Bayro-Corrochano E., Scheuermann G. Geometric Algebra Computing: in Engineering and Computer Science

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Geometric Algebra Computing

Eduardo Bayro-Corrochano



Gerik Scheuermann

Editors

Geometric

Algebra

Computing

in Engineering

and Computer Science

Editors

Prof. Eduardo Bayro-Corrochano

Dept. Electrical Eng. &

Computer Science

CINVESTAV

Unidad Guadalajara

Av. Cientíﬁca 1145

45015 Colonia el Bajío,

Zapopan, JAL

Mexico

edb@gdl.cinvestav.mx

http://www.gdl.cinvestav.mx/edb

Prof. Dr. Gerik Scheuermann

Inst. Informatik

Universität Leipzig

04009 Leipzig

Germany

scheuermann@informatik.uni-leipzig.de

ISBN 978-1-84996-107-3 e-ISBN 978-1-84996-108-0

DOI 10.1007/978-1-84996-108-0

Springer London Dordrecht Heidelberg New York

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

Library of Congress Control Number: 2010926690

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contained in this book and cannot accept any legal responsibility or liability for any errors or omissions

that may be made.

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Preface

This book presents new results on applications of geometric algebra. The time when

researchers and engineers were starting to realize the potential of quaternions for ap-

plications in electrical, mechanic, and control engineering passed a long time ago.

Since the publication of Space-Time Algebra by David Hestenes (1966) and Clifford

Algebra to Geometric Calculus: A Uniﬁed Language for Mathematics and Physics

by David Hestenes and Garret Sobczyk (1984), consistent progress in the appli-

cations of geometric algebra has taken place. Particularly due to the great devel-

opments in computer technology and the Internet, researchers have proposed new

ideas and algorithms to tackle a variety of problems in the areas of computer science

and engineering using the powerful language of geometric algebra. In this process,

pioneer groups started the conference series entitled “Applications of Geometric

Algebra in Computer Science and Engineering” (AGACSE) in order to promote the

research activity in the domain of the application of geometric algebra. The ﬁrst

conference, AGACSE’1999, organized by Eduardo Bayro-Corrochano and Garret

Sobczyk, took place in Ixtapa-Zihuatanejo, Mexico, in July 1999. The contribu-

tions were published in Geometric Algebra with Applications in Science and Engi-

neering, Birkhäuser, 2001. The second conference, ACACSE’2001, was held in the

Engineering Department of the Cambridge University on 9–13 July 2001 and was

organized by Leo Dorst, Chris Doran, and Joan Lasenby. The best conference contri-

butions appeared as a book entitled Applications of Geometric Algebra in Computer

Science and Engineering, Birkhäuser, 2002. The third conference, AGACSE’2008,

took place in August 2008 in Grimma, Leipzig, Germany. The conference chairs,

Eduardo Bayro-Corrochano and Gerik Sheuermann, edited this book using selected

contributions that were peer-reviewed by at least two reviewers.

In the history of science, theories would have not been developed at all with-

out essential mathematical concepts. In various periods of the history of mathemat-

ics and physics, there is clear evidence of stagnation, and it is only thanks to new

mathematical developments that astonishing progress has taken place. Furthermore,

researchers unavoidably cause fragmented knowledge in their various attempts to

combine different mathematical systems. We realize that each mathematical sys-

tem brings about some parts of geometry; however, together, they constitute a sys-

tem that is highly redundant due to an unnecessary multiplicity of representations

vi Preface

for geometric concepts. In contrast, in the geometric algebra language, most of the

standard matter taught in engineering and computer science can be advantageously

reformulated without redundancies and in a highly condensed fashion.

This book presents a selection of articles about the theory and applications of the

advanced mathematical language geometric algebra which greatly helps to express

the ideas and concepts and to develop algorithms in the broad domains of computer

science and engineering. The contributions are organized in seven parts.

The ﬁrst part presents screw theory in geometric algebra, the parameterization

of 3D conformal transformations in conformal geometric algebra, and an overview

of applications of geometric algebra. The second part includes thorough studies on

Cliffor–Fourier transforms: the two-dimensional Clifford windowed Fourier trans-

form; the cylindrical Fourier transform; applications of the 3D geometric algebra

Fourier transform in graphics engineering; the 4D Clifford–Fourier transform for

color image processing; and the use of the Hilbert transforms in Clifford analysis

for signal processing. In the third part, self-organizing geometric neural networks

are utilized for 2D contour and 3D surface reconstruction in medical image process-

ing. The clustering and classiﬁcation are handled using geometric neural networks

and associative memories designed in the conformal geometric algebra. This part

concludes with a retrospective of the quaternion wavelet transform, including an

application for stereo vision. The fourth part for computer vision starts with a new

cone-pixel camera using a convex hull and twists in conformal geometric algebra.

The next work introduces a model-based approach for global self-localization using

active stereo vision and Gaussian spheres. In the ﬁfth part, the geometric character-

ization of M-conformal mappings is discussed, and a study of ﬂuid ﬂow problems

is carried out in depth using quaternionic analysis. The sixth part shows the im-

pressive space group visualizer for all 230 3D groups using the software packet

for geometric algebra computations CLUCalc. The second author studies geometric

algebra formalism as an alternative to distributed representation models; here con-

volutions are replaced by geometric products, and, as a result, a natural language

for visualization of higher concepts is proposed. Another author studies computa-

tional complexity reductions using Clifford algebras and shows that graph problems

of complexity class NP are polynomial in the number of Clifford operations re-

quired. The seventh part includes new developments in efﬁcient geometric algebra

computing: The ﬁrst author presents an efﬁcient blade factorization algorithm to

produce faster implementations of the Join; with the software packet GALOOP, the

second author symbolically reduces involved formulas of conformal geometric al-

gebra, generating suitable code for computing using hardware accelerators. Another

chapter shows applications of Grobner bases in robotics, formulated in the language

of Clifford algebras, in engineering to the theory of curves, including Fermat and

Bezier cubics, and in the interpolation of functions used in ﬁnite element theory.

We are very thankful to all book contributors, who are working persistently to

advance the applications of geometric algebra. We do hope that the reader will ﬁnd

this collection of contributions in a broad scope of the areas of engineering and

computer science very stimulating and encouraging. We hope that, as a result, we

will see our community growing and beneﬁtting from new and promising scientiﬁc

Preface vii

contributions. Finally, we thank also for the support to this book project given by

CINVESTAV Unidad Guadalajara and CONACYT Project 2007-1 82084.

CINVESTAV, Guadalajara, México

Universität Leipzig,

Institut für Informatik, Germany

Eduardo Bayro-Corrochano

Gerik Sheuermann

Contents

Part I Geometric Algebra

New Tools for Computational Geometry and Rejuvenation of Screw

Theory ................................... 3

David Hestenes

1 Introduction ............................. 3

2 Universal Geometric Algebra . ................... 4

3 Group Theory with Geometric Algebra ............... 6

4 Euclidean Geometry with Conformal GA .............. 8

5 Invariant Euclidean Geometry . ................... 10

6 ProjectiveGeometry......................... 13

7 Covariant Euclidean Geometry with Conformal Splits ....... 14

8 Rigid Displacements ......................... 18

9 FramingaRigidBody........................ 20

10 RigidBodyKinematics ....................... 22

11 Rigid Body Dynamics ........................ 24

12 Screw Theory ............................ 26

13 ConformalSplitandMatrixRepresentation............. 28

14 LinkedRigidBodies&Robotics .................. 31

References . ............................. 33

Tutorial: Structure-Preserving Representation of Euclidean Motions

Through Conformal Geometric Algebra ................ 35

Leo Dorst

1 Introduction ............................. 36

2 Conformal Geometric Algebra ................... 36

2.1 Trick 1: Representing Euclidean Points in Minkowski Space 36

2.2 Trick 2: Orthogonal Transformations as Multiple

Reﬂections in a Sandwiching Representation ........ 39

2.3 Trick3:ConstructingElementsbyAnti-Symmetry..... 42

2.4 Trick 4: Dual Speciﬁcation of Elements Permits Intersection 43