Maths - Clifford / Geometric Algebra

Maths - Clifford / Geometric Algebra - Forum Discussion

On the web page I originally asked:

If a term such as e1^ e1 occurs then this can be replaced by a scalar value, this is because e1^ e1 = 1 or -1 depending on how it is defined. There seems to be a contradiction here because by the anti-commute rule then we should have e1^ e1 = - e1^ e1 which would only be true if e1=0?

This was kindly answered on the forum as follows:

By: Nobody/Anonymous - nobody
geometric algebra page
2006-02-13 23:32

e1^e1 = -e1^e1

implies that e1^e1=0 not that e1=0

since for scalars a,b

(a e1)^(b e1) = (ab) e1^e1 = -(ab) e1^e1

the a, b cancel, telling us that the above property has to do not with the magnitude of the vectors, but the directions

for instance, think about the product of two vectors

A prod B = |A||B| sin(angle between A,B)

when A,B are in the same *direction*, A prod B = 0

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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation.

Other Math Books

Commercial Software Shop

Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.

Mathmatica

Can you help?

Please send me any improvements to here. I would appreciate ideas to make the pages more useful including error correction, ideas for new pages, improvements to wording. It helps if you quote the full URL of the page.

Could anyone let me know of a good proof that a quaternion multiplication can be used to represent a rotation in 3 dimensions, I'm not looking for the shortest proof, but the most easily understood.

Terminology and Notation

Specific to this page here:

program

I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here:

http://sourceforge.net/projects/mjbworld/

This site may have errors. Don't use for critical systems.

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see also:	kinematics dynamics
Correspondence about this page	question on sci.math newsgroup discussion with Fred Lunnon
Book Shop - Further reading. Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.	Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling. New Foundations for Classical Mechanics (Fundamental Theories of Physics). This is very good on the geometric interpretation of this algebra. It has lots of insights into the mechanics of solid bodies. I still cant work out if the position, velocity, etc. of solid bodies can be represented by a 3D multivector or if 4 or 5D multivectors are required to represent translation and rotation. Other Math Books
Commercial Software Shop Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.	Mathmatica
Can you help? Please send me any improvements to here. I would appreciate ideas to make the pages more useful including error correction, ideas for new pages, improvements to wording. It helps if you quote the full URL of the page.	Could anyone let me know of a good proof that a quaternion multiplication can be used to represent a rotation in 3 dimensions, I'm not looking for the shortest proof, but the most easily understood.
Terminology and Notation Specific to this page here:
program I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here:	http://sourceforge.net/projects/mjbworld/