Introduction to Geometric Computing

Computing is quickly making much of geometry intriguing not only for philosophers and mathematicians, but also for scientists and engineers. What is the core set of topics that a practitioner needs to study before embarking on the design and implementation of a geometric system in a specialized discipline? This book attempts to find the answer, and it is the first book to include geometric issues, implementation issues, base software layers, and geometry background. Every programmer tackling a geometric computing problem encounters design decisions that need to be solved. This book reviews the geometric theory then applies it in an attempt to find that elusive "right" design.

First book to include geometric issues, implementation issues, base software layers, and geometry background Includes supplementary material: sn.pub/extras

Klappentext

The geometric ideas in computer science, mathematics, engineering, and physics have considerable overlap and students in each of these disciplines will eventually encounter geometric computing problems. The topic is traditionally taught in mathematics departments via geometry courses, and in computer science through computer graphics modules. This text isolates the fundamental topics affecting these disciplines and lies at the intersection of classical geometry and modern computing.

The main theme of the book is the definition of coordinate-free geometric software layers for Euclidean, spherical, projective, and oriented-projective geometries. Results are derived from elementary linear algebra and many classical computer graphics problems (including the graphics pipeline) are recast in this new language. Also included is a novel treatment of classical geometric and solid modeling problems. The definition of geometric software layers promotes reuse, speeds up debugging, and prepares the ground for a thorough discussion of advanced topics.

Start-up programs are provided for many programming exercises making this an invaluable book for computer science lecturers as well as software developers and researchers in the computer graphics industry.

Zusammenfassung
What may not be clear to individual programmers is that these design decisions have already been contemplated by others who have gone down some system design path only to discover (usually much later) that the design decisions that were made were lacking in some respect.

Inhalt
Euclidean Geometry.- 2D Computational Euclidean Geometry.- Geometric Predicates.- 3D Computational Euclidean Geometry.- Affine Transformations.- Affine Intersections.- Genericity in Geometric Computing.- Numerical Precision.- Non-Euclidean Geometries.- 1D Computational Spherical Geometry.- 2D Computational Spherical Geometry.- Rotations and Quaternions.- Projective Geometry.- Homogeneous Coordinates for Projective Geometry.- Barycentric Coordinates.- Oriented Projective Geometry.- Oriented Projective Intersections.- Coordinate-Free Geometry.- Homogeneous Coordinates for Euclidean Geometry.- Coordinate-Free Geometric Computing.- to CGAL.- Raster Graphics.- Segment Scan Conversion.- Polygon-Point Containment.- Illumination and Shading.- Raster-Based Visibility.- Ray Tracing.- Tree and Graph Drawing.- Tree Drawing.- Graph Drawing.- Geometric and Solid Modeling.- Boundary Representations.- The Halfedge Data Structure and Euler Operators.- BSP Trees in Euclidean and Spherical Geometries.- Geometry-Free Geometric Computing.- Constructive Solid Geometry.- Vector Visibility.- Visibility from Euclidean to Spherical Spaces.- Visibility in Space.