometric computations. The authors thoroughly cover topics like interval arithmetic operations, implementation of interval arithmetic, inclusion functions, and natural interval extensions. They discuss the importance of Skelboe's Principle and present inner approximations for linear functions. Centered forms and other inclusion techniques are also explored, along with subdivision methods for range estimation.
Next, the book introduces Interval Newton Methods, which play a crucial role in solving geometric equations and functions with enhanced precision. The authors discuss different versions of Interval Newton Methods and present the Hansen-Sengupta Version, including the existence test.
A significant part of the book is dedicated to Intersection Tests, which are essential in numerous geometric applications. Various intersection tests for line segments, boxes, rectangles, triangles, tetrahedra, ellipses, and spheres are meticulously explored. The authors provide detailed algorithms and numerical examples to illustrate the practicality and reliability of these intersection tests.
The book also presents the SCCI-Hybrid Method for 2D-curve tracing, which is a robust approach to handle curve tracing in computer graphics and computational geometry. The method is explained in-depth, and several examples are provided to demonstrate its effectiveness.
Moreover, the book explores Interval Versions of Bernstein Polynomials, Bezier Curves, and the de Casteljau Algorithm, which are fundamental in computer graphics and curve representation. The authors introduce interval polynomials and interval Bernstein polynomials, extending the concepts to Bezier curves and the de Casteljau Algorithm.
In the final sections, the book tackles the robust computations of selected discrete problems, including convex-hull computations in 2D and exact computation of Delaunay and Power Triangulations. The authors present algorithms and practical examples to achieve accurate results in these discrete geometric problems. Additionally, the book covers exact and robust line simplification, essential for efficient data processing and analysis.
In conclusion, "Geometric Computations with Interval and New Robust Methods" is a comprehensive and authoritative guide for researchers, professionals, and students interested in the field of geometric computations. By addressing errors and providing robust techniques, this book offers valuable insights and practical tools for computer graphics, GIS, and computational geometry applications.