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2001 | Buch

Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry Kapitel lesen Erstes Kapitel lesen

Geometric Algebra with Applications in Science and Engineering

herausgegeben von: Dr. Eduardo Bayro Corrochano, Prof. Garret Sobczyk

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

The goal of this book is to present a unified mathematical treatment of diverse problems in mathematics, physics, computer science, and engineer­ ing using geometric algebra. Geometric algebra was invented by William Kingdon Clifford in 1878 as a unification and generalization of the works of Grassmann and Hamilton, which came more than a quarter of a century before. Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. The approach to Clifford algebra adopted in most of the ar­ ticles here was pioneered in the 1960s by David Hestenes. Later, together with Garret Sobczyk, he developed it into a unified language for math­ ematics and physics. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from 1966 to 1967. He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education. Geometric algebra provides a rich, general mathematical framework for the develop­ ment of multilinear algebra, projective and affine geometry, calculus on a manifold, the representation of Lie groups and Lie algebras, the use of the horosphere and many other areas. This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers.

Inhaltsverzeichnis

Frontmatter

Advances in Geometric Algebra

Frontmatter
Chapter 1. Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry
Abstract
My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computer-aided geometry, robotics, computer vision and the like. A detailed description and analysis of the model is soon to be published elsewhere, so I can concentrate on highlights here, although with a slightly different formulation that I find more convenient for applications. Also, I can assume that this audience is familiar with Geometric Algebra, so we can proceed rapidly without belaboring the basics.
David Hestenes
Chapter 2. Universal Geometric Algebra
Abstract
Since Grassmann’s original work “Ausdehnungslehre” in 1844, and William Kingdom Clifford’s later discovery of “geometric algebra” in 1878, the mathematical community has been puzzled by exactly how these works fit into the main stream of mathematics. Certainly the importance of these works in the mathematics at the end of the 20th Century has been recognized, but there has been no general agreement about where and how the methods should be utilized. In this chapter, I wish to show how the works of Grassmann and Clifford can be integrated into the mainstream of mathematics in such a way as to require as little as possible changes to the main body of mathematics as we know it today. As has been often repeated by Hestenes and others, geometric algebra should be seen as a great unifier of the geometric ideas of mathematics.
Garret Sobczyk
Chapter 3. Realizations of the Conformal Group
Abstract
Perhaps one of the first to consider the problems of projective geometry was Leonardo da Vinci (1452-1519). However, projective geometry as a self-contained discipline was not developed until the work “Traite des propriés projectives des figure” of the French mathematician Poncelet (1788-1867), published in 1822. The extrordinary generality and simplicity of projective geometry led the English mathematician Cayley to exclaim: “Projective Geometry is all of geometry” [16]. D. Hestenes in [8] showed how the methods of projective geometry, formulated in geometric algebra, can be effectively used to study basic properties of the conformal group. The purpose of this article is to further explore the deep relationships that exist beween projective geometry and the conformal group.
Jose Maria Pozo, Garret Sobczyk
Chapter 4. Hyperbolic Geometry
Abstract
Hyperbolic geometry is an important branch of mathematics and physics. For hyperbolic n-space, there are five important analytic models: the Poincaré ball model, the Poinca re half-space model, the Klein ball model, the hemisphere model and the hyperboloid model. The hyperboloid model is defined to be one branch H n of the set
$$ \left\{ {x{ \in ^{{n,1}}}|x \cdot x = - 1} \right\}. $$
Hongbo Li

Theorem Proving

Frontmatter
Chapter 5. Geometric Reasoning With Geometric Algebra
Abstract
Geometric (Clifford) algebra was motivated by geometric considerations and provides a comprehensive algebraic formalism for the expression of geometric ideas [11]. Recent research has shown that this formalism may be effectively used in algebraic approaches for automated geometric reasoning [7, 12, 14, 20, 24]. Starting with an introduction to Clifford algebra for n-dimensional Euclidean geometry, this chapter is mainly concerned with the automatic proving of theorems in geometry and identities in Clifford algebra. We explain how to express geometric concepts and relations and how to formulate geometric problems in the language of Clifford algebra. Several examples are given to illustrate a simple mechanism for deriving Clifford algebraic representations of constructed points, or other geometric objects, and how the representations may be used for proving theorems automatically. With explicit representations of geometric objects and simple substitutions, proving a theorem is reduced finally to verifying whether a Clifford algebraic expression is equal to 0. The latter is accomplished in our case by the techniques of term-rewriting for any fixed n.
Dongming Wang
Chapter 6. Automated Theorem Proving
Abstract
In modern algebraic methods for automated geometry theorem proving, Wu’s characteristic set method (Wu, 1978, 1994; Chou, 1988) and the Gröbner basis method (Buchberger, Collins and Kutzler, 1988; Kutzler and Stifter, 1986; Kapur, 1986) are two basic ones. In these methods, the first step is to set up a coordinate system, and represent the geometric entities and constraints in the hypothesis of a theorem by coordinates and polynomial equations. The second step is to compute a characteristic set or Gröbner basis by algebraic manipulations among the polynomials. The third step is to verify the conclusion of the theorem by using the characteristic set or Gröbner basis.
Hongbo Li

Computer Vision

Frontmatter
Chapter 7. The Geometry Algebra of Computer Vision
Abstract
In this chapter we present a mathematical approach for the computation of problems in computer vision which is based on geometric algebra. We will show that geometric algebra is a well-founded and elegant language for expressing and implementing those aspects of linear algebra and projective geometry that are useful for computer vision. Since geometric algebra offers both geometric insight and algebraic computational power, it is useful for tasks such as the computation of projective invariants, camera calibration and recovery of shape and motion. We will mainly focus on the geometry of multiple uncalibrated cameras
Eduardo Bayro Corrochano, Joan Lasenby
Chapter 8. Using Geometric Algebra for Optical Motion Capture
Abstract
Optical motion capture refers to the process by which accurate 3D data from a moving subject is reconstructed from the images in two or more cameras. In order to achieve this reconstruction it is necessary to know how the cameras are placed relative to each other, the internal characteristics of each camera and the matching points in each image. The goal is to carry out this process as automatically as possible. In this paper we will outline a series of calibration techniques which use all of the available data simultaneously and produce accurate reconstructions with no complicated calibration equipment or procedures. These techniques rely on the use of geometric algebra and the ability therein to differentiate with respect to multivectors and linear functions.
Joan Lasenby, Adam Stevenson
Chapter 9. Bayesian Inference and Geometric Algebra: An Application to Camera Localization
Abstract
Geometric algebra is an extremely powerful language for solving complex geometric problems in engineering [4, 8]. Its advantages are particularly clear in the treatment of rotations. Rotations of a vector are performed by the double-sided application of a rotor, which is formed from the geometric product of an even number of unit vectors. In three dimensions a rotor is simply a normalised element of the even subalgebra of G 3, the geometric algebra of three dimensional space. In this paper we are solely interested in rotations in space, and henceforth all reference to rotors can be assumed to refer to the 3-d case. Rotors have a number of useful features. They can be easily parameterised in terms of the bivector representing the plane of rotation. Their product is a very efficient way of computing the effect of compound rotations, and is numerically very stable.
Chris Doran
Chapter 10. Projective Reconstruction of Shape and Motion Using Invariant Theory
Abstract
In this chapter we present a geometric approach for the computation of shape and motion using projective invariants in the geometric algebra framework [6, 7].
Eduardo Bayro Corrochano, Vladimir Banarer

Robotics

Frontmatter
Chapter 11. Robot Kinematics and Flags
Abstract
In robotics the group of proper rigid transformations of 3-dimensional space is of central importance. The relevant Clifford algebra in this case is a degenerate one with three generators that square to — 1 and a single generator that squares to 0. The algebra contains a copy of the group’s double cover.
J. M. Selig
Chapter 12. The Clifford Algebra and the Optimization of Robot Design
Abstract
The goal of this chapter is a computer aided design environment that assists the inventor to formulate a task and evaluate candidate devices. The task trajectory of a robot is specified as a set of homogeneous transforms that define key frames for a desired end-effector trajectory. These key frames are converted to double quaternions and interpolated by generalizing well known techniques for Bezier interpolation of quaternions. The result is an efficient interpolation algorithm.
Shawn G. Ahlers, John Michael McCarthy
Chapter 13. Applications of Lie Algebras and the Algebra of Incidence
Abstract
We present the fundamentals of Lie algebra and the algebra of incidence in the n-dimensional affine plane. The difference between our approach and previous contributions, [5, 4, 2] is twofold. First, our approach is easily accessible to the reader because there is a direct translation of the familiar matrix representations to our representation using bivectors from the appropriate geometric algebra. Second, our “hands on” approach provides examples from robotics and image analysis so that the reader can become familiar with the computational aspects of the problems involved. This chapter is to some extent complimentary to the above mentioned references. Lie group theory is the appropriate tool for the study and analysis of the action of a group on a manifold. Geometric algebra makes it possible to carry out computations in a coordinate-free manner by using a bivector representation of the most important Lie algebras [5]. Using the bivector representation of a Lie operator, we can easily compute a variety of invariants useful in robotics and image analysis. In our study of rigid motion in the n-dimensional affine plane, we use both the structure of the Lie algebra alongside the operations of meet and join from incidence algebra.
Eduardo Bayro Corrochano, Garret Sobczyk

Quantum and Neural Computing, and Wavelets

Frontmatter
Chapter 14. Geometric Algebra in Quantum Information Processing by Nuclear Magnetic Resonance
Abstract
The relevance of information theoretic concepts to quantum mechanics has been apparent ever since it was realized that the Einstein-Podolsky-Rosen paradox does not violate special relativity because it cannot be used to transmit information faster than light [22, 39]. Over the last few years, physicists have begun to systematically apply these concepts to quantum systems. This was initiated by the discovery, due to Benioff [3], Feynman [25] and Deutsch [17], that digital information processing and even universal computation can be performed by finite state quantum systems. Their work was originally motivated by the fact that as computers continue to grow smaller and faster, the day will come when they must be designed with quantum mechanics in mind (as Feynman put it, “there’s plenty of room at the bottom”). It has since been found, however, that quantum information processing can accomplish certain cryptographic, communication, and computational feats that are widely believed to be classically impossible [5, 9, 19, 23, 40, 53], as shown for example by the polynomial-time quantum algorithm for integer factorization due to Shor [45]. As a result, the field has now been the subject of numerous popular accounts, including [1, 11, 37, 60]. But despite these remarkable theoretical advances, one outstanding question remains: Can a fully programmable quantum computer actually be built?
Timothy F. Havel, David G. Cory, Shyamal S. Somaroo, Ching-Hua Tseng
Chapter 15. Geometric Feedforward Neural Networks and Support Multivector Machines
Abstract
The representation of the external world in biological creatures appears to be definable in terms of geometry. We can formalize the relationships between the physical signals of external objects and the internal signals of a biological creature by using extrinsic vectors coming from the world and intrinsic vectors representing the world internaly. We can also assume that the external world and the internal world have different reference coordinate systems. If we consider the acquisition and coding of knowledge as a distributed and differentiated process, it is imaginable that there should e-xist various domains of knowledge representation obeying different metrics which can be modelled using different vectorial basis. How it is possible that nature could have acquired through evolution such tremendous representation power for dealing with complicated geometric signal processing [13]? Pellionisz and Llinàs [15, 16] claim in a stimulating series of articles that the formalization of the geometrical representation seems to be the dual expression of extrinsic physical cues performed by the intrinsic central nervous system vectors. These vectorial representations, related to reference frames intrinsic to the creature, are covariant for perception analysis and contravariant for action synthesis. The authors explain that the geometric mapping between these two vectorial spaces can be implemented by a neural network which performs as a metric tensor [16].
Eduardo Bayro Corrochano, Refugio Vallejo
Chapter 16. Image Analysis Using Quaternion Wavelets
Abstract
The idea for Quaternion Wavelets comes from the need to represent evolving objects without the use of sequencial pictures of the object in different positions. We present a short introduction to ordinary wavelets, emphazising those concepts that are needed to translate the ideas to a quaternion framework. We also set up the quaternionic framework for the theory, because even when it is known, it has been used in so many different ways that a coherent picture becomes very difficult.
Leonardo Traversoni

Applications to Engineering and Physics

Frontmatter
Chapter 17. Objects in Contact: Boundary Collisions as Geometric Wave Propagation
Abstract
The motivation behind this work is to make the computation of collision-free motions of robots efficiently computable. For translational motions, the boundary of permissible translations of a reference point is obtained from the obstacles and the robot by a kind of dilation, ‘thickening’ the obstacle (see below for details) to produce the forbidden states in the configuration space of translations. The intuitive similarity of this operation to convolution suggests that we might be able to find a kind of Fourier transformation, in the sense that we might separate the shapes into independent ‘spectral components’ and combine those simply; after which the collision boundary would be obtained by the inverse transformation. This would enable the development of a ‘systems theory’ for collisions.
Leo Dorst
Chapter 18. Modern Geometric Calculations in Crystallography
Abstract
The aim of mathematical crystallography is the classification of periodic structures by means of different equivalence relationships, yielding the well known crystallographic classes and Bravais lattices [1]. Periodicity (crys-tallinity) has been the paradigm of classical crystallography. Recently, more systematical attention has been paid to structures which are not orthodox crystals. For example, some generalizations, involving curved spaces with non-Euclidean metrics, were developed for the understanding of random and liquid crystalline structures [2]. However, the first step away from orthodox crystalline order, represented by the 230 crystallographic space groups, was motivated by the appearance of quasicrystals in 1984 [3]. Since then, crystallography has been the subject of deep revisions. Quasicrystals are metallic alloys whose diffraction patterns exhibit sharp spots (like a crystal) but non-crystallographic symmetry. This means that the lattice underlying the atomic structure cannot be periodic. So, crystallography faces a non-crystalline but perfectly ordered structure. There are also many other directions in which classical crystallography can be generalized, by relaxing or altering various requirements, to include structures which are ordered but do not follow the exact paradigm of crystallization [4].
G. Aragon, J. L. Aragon, F. Davila, A. Gomez, M. A. Rodriguez
Chapter 19. Quaternion Optimization Problems in Engineering
Abstract
The interconnection between algebraic and geometric descriptions of space-time properties has attracted and ravished mathematicians since the time of Euclid. The last two centuries have been marked by several great contributions to this subject; among them Clifford and Grassmann algebras and Hamilton’s quaternions. Quaternions were invented by Hamilton to simplify mathematical modelling of rigid body motion in three dimensions. A fascinating history of quaternions is presented in many books, see, for instance, [1]. The joint work of many mathematicians revealed the fundamental connections of Clifford’s, Grassmann’s, and Hamilton’s approaches. The result of all this work is Geometric Algebra (see [13]).
Ljudmila Meister
Chapter 20. Clifford Algebras in Electrical Engineering
Abstract
The Maxwell-Lorentz equations that underpin all of electrical engineering are intrinsically relativistic. Even in problems confined to nonrelativistic velocities, the fundamental relativistic symmetries of the underlying theory imply important relations that often hold the keys to solutions of electromagnetic problems and may suggest significant insights into the underlying phenomena.
William E. Baylis
Chapter 21. Applications of Geometric Algebra in Physics and Links With Engineering
Abstract
While the early applications of geometric algebra (GA) were confined to physics, there has been significant progress over recent years in applying geometric algebra to areas of engineering and computer science. The beauty of using the same language for these applications is that both engineers and physicists should be able to understand the work done in each others fields. It is the aim of this paper to give brief outlines of the use of GA in the a-reas of relativity, quantum mechanics and gravitation — all using tools with which anyone working with GA should be familiar. Taking one particular area, multiparticle quantum mechanics, it is shown that the same mathematics may have some interesting applications in the fields of computer vision and robotics.
Anthony Lasenby, Joan Lasenby

Computational Methods in Clifford Algebras

Frontmatter
Chapter 22. Clifford Algebras as Projections of Group Algebras
Abstract
Clifford algebras appeared as a result of the natural desire of mathematicians to extend a finite-dimensional vector space to an algebraic structure where the inner and outer products are defined in terms of a single geometric multiplication [14], [26], [1], [25]. This idea was most attractively developed by D.Hestenes [20], [21]), and was immediately accepted by some physicists.
Vladimir M. Chernov
Chapter 23. Counterexamples for Validation and Discovering of New Theorems
Abstract
This chapter describes an experiment which took place over the two year period 1997-1999. The chapter consists of an adaptation of excerpts from the two Web Pages http://www.hit.fi/~lounesto/counterexamples.htm http://www,hit.fi/~lounesto/sci.math.htm.
Pertti Lounesto
Chapter 24. The Making of GABLE: A Geometric Algebra Learning Environment in Matlab
Abstract
Geometric algebra extends Clifford algebra with geometrically meaningful operators with the purpose of facilitating geometrical computations. Present textbooks and implementation do not always convey this geometrical flavor or the computational and representational convenience of geometric algebra, so we felt a need for a computer tutorial in which representation, computation and visualization are combined to exhibit the intuition and the techniques of geometric algebra. Current software packages are either Clifford algebra only (CLICAL [11] and CLIFFORD [3]) or do not include graphics [2], so we decided to build our own. The result is GABLE (Geometric AlgeBra Learning Environment), a hands-on tutorial on geometric algebra that is suited for undergraduate students [7].
Stephen Mann, Leo Dorst, Tim Bouma
Chapter 25. Helmstetter Formula and Rigid Motions with CLIFFORD
Abstract
CLIFFORD is a Maple package for symbolic computations in Clifford algebras Cl(B) of an arbitrary symbolic or numeric bilinear form B. The purpose of this paper is to show usability and power of CLIFFORD when performing computer-based proofs and explorations of mathematical aspects of Clifford algebras and their applications. It is intended as an invitation to engineers, computer scientists, and robotics to use Clifford algebra methods as opposed to coordinate/matrix methods. CLIFFORD has been designed as a tool to promote and facilitate explorative mathematics among non Clifford-algebra specialists. As an example of the power of CLIFFORD, we restate a formula due to Helmstetter which relates the product in Cl(g), the Clifford algebra of the symmetric part of B, to the product in Cl(B). Then, with CLIFFORD, we prove it in dimension 3. Clifford algebras of a degenerate quadratic form provide a convenient tool with which to study groups of rigid motions in ℝ3. Using CLIFFORD we will actually explicitly describe all elements of Pin(3) and Spin(3). Rotations in ℝ3 can then be generated by unit quaternions realized as even elements in Cl 0,3 + Simple computations using quaternions are then performed with CLIFFORD. Throughout this paper we illustrate actual CLIFFORD commands and steps undertaken to solve the problems.
Rafal Ablamowicz
Backmatter
Metadaten
Titel
Geometric Algebra with Applications in Science and Engineering
herausgegeben von
Dr. Eduardo Bayro Corrochano
Prof. Garret Sobczyk
Copyright-Jahr
2001
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0159-5
Print ISBN
978-1-4612-6639-6
DOI
https://doi.org/10.1007/978-1-4612-0159-5