Introduction to Geometric Algebra Computing

Computing with Circles and Lines

Dietmar Hildenbrand

• Publisher: Chapman and Hall/CRC
• Year: 2020
• ISBN:
  9781351648219
• Edition: 1
• Topic: Ciencia de la computación
• Subtopic: Gráficos computacionales

From the Foreword:

"Dietmar Hildenbrand's new book, Introduction to Geometric Algebra Computing, in my view, fills an important gap in Clifford's geometric algebra literature…I can only congratulate the author for the daring simplicity of his novel educational approach taken in this book, consequently combined with hands on computer based exploration. Without noticing, the active reader will thus educate himself in elementary geometric algebra algorithm development, geometrically intuitive, highly comprehensible, and fully optimized."

--Eckhard Hitzer, International Christian University, Tokyo, Japan

Geometric Algebra is a very powerful mathematical system for an easy and intuitive treatment of geometry, but the community working with it is still very small. The main goal of this book is to close this gap with an introduction to Geometric Algebra from an engineering/computing perspective.

This book is intended to give a rapid introduction to computing with Geometric Algebra and its

power for geometric modeling. From the geometric objects point of view, it focuses on the most basic ones, namely points, lines and circles. This algebra is called Compass Ruler Algebra, since it is comparable to working with a compass and ruler. The book explores how to compute with these geometric objects, and their geometric operations and transformations, in a very intuitive way.

The book follows a top-down approach, and while it focuses on 2D, it is also easily expandable to 3D computations. Algebra in engineering applications such as computer graphics, computer vision and robotics are also covered.

CHAPTER 1

Introduction

CONTENTS

1.1 Geometric Algebra

1.2 Geometric Algebra Computing

1.3 Outline

1.3.1 SECTION I: Tutorial

1.3.2 SECTION II: Mathematical Foundations

1.3.3 SECTION III: Applications

1.3.4 SECTION IV: Geometric Algebra at School

This book serves as an introduction to Geometric Algebra from a computing/engineering perspective. Its goal is to develop an appetite for delving into more. It pushes the reader into the cold water of swimming with Geometric Algebra right away, showing how to do things and what can be done, without the often burdensome overhead of rigid mathematical definitions.

1.1 Geometric Algebra

Geometric Algebra is a mathematical framework that makes it easy to describe geometric concepts and operations. It allows us to develop algorithms fast and in an intuitive way.

Geometric Algebra is based on the work of the German high school teacher Hermann Grassmann and his vision of a general mathematical language for geometry. His very fundamental work, called Ausdehnungslehre [14], was little noted in his time. Today, however, Grassmann is more and more respected as one of the most important mathematicians of the 19th century. William Clifford [5] combined Grassmann’s exterior algebra and Hamilton’s quaternions in what we call Clifford algebra or Geometric Algebra¹.

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¹ David Hestenes writes in his article [23] about the genesis of Geometric Algebra: Even today mathematicians typically typecast Clifford Algebra as the algebra of a quadratic form, with no awareness of its grander role in unifying geometry and algebra as envisaged by Clifford himself when he named it Geometric Algebra. It has been my privilege to pick up where Clifford left off to serve, so to speak, as principal architect of Geometric Algebra and Calculus as a comprehensive mathematical language for physics, engineering and computer science.

Pioneering work has been done by David Hestenes, 50 years ago. His book Space-Time Algebra [15] was the starting point for his development of Geometric Algebra into a unified mathematical language for physics, engineering and mathematics [16, 24] [20]. Especially interesting for this book is his work on Conformal Geometric Algebra (CGA) [17] [50]: the Compass Ruler Algebra (CRA), mainly treated in this book, is simply the Conformal Geometric Algebra in 2D.

The main advantage of Geometric Algebra is its easy and intuitive treatment of geometry. This is why the focus of this book is on the introduction of Geometric Algebra based on the computing with the most basic geometric objects, namely points, lines and circles. While we are computing in 2D space, the underlying algebra is the 4D Compass Ruler Algebra with a close link between algebra and the geometry of these basic geometric objects. While focusing on 2D, it is easily expandable to 3D computations as used, for instance, in the books [26], [57], [1] and [8].

1.2 Geometric Algebra Computing

Especially since the introduction of Conformal Geometric Algebra there has been an increasing interest in using Geometric Algebra in engineering. The use of Geometric Algebra in engineering applications relies heavily on the availability of an appropriate computing technology. The main problem of Geometric Algebra Computing is the exponential growth of data and computations compared to linear algebra, since the multivector² of an n-dimensional Geometric Algebra is 2ⁿ-dimensional. For the 5-dimensional Conformal Geometric Algebra, the multivector is already 32-dimensional.

An approach to overcome the runtime limitations of Geometric Algebra has been through optimized software solutions. Tools have been developed for high-performance implementations of Geometric Algebra algorithms such as the C++ software library generator Gaigen 2 from Daniel Fontijne and Leo Dorst of the University of Amsterdam [11], GMac from Ahmad Hosney Awad Eid of Suez Canal University [10], the Versor library [6] from Pablo Colapinto, the C++ expression template library Gaalet [63] from Florian Seybold of the University of Stuttgart, and our GAALOP compiler [29], which can also be used as a precompiler for languages such as C/C++, CUDA, OpenCL and C++ AMP. The big potential of optimizations of Geometric Algebra algorithms can be very well demonstrated with the inverse kinematics algorithm of [30] [25], which was in 2006 the first Geometric Algebra application that was faster than the standard implementation.

The book Foundations of Geometric Algebra Computing [26] defines Geometric Algebra Computing as the geometrically intuitive development of algorithms using Geometric Algebra with a focus on their efficient implementation. It describes Geometric Algebra Computing in a very fundamental way, since it breaks down the computing of Geometric Algebra algorithms to the most basic arithmetic operations.

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² The main algebraic element of Geometric Algebra (please refer to Chapt. 2)

This book on hand makes use of GAALOP [29] as a free and easy to handle tool in order to compute and visualize with Geometric Algebra. The book is suitable as a starting point for the understanding of Geometric Algebra for everybody interested in it as a new powerful mathematical system, especially for students, engineers and researchers in engineering, computer science and mathematics.

1.3 OUTLINE

This book is organized in the following sections:

I Tutorial

II Mathematical Foundations

III Applications

IV Geometric Algebra at School

SECTION I is a tutorial on how to work with Geometric Algebra, especially with Compass Ruler Algebra and its geometric objects, namely circles, lines and point pairs. SECTION II is for readers, now interested in the mathematical background of what they did in SECTION I. Readers, more interested in applications, are able to directly switch to SECTION III with applications in the areas of robotics, computer vision and computer graphics. SECTION IV gives some considerations about using Geometric Algebra already at school and about Space-Time Algebra in honor of the work of David Hestenes and especially to the 50th anniversary of his book about this algebra.

1.3.1 SECTION I: Tutorial

Chapt. 2 presents Compass Ruler Algebra in a nutshell as an algebra of circles, lines and point pairs. It summarizes the algebraic expressions needed for the tutorial in Chapt. 3 in order to describe the geometric objects and their intersections, the angles and distances between them as well as their reflections, rotations and translations.

Chapt. 3 is an easy to understand tutorial ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Foreword
• Preface
• Acknowledgments
• Chapter 1 ■ Introduction
• Section I Tutorial
• Section II Mathematical Foundations
• Section III Applications
• Section IV Geometric Algebra at School
• Bibliography
• Index