Maths - 3D Clifford Algebra - Arithmetic

By: d_pete (d_pete) - 2007-02-23 07:38
I've found your Euclidean Space pages most interesting, especially the pages about Clifford Algebra. 
 
You've asked for a little help with the multiplication tables for the outer product on the page https://www.euclideanspace.com/maths/algebra/clifford/d3/arithmetic/innerAndOuter.htm and elsewhere. 
 
Yes you're right, most of the terms are zero.  
 
However the first row and column where you multiply by the identity have all been set to zero when they shouldn't be. 
 
Quote "I think that a vector, outer multiplied by itself, will be zero? I cant find an identity to prove this ..." 
 
This follows from antisymmetry a^b = -b^a so a^a = -a^a and hence a^a = 0. Intuitively, if you think of sweeping one vector along the other to get the area then taking one vector along itself will not produce any area. 
 
Similarly you can inutuitively think of a^b^c as the volume from sweeping the area a^b along c. So (a^b)^a is zero since sweeping a^b along creates no volume. Alternatively since the outer product is associative and anti-symmetric a^b^a = -a^a^b = 0. 
 
On another page (I can't find it now) you have asked why (e2^e1)^e1 is e2 I think, while e1^e1=0 
but as the product is associative your assertion that it's e2 is wrong. 
 
On some of your multiplication tables you have for example e1^e31 = -e3 but this should be 0 
 
e1^e31 = e1^e3^e1 = -e1^e1^e3 = 0^e3 = 0 
 
Similarly you have e12^e31 = e23 but this should also be 0.  
 
e12^e31 = e1^e2^e3^e1 and similarly to above the e1^e1 gives 0 
 
In 3D any four vectors must be linearly dependant so a^b^c^d must be zero. 
 
Finally you have e1^e23 = 0 when it should be e123 = e1^e2^e3 
 
I suspect there may some mixing up with the Geometric product going on in places. 
 
Ian Bell's website has an explicit outer product multiplication table (amongst others), but he multiplies from the top row first so some of the signs will differ from yours.  
 
It's also quite hard on the eyes. 
 
http://www.iancgbell.clara.net/maths/geoalg1.htm 
 
I hope you find this useful and take it as a compliment to your website that I care enough to want to see things corrected. 
 
Thanks 
 
Pete 
 


By: Martin Baker (martinbakerProject Admin) - 2007-02-23 10:40
Hello Pete, 
 
> However the first row and column where you multiply by the identity 
> have all been set to zero when they shouldn't be. 
 
Thanks very much for letting me know about this; you've made me think (a good thing!) what I did, at some stage ,was to apply the following equations to all the entries in the multiplication table: 
a^b = (a*b - b*a)/2 = non-zero for antisymmetric terms 
a.b = (a*b + b*a)/2 = non-zero for symmetric terms 
a*b = a^b + a.b 
However these equations are valid when a and b are vectors, not scalars, so I'll correct that. I'll also try to find if there are more general versions of these equations. 
I'll also correct the other errors you pointed out in the table. 
 
> This follows from antisymmetry a^b = -b^a so a^a = -a^a and hence 
> a^a = 0.  
Thanks I'll add this, I'll also try to link the outer product back to Grassmann algebra. 
 
> I hope you find this useful and take it as a compliment to your website 
> that I care enough to want to see things corrected. 
 
I do and its good timing as well as I've just started to work through the Clifford Algebra pages to include algebras generated from different combinations of vector dimensions that square to -ve and zero as well as +ve. I'm also planning to change the bases for the 4D algebras to be a superset of the 3D case, so that it covers G(3,1,0) - for space-time, and G(3,0,1) - for dual quaternions. I'm hoping that this will underpin the pages on dual quaternions and Motors. 
 
So hopefully, with the changes that you have sent here, the pages should be improving. 
 
Martin


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flag flag flag flag flag flag 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.

 

flag flag flag flag flag flag 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.

 

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