Quaternions and Rotation Sequences

Author: J. B. Kuipers
Publisher: Princeton University Press
ISBN: 0691211701
Category : Mathematics
Languages : en
Pages : 396

Book Description
Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the nineteenth century--a feat he celebrated by carving the founding equations into a stone bridge--mathematicians and engineers have been fascinated by these mathematical objects. Today, they are used in applications as various as describing the geometry of spacetime, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J. B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations. The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.

Quaternions and Rotation Sequences

Author: J. B. Kuipers
Publisher: Princeton University Press
ISBN: 9780691102986
Category : Mathematics
Languages : en
Pages : 398

Book Description
The book is an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The first part present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the last part, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.

Quaternions and Rotation Sequences

Author: Jack B. Kuipers
Publisher:
ISBN: 9780691058726
Category : Mathematics
Languages : en
Pages : 371

Rotations, Quaternions, and Double Groups

Author: Simon L. Altmann
Publisher: Courier Corporation
ISBN: 0486317730
Category : Mathematics
Languages : en
Pages : 315

Book Description
This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.

Quaternions and Rotation Sequences

Author: J. B. Kuipers
Publisher: Princeton University Press
ISBN: 0691102988
Category : Mathematics
Languages : en
Pages : 393

Visualizing Quaternions

Author: Andrew J. Hanson
Publisher: Elsevier
ISBN: 0080474772
Category : Mathematics
Languages : en
Pages : 530

Book Description
Introduced 160 years ago as an attempt to generalize complex numbers to higher dimensions, quaternions are now recognized as one of the most important concepts in modern computer graphics. They offer a powerful way to represent rotations and compared to rotation matrices they use less memory, compose faster, and are naturally suited for efficient interpolation of rotations. Despite this, many practitioners have avoided quaternions because of the mathematics used to understand them, hoping that some day a more intuitive description will be available.The wait is over. Andrew Hanson's new book is a fresh perspective on quaternions. The first part of the book focuses on visualizing quaternions to provide the intuition necessary to use them, and includes many illustrative examples to motivate why they are important—a beautiful introduction to those wanting to explore quaternions unencumbered by their mathematical aspects. The second part covers the all-important advanced applications, including quaternion curves, surfaces, and volumes. Finally, for those wanting the full story of the mathematics behind quaternions, there is a gentle introduction to their four-dimensional nature and to Clifford Algebras, the all-encompassing framework for vectors and quaternions. - Richly illustrated introduction for the developer, scientist, engineer, or student in computer graphics, visualization, or entertainment computing. - Covers both non-mathematical and mathematical approaches to quaternions.

From Natural Numbers to Quaternions

Author: Jürg Kramer
Publisher: Springer
ISBN: 3319694294
Category : Mathematics
Languages : en
Pages : 288

Book Description
This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory.

A Tribble's Guide to Space

Author: Alan C. Tribble
Publisher: Princeton University Press
ISBN: 9780691027630
Category : Science
Languages : en
Pages : 198

Book Description
Space--the final frontier. It's as little as fifty miles away, and yet it is considered one of the most dangerous and remote of places. Popular television shows such as Star Trek and movies such as Apollo 13 and October Sky have fired the imaginations of would-be explorers. Alan Tribble has worked on the design and development of dozens of spacecraft, including the Space Shuttle and the International Space Station. A Tribble's Guide to Space is a how-to book that is firmly grounded in the realities of current state-of-the-art space engineering while tapping into the power of imagination that drives us to explore. Alan Tribble offers a delightful guide to the nuts and bolts of space exploration, from how to get there to how to phone home and how to survive the harsh environment of space. Using examples from famous space missions, both factual and fictional, Tribble tackles fascinating real-world problems encountered in space exploration. Why do submarine hatches open inward and spacecraft hatches open outward? What do curveballs and satellites have in common? Why did Scotty, the chief engineer of the fictional USS Enterprise, always "need more power?" Why did the fire that destroyed Apollo I on the launch pad burn so furiously that no one could react fast enough to save the lives of the crew? In answering these questions, Tribble examines getting to space, from the physics of motion to the practical implications of Einstein's theory of relativity. He explores the basics of spaceships and starships, from power usage to navigation--all from the viewpoint of a spacecraft designer. This book describes rocket science and more in a manner that captures the common excitement shared by anyone with an interest in space exploration. Armchair astronauts, engineers who work in a space company and anyone who has looked up at the sky and wondered how we will get there will find reading this book a wonderful and enlightening experience.

Understanding Quaternions

Author: Peng Du
Publisher: Nova Science Publishers
ISBN: 9781536185539
Category : Mathematics
Languages : en
Pages : 197

Book Description
"Quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. They form an interesting algebra where each object contains 4 scalar variables, instead of Euler angles, which is useful to overcome the gimbal lock phenomenon when treating the rotation of objects. This book is about the mathematical basics and applications of quaternions. The first four chapters mainly concerns the mathematical theories, while the latter three chapters are related with three application aspects. It is expected to provide useful clues for researchers and engineers in the related area. In detail, this book is organized as follows: In Chapter 1, mathematical basics including the quaternion algebra and operations with quaternions, as well as the relationships of quaternions with other mathematical parameters and representations are demonstrated. In Chapter 2, how quaternions are formulated in Clifford Algebra, how it is used in explaining rotation group in symplectic vector space and parallel transformation in holonomic dynamics are presented. In Chapter 3, the wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In Chapter 4, hyperbolic Lobachevsky and spherical Riemann models, parameterized coordinates with spherical and cylindric symmetry are studied. In Chapter 5, ship hydrodynamics with allowance of trim and sinkage is investigated and validated with experiments. In Chapter 6, the ballast flying phenomenon based on Discrete Discontinuous Analysis is presented. In Chapter 7, a numerical study is proposed to analyze the effect of the caisson sliding subjected to a hydrodynamic loading in the stability of the rear side of the rubble mound breakwater"--

On Quaternions and Octonions

Author: John H. Conway
Publisher: CRC Press
ISBN: 1439864187
Category : Mathematics
Languages : en
Pages : 172

Book Description
This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less f