Guide to Geometric Algebra in Practice

ISBN-13: 9780857298102 / Angielski / Twarda / 2011 / 458 str.

Leo Dorst; Joan Lasenby

Opis

Geometric algebra (GA), also known as Clifford algebra, is a powerful unifying framework for geometric computations that extends the classical techniques of linear algebra and vector calculus in a structural manner. Its benefits include cleaner computer-program solutions for known geometric computation tasks, and the ability to address increasingly more involved applications.This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software and hardware tools. Contributions are included from an international community of experts spanning a broad range of disciplines.Topics and features: provides hands-on review exercises throughout the book, together with helpful chapter summaries; presents a concise introductory tutorial to conformal geometric algebra (CGA) in the appendices; examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing; reviews the employment of GA in theorem proving and combinatorics; discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA; proposes applications of coordinate-free methods of GA for differential geometry.This comprehensive guide/reference is essential reading for researchers and professionals from a broad range of disciplines, including computer graphics and game design, robotics, computer vision, and signal processing. In addition, its instructional content and approach makes it ideal for course use and students who need to learn the value of GA techniques.

Spis treści

How to Read this Guide to Geometric Algebra in Practice
Leo Dorst and Joan Lasenby

Part I: Rigid Body Motion

Rigid Body Dynamics and Conformal Geometric Algebra
Anthony Lasenby, Robert Lasenby and Chris Doran

Estimating Motors from a Variety of Geometric Data in 3D Conformal Geometric Algebra
Robert Valkenburg and Leo Dorst

Inverse Kinematics Solutions Using Conformal Geometric Algebra
Andreas Aristidou and Joan Lasenby

Reconstructing Rotations and Rigid Body Motions from Exact Point Correspondences through Reflections
Daniel Fontijne and Leo Dorst

Part II: Interpolation and Tracking

Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra using Polar Decomposition
Leo Dorst and Robert Valkenburg

Attitude and Position Tracking / Kinematics
L.P Candy and J Lasenby

Calibration of Target Positions using Conformal Geometric Algebra
Robert Valkenburg and Nawar Alwesh

Part III: Image Processing

Quaternion Atomic Function for Image Processing
Eduardo Bayro-Corrochano and Ulises Moya-Sánchez

Color Object Recognition Based on a Clifford Fourier Transform
Jose Mennesson, Christophe Saint-Jean and Laurent Mascarilla

Part IV: Theorem Proving and Combinatorics

On Geometric Theorem Proving with Null Geometric Algebra
Hongbo Li and Yuanhao Cao

On the Use of Conformal Geometric Algebra in Geometric Constraint Solving
Philippe Serré, Nabil Anwer and JianXin Yang

On the Complexity of Cycle Enumeration for Simple Graphs
René Schott and G. Stacey Staples

Part V: Applications of Line Geometry

Line Geometry in Terms of the Null Geometric Algebra over R^3,3, and Application to the Inverse Singularity Analysis of Generalized Stewart Platforms
Hongbo Li and Lixian Zhang

A Framework for n-dimensional Visibility Computations
L. Aveneau, S. Charneau, L Fuchs and F. Mora

Part VI: Alternatives to Conformal Geometric Algebra

On the Homogeneous Model of Euclidean Geometry
Charles Gunn

A Homogeneous Model for 3-Dimensional Computer Graphics Based on the Clifford Algebra for R³
Ron Goldman

Rigid-Body Transforms using Symbolic Infinitesimals
Glen Mullineux and Leon Simpson

Rigid Body Dynamics in a Constant Curvature Space and the ‘1D-up’ Approach to Conformal Geometric Algebra
Anthony Lasenby

Part VII: Towards Coordinate-Free Differential Geometry

The Shape of Differential Geometry in Geometric Calculus
David Hestenes

On the Modern Notion of a Moving Frame
Elizabeth L. Mansfield and Jun Zhao

Tutorial: Structure Preserving Representation of Euclidean Motions through Conformal Geometric Algebra
Leo Dorst

Blurb

Geometric algebra (GA), also known as Clifford algebra, is a powerful unifying framework for geometric computations that extends the classical techniques of linear algebra and vector calculus in a structural manner. Its benefits include cleaner computer-program solutions for known geometric computation tasks, and the ability to address increasingly more involved applications.

This highly practical Guide to Geometric Algebra in Practice reviews algebraic techniques for geometrical problems in computer science and engineering, and the relationships between them. The topics covered range from powerful new theoretical developments, to successful applications, and the development of new software tools. Contributions are included from an international community of experts spanning a broad range of disciplines.

Topics and features:

• Provides hands-on review exercises throughout the book, together with helpful chapter summaries
• Presents a concise introductory tutorial to conformal geometric algebra (CGA)
• Examines the application of CGA for the description of rigid body motion, interpolation and tracking, and image processing
• Reviews the employment of GA in theorem proving and combinatorics
• Discusses the geometric algebra of lines, lower-dimensional algebras, and other alternatives to 5-dimensional CGA
• Proposes applications of coordinate-free methods of GA for differential geometry

This comprehensive guide/reference is essential reading for researchers and professionals from a broad range of disciplines, including computer graphics and game design, robotics, computer vision, and signal processing. In addition, its instructional content and approach makes it suitable for course use and students who need to learn the value of GA techniques.

Dr. Leo Dorst is Universitair Docent (tenured assistant professor) in the Faculty of Sciences, University of Amsterdam, The Netherlands. Dr. Joan Lasenby is University Senior Lecturer in the Engineering Department of Cambridge University, U.K.