Maths - Clifford Algebra - Join

We are looking for a similar operation to union in set theory or the 'OR' operation in logic.

However such an operation would be non-linear and unstable. For example, if combining two vectors, the union would be these two vectors but not the vectors inbetween. How would we represent such a combination? We therefore change the definition slightly so that the join consists of the plane containing both vectors (both the vectors and the plane go through the origin).

Join Example

Lets start with the simple case of the basis vectors e1 and e2. The join is e1^e2 this is a plane and can't be simplified further.

join

So how can we generalise this?

Duality of Meet and Join

We can calculate the meet if we know the join or visa-versa. However the dual depends on the blade being considered, this means we can't get an expression for the complete multivector.

(AMB)* = B* U A*

where:

Since the meet and join are both hard to calculate, this relationship does not necessarily help to calculate the join of a given blade.

Algorithm for calculating Join

It is easyest to calculate meet and join together, see meet page.

 

 

 

 

 

 


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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.

This book has some explanation of an algorithm for calculating meet and join (section 21.7) but I don't claim to understand it .

flag flag flag flag flag flag Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry. This book stresses the Geometry in Geometric Algebra, although it is still very mathematically orientated. Programmers using this book will need to have a lot of mathematical knowledge. Its good to have a Geometric Algebra book aimed at computer scientists rather than physicists. There is more information about this book here.

 

Terminology and Notation

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