Dual Quaternions and Their Associated Clifford Algebras

Clifford algebra for dual quaternions has emerged recently as an alternative to standard matrix algebra as a computational framework for computer graphics. This book presents dual quaternions and their associated Clifford algebras in a new light, accessible to and geared toward the computer graphics community.

Collecting all the associated formulas and theorems in one place, this book provides an extensive and rigorous treatment of dual quaternions, as well as showing how two models of Clifford algebra emerge naturally from the theory of dual quaternions. Each section comes complete with a set of exercises to help readers sharpen and practice their understanding.

This book is accessible to anyone with a basic knowledge of quaternion algebra and is of particular use to forward-thinking members of the computer graphics community.

Clifford代數對於雙四元數最近作為一種替代標準矩陣代數的計算框架在電腦圖形學中嶄露頭角。本書以一種新的方式呈現了雙四元數及其相關的Clifford代數，並且專為電腦圖形學界的人士提供了易於理解和適用的內容。

本書將所有相關的公式和定理集中在一個地方，全面而嚴謹地介紹了雙四元數，同時展示了兩種Clifford代數模型如何自然地從雙四元數理論中產生。每個章節都附有一套練習題，幫助讀者鞏固和練習他們的理解能力。

本書對於具備基本四元數代數知識的任何人都是易於理解的，對於具有前瞻性的電腦圖形學界成員尤其有用。

Ronald Goldman is a Professor of Computer Science at Rice University in Houston, Texas. Professor Goldman received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973.

Professor Goldman's current research concentrates on the mathematical representation, manipulation, and analysis of shape using computers. His work includes research in computer-aided geometric design, solid modeling, computer graphics, subdivision, polynomials, and splines. His most recent focus is on the uses of quaternions, dual quaternions, and Clifford algebras in computer graphics.

Dr. Goldman has published over two hundred research articles in journals, books, and conference proceedings. He has also authored two books on computer graphics and geometric modeling: Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling and An Integrated Introduction to Computer Graphics and Geometric Modeling. In addition, he has written an extended monograph on quaternions: Rethinking Quaternions: Theory and Computation.

Dr. Goldman is currently an Associate Editor of Computer Aided Geometric Design.

Before returning to academia, Dr. Goldman worked for ten years in industry solving problems in computer graphics, geometric modeling, and computer-aided design. He served as a Mathematician at Manufacturing Data Systems Inc., where he helped to implement one of the first industrial solid modeling systems. Later he worked as a Senior Design Engineer at Ford Motor Company, enhancing the capabilities of their corporate graphics and computer-aided design software. From Ford he moved on to Control Data Corporation, where he was a Principal Consultant for the development group devoted to computer-aided design and manufacture. His responsibilities included database design, algorithms, education, acquisitions, and research.

Dr. Goldman left Control Data Corporation in 1987 to become an Associate Professor of Computer Science at the University of Waterloo in Ontario, Canada. He joined the faculty at Rice University in Houston, Texas as a Professor of Computer Science in July 1990.

Ronald Goldman是德克薩斯州休斯頓萊斯大學的計算機科學教授。Goldman教授於1968年從麻省理工學院獲得數學學士學位，並於1973年在約翰霍普金斯大學獲得數學碩士和博士學位。

Goldman教授目前的研究集中在使用計算機對形狀進行數學表示、操作和分析。他的研究包括計算機輔助幾何設計、實體建模、計算機圖形學、細分、多項式和樣條。他最近的重點是在計算機圖形學中使用四元數、雙四元數和克利福德代數。

Goldman博士在期刊、書籍和會議論文中發表了200多篇研究文章。他還撰寫了兩本關於計算機圖形學和幾何建模的書籍:《金字塔算法: 一種動態規劃方法用於曲線和曲面的幾何建模》和《計算機圖形學和幾何建模的綜合介紹》。此外，他還撰寫了一本關於四元數的專題論文:《重新思考四元數: 理論和計算》。

Goldman博士目前是《計算機輔助幾何設計》的副編輯。

在回到學術界之前，Goldman博士在工業界工作了十年，解決了計算機圖形學、幾何建模和計算機輔助設計方面的問題。他曾在製造數據系統公司擔任數學家，協助實施了其中一個最早的工業實體建模系統。後來，他在福特汽車公司擔任高級設計工程師，增強了他們的企業圖形和計算機輔助設計軟件的功能。從福特汽車公司，他轉到Control Data Corporation，擔任計算機輔助設計和製造開發小組的首席顧問。他的職責包括數據庫設計、算法、教育、收購和研究。

Goldman博士於1987年離開Control Data Corporation，成為加拿大安大略省滑鐵盧大學的計算機科學副教授。他於1990年7月加入德克薩斯州休斯頓的萊斯大學計算機科學系，擔任教授職位。