The Power of Geometric Algebra Computing

For Engineering and Quantum Computing

Dietmar Hildenbrand

• Éditeur: Chapman and Hall/CRC
• Année: 2021
• ISBN:
  9781000461237
• Édition: 1
• Sujet: Computer Science
• Sous-sujet: Computer Graphics

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 from a computing perspective in presenting the power of Geometric Algebra Computing for engineering applications and quantum computing.

The Power of Geometric Algebra Computing is based on GAALOPWeb, a new user-friendly, web-based tool for the generation of optimized code for different programming languages as well as for the visualization of Geometric Algebra algorithms for a wide range of engineering applications.

Key Features:

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• Introduces a new web-based optimizer for Geometric Algebra algorithms

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• Supports many programming languages as well as hardware

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• Covers the advantages of high-dimensional algebras

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• Includes geometrically intuitive support of quantum computing

This book includes applications from the fields of computer graphics, robotics and quantum computing and will help students, engineers and researchers interested in really computing with Geometric Algebra.

CHAPTER 1

Introduction

DOI: 10.1201/9781003139003-1

CONTENTS

• 1.1Geometric Algebra
• 1.2Geometric Algebra Computing
• 1.3Outline

Geometric Algebra is a very powerful mathematical language combining geometric intuitivity with the potential of high runtime-performance of the implementations. This book on hand is based on GAALOPWeb, a new user-friendly, web-based tool for the generation of optimized code for different programming languages as well as for the visualization of Geometric Algebra algorithms for a wide range of engineering applications. It includes applications from the fields of computer graphics, robotics and quantum computing.

The book Foundations of Geometric Algebra Computing [29] describes GAALOP (see Chapt. 4) in a very fundamental way, since it breaks down the computing of Geometric Algebra algorithms into the most basic arithmetic operations. This book on hand makes use of its web-based extention GAALOPWeb (see Chapt. 4).

This book is suitable as a starting point for computing with Geometric Algebra for everybody interested in it as a new powerful mathematical system, especially for students, engineers and researchers in engineering, computer science, quantum computing and mathematics.

1.1 Geometric Algebra

The main advantage of Geometric Algebra is its easy and intuitive treatment of geometry.

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 [24], 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 [11] combined Grassmann's exterior algebra and Hamilton's quaternions in what we call Clifford algebra or Geometric Algebra¹. Pioneering work has been done by David Hestenes. Especially interesting for this book is his work on Conformal Geometric Algebra (CGA) [26][50].

1.2 Geometric Algebra Computing

Especially since the introduction of Conformal Geometric Algebra (see Sect. 2.2) 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 achieved 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 [21], GMac from Ahmad Hosney Awad Eid of Suez Canal University [20], the Versor library [12] from Pablo Colapinto and the C++ expression template library Gaalet [60] from Florian Seybold of the University of Stuttgart. BiVector.net (see Figure 1.1) gives a good overview over software solutions for Geometric Algebra. It provides code generators for C++, C#, Python, Rust as well as starting points for libraries for Python, C/C++, Julia and the JavaScript visualization tool Ganja (see Sect. 2.4).

Figure 1.1 Screenshot of https://bivector.net.

Our GAALOP compiler [32] is not specific for one programming language but supports many of them. It can be used as a compiler for languages such as C/C++, C++ AMP, OpenCL and CUDA [29][31] as well as Python, Matlab, Mathematica, Julia or Rust. Please find details about GAALOP in Chapt. 3 and about its web-based extension GAALOPWeb in Chapt. 4.

¹ David Hestenes writes in his article [27] 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.

² The main algebraic element of Geometric Algebra (please refer to Sect. 2.2)

1.3 Outline

Chapt. 2 presents the most important Geometric Algebras for engineering, namely Conformal Geometric Algebra (CGA), Compass Ruler Algebra (CRA) and Projective Geometric Algebra (PGA).

Chapt. 3 is dealing with GAALOP and Chapt. 4 with GAALOPWeb, our new user-friendly, web-based version of GAALOP for a wide range of engineering applications based on Geometric Algebra algorithms. It makes it much easier for users to generate optimized source code without any software installation. GAALOPScript, the language to describe Geometric Algebra algorithms for the handling with GAALOPWeb is presented in Sect. 3.2. GAALOPWeb supports the user with visualizations of Geometric Algebra algorithms as demonstrated in Sect. 4.3. The visualization is based on Ganja as described in Sect. 2.4.

In the following chapters some applications of GAALOPWeb for different programming languages are presented. GAALOPWeb for C/C++ is presented in Chapt. 5 and GAALOPWeb for Python in Chapt. 6. A Molecular Distance Application using GAALOPWeb for Mathematica is shown in Chapt. 7 and an application of robot kinematics based on GAALOPWeb for Matlab in Chapt. 8.

The power of high-dimensional Geometric Algebras, as presented in Chapt. 9, comes with their ability to easily handle complex geometric objects. While the default Geometric Algebra of GAALOPWeb is the 5D Conformal Geometric Algebra with points, spheres, planes, circles and lines as geometric objects, three additional algebras are presented. The handling of conics in GAALOPWeb is shown in Chapt. 10. Quadrics as another example of geometric objects, which are interesting for applications, are handled in Chapt. 11 and cubics in Chapt. 12.

Geometric Algebra has an inherent potential for parallelization. This can be very well seen in an intermediate language of GAALOP called GAPP. Chapt. 13 describes GAALOPWeb for this language which is very well suitable for hardware implementations. Chapt. 14 shows, how GAALOPWeb for GAPPCO is dealing with the Geometric Algebra hardware design GAPPCO. Chapt. 15 describes a new hardware design called GAPPCO II.

Also quantum computing benefits a lot from Geometric Algebra. This is due to its ability to describe the quantum bit (qubit) operations as geometric transfo...