Geometric Algebra book requests for January 2012

I sent our department’s library coordinator the following list of book requests to be sent to the Ateneo de Manila University Library.  I culled the list from my Geometric Algebra Facebook page:

  1. “Geometric Algebra Computing: In Engineering and Computer Science” (2010) by Bayro-Corrochano, Gerik Scheuermann
  2. “Classical Groups and Geometric Algebra” (2001) by Larry C. Grove
  3. “Geometric Algebra and Applications to Physics (2006) by Bidyut Kumar Datta and Venzo de Sabbata
  4. “A New Approach to Differential Geometry using Clifford’s Geometric Algebra” (2012) by John Snygg
  5. “Quaternions, Clifford Algebras, and Relativistic Physics” (2007) by Patrick R. Girard
  6. “Clifford Algebras and Their Application in Mathematical Physics” (1993) by Gerhard Jank, Klaus Habetha, and Volker Dietrich
  7. “Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry” (2007) by Leo Dorst, Daniel Fontijne, and Stephen Mann

JOSA A 2012: Polarization ellipse and Stokes parameters in geometric algebra

Adler G. Santos, Quirino M. Sugon, Jr., and Daniel J. McNamara, “Polarization ellipse and Stokes parameters in geometric algebra,” J. Opt. Soc. Am. A 29, 89-98 (2012)

Abstract

In this paper, we use geometric algebra to describe the polarization ellipse and Stokes parameters. We show that a solution to Maxwell’s equation is a product of a complex basis vector in Jackson and a linear combination of plane wave functions. We convert both the amplitudes and the wave function arguments from complex scalars to complex vectors. This conversion allows us to separate the electric field vector and the imaginary magnetic field vector, because exponentials of imaginary scalars convert vectors to imaginary vectors and vice versa, while exponentials of imaginary vectors only rotate the vector or imaginary vector they are multiplied to. We convert this expression for polarized light into two other representations: the Cartesian representation and the rotated ellipse representation. We compute the conversion relations among the representation parameters and their corresponding Stokes parameters. And finally, we propose a set of geometric relations between the electric and magnetic fields that satisfy an equation similar to the Poincaré sphere equation.

© 2012 Optical Society of America

Original Manuscript: July 7, 2011
Revised Manuscript: October 24, 2011
Manuscript Accepted: October 27, 2011
Published: December 12, 2011

Geometric algebra now has a Facebook page

William Kingdon Clifford

William Kingdon Clifford

I created a Facebook page for geometric algebra: http://www.facebook.com/geometricalgebra.    It is difficult to post the latest updates on geometric algebra and Clifford algebra through this blog.  Facebook makes it easier for me to just simply copy the webpage’s url and first paragraph, and voila! its a blog post, with a preview on the actual view of the original webpage.  So if you need to get the latest updates on geometric algebra, subscribe to the Facebook page.

The mission and vision of the geometric algebra Facebook page is to collect in one repository the web pages I cull from Google Alerts for CA and GA.  If I need to lengthy comment on a particular entry, I shall do it in this blog.  I will try to make this blog more personal–my thoughts on the subject–rather than duplicate information from somewhere else.  Google does not love copycats, as I learned from books on Search Engine Optimization.  What this blog will do is to organize the references in the Facebook page, particularly the books and journal articles, which are handy when writing research papers.  The books I shall really keep track and ask the Ateneo de Manila University Library to purchase them.  My aim is to make the Ateneo Library host the best geometric algebra  collection in this part of the world.

Modern Geometric Algebra: A (Very Incomplete!) Survey by Jeff Suzuki

from the Mathematics Teacher

August 2009, volume 103, issue 1, page 26

Abstract

Representing products as rectangles can be used to introduce, connect, and reinforce concepts across mathematics.

Keywords:
Grades 9-12
Article

Geometric Algebra and Applications to Physics: a book by Venzo de Sabbata and Bidyut Kumar Datta

from Amazon:

Product Description

Bringing geometric algebra to the mainstream of physics pedagogy, Geometric Algebra and Applications to Physics not only presents geometric algebra as a discipline within mathematical physics, but the book also shows how geometric algebra can be applied to numerous fundamental problems in physics, especially in experimental situations.

This reference begins with several chapters that present the mathematical fundamentals of geometric algebra. It introduces the essential features of postulates and their underlying framework; bivectors, multivectors, and their operators; spinor and Lorentz rotations; and Clifford algebra. The book also extends some of these topics into three dimensions. Subsequent chapters apply these fundamentals to various common physical scenarios. The authors show how Maxwell’s equations can be expressed and manipulated via space-time algebra and how geometric algebra reveals electromagnetic waves’ states of polarization. In addition, they connect geometric algebra and quantum theory, discussing the Dirac equation, wave functions, and fiber bundles. The final chapter focuses on the application of geometric algebra to problems of the quantization of gravity.

By covering the powerful methodology of applying geometric algebra to all branches of physics, this book provides a pioneering text for undergraduate and graduate students as well as a useful reference for researchers in the field.

Harcover, 184 pages, Taylor and Francis, 1st Edition, 2 December 2006.

Conformal geometric algebra in Wikipedia

from JHeald of Wikipedia

In mathematics, more specifically in the area of computational geometryconformal geometric algebra (CGA) is the geometric algebra(Clifford algebra) that results if an n-dimensional Euclidean (or pseudo-Euclidean) space ℝp,q is projectively mapped into ℝp+1,q+1 in a particular way, and a geometric algebra is then constructed over that n+2 dimensional space. This allows translations of the n-dimensional space, as well as rotations, to be represented using elements of a geometric algebra; and it is found that spheres and circles as well as lines and planes gain particularly natural, as well as computationally amenable, representations.

The projective mapping from ℝp,q to ℝp+1,q+1 is constructed so that the higher dimensional space contains the full projective space of the underlying original space; so that generalized spheres in the underlying space map onto (hyper-)planes orthogonal to a particular vector inn+2 dimensional space (which can thus be used to represent them); and so that the effect of a translation (or in fact any conformal mapping) of the underlying space corresponds to the effect of an n+2 dimensional rotation in the higher dimensional space.

Introduction of the geometric algebra of this space, based on the Clifford product of vectors, allows such transformations to be represented using the algebra’s characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently using the Clifford product rule. A consequence of the sandwich transformation structure is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. Some intersection operations also acquire a very tidy algebraic form: for example, applying the wedge product to the vectors corresponding to two spheres produces abivector corresponding to their circle of intersection; alternatively the tetra-vector of the hypersurface which is the dual of such a vector can be decomposed as the successive wedge product of vectors representing four points on the sphere.

Clifford Algebras, Clifford Analysis and their applications, ICNPAA Congress: Mathematical Problems in Engineering, Aerospace and Sciences, July 11-14,2012, Vienna, Austria

From GA Net Updates:

The session aims to present the latest advances in the field of Clifford (geometric) algebras and their applications in mathematics, physics, engineering and other applied sciences. The proposed session intends to gather experts working on various actually important aspects of Clifford algebras and to explore new connections between different research areas. We expect a fruitful exchange of new ideas and collaboration regarding the research development of this discipline among the participants.

We invite scientists and engineers working by means of the theory of integral equations to contribute to the session. You will find all information on the congress website:

http://icnpaa.com/index.php/icnpaa/2012

Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (The Morgan Kaufmann Series in Computer Graphics)

by Leo Dorst; Daniel Fontijne; Stephen Mann (download ebook from System Things)

Until recently, roughly all of the interactions in between objects in practical 3D worlds have been formed upon calculations achieved regulating linear algebra. Linear algebra relies heavily upon coordinates, however, that can have many geometric programming tasks really specific as good as complex-often the lot of bid is compulsory to move about even medium opening enhancements. Although linear algebra is an fit approach to mention low-level computations, it is not the befitting high-level denunciation for geometric programming.

Geometric Algebra for Computer Science presents the constrained pick to the stipulations of linear algebra. Geometric algebra, or GA, is the compact, time-effective, as good as performance-enhancing approach to paint the geometry of 3D objects in mechanism programs. In this book we will find an pass to GA that will give we the clever sense of the attribute to linear algebra as good as the stress for your work. You will sense how to operate GA to paint objects as good as perform geometric operations upon them. And we will proceed mastering proven techniques for creation GA an constituent partial of your applications in the approach that simplifies your formula but negligence it down.

Features

  • Explains GA as the healthy prolongation of linear algebra as good as conveys the stress for 3D programming of geometry in graphics, vision, as good as robotics.
  • Systematically explores the concepts as good as techniques that have been pass to representing facile objects as good as geometric operators regulating GA.
  • Covers in item the conformal model, the available approach to exercise 3D geometry regulating the 5D illustration space.
  • Presents in outcome approaches to creation GA an constituent partial of your programming.
  • Includes countless drills as good as programming exercises beneficial for both students as good as practitioners.
  • Companion web site includes links to GAViewer, the module that will concede we to correlate with many of the 3D total in the book, as good as Gaigen 2, the height for the exegetic programming exercises that interpretation any chapter.

Computer Algebra and Geometric Algebra with Applications: 6th International Workshop, IWMM 2004, Shanghai, China, May 19-21, 2004

Gerald Sommer, Hongbo Li, Peter J. Olver | Springer | 2011-08-02 | 458 pages | English | PDF

This book constitutes the thoroughly refereed joint post-proceedings of the 6th International Workshop on Mathematics Mechanization, IWMM 2004, held in Shanghai, China in May 2004 and the International Workshop on Geometric Invariance and Applications in Engineering, GIAE 2004, held in Xian, China in May 2004.

The 30 revised full papers presented were rigorously reviewed and selected from 65 presentations given at the two workshops. The papers are devoted to topics such as applications of computer algebra in celestial and engineering multibody systems, differential equations, computer vision, computer graphics, and the theory and applications of geometric algebra in geometric reasoning, robot vision, and computer graphics.

Source: eBookee

Oblique superposition of elliptically polarized lightwaves using geometric algebra: is energy-momentum conserved?

Michelle Wynne C. Sze, Quirino M. Sugon, Jr., and Daniel J. McNamara, “Oblique superposition of two elliptically polarized lightwaves using geometric algebra: is energy–momentum conserved?,” J. Opt. Soc. Am. A 27, 2468-2479 (2010). doi:10.1364/JOSAA.27.002468

Abstract

In this paper, we use Clifford (geometric) algebra Cl3,0 to verify if electromagnetic energy–momentum density is still conserved for oblique superposition of two elliptically polarized plane waves with the same frequency. We show that energy–momentum conservation is valid at any time only for the superposition of two counter-propagating elliptically polarized plane waves. We show that the time-average energy–momentum of the superposition of two circularly polarized waves with opposite handedness is conserved regardless of the propagation directions of the waves. And, we show that the resulting momentum density of the superposed waves generally has a vector component perpendicular to the momentum densities of the individual waves.

© 2010 Optical Society of America

History
Original Manuscript: July 29, 2010
Manuscript Accepted: September 18, 2010
Revised Manuscript: September 12, 2010
Published: October 22, 2010

Read more: Ateneo Physics News

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