Maths - Clifford Algebra - Geometrical Interpretation

These pages explain how geometric algebra can be used to represent geometric objects. We start with vectors (quantities with direction and magnitude) we then go on to combine these to give planes and higher order objects. The various types of element are discussed on this page.

bivector

A vector can represent a line which starts at the origin and ends at some given point, a bivector can represent a plane, which in turn can represent a direction of rotation.

All of these quantities go through the origin, this is due to the linear nature of the algebra that we are working with, in many cases this is not a problem. For example, we often want to calculate rotations around the origin, if we need displacements from the origin it is often easiest to deal with these separately. It is possible to use geometric algebra to work with elements that do not go through the origin, this is explained on the pages that deal with projective and conformal space, but for now we will keep things simple and concentrate on elements that go through the origin.

Join

If we have two vectors then they define a plane (assuming the vectors are different from each other). note: for now all these quantities go through the origin. That is, there is only one plane which contains both vectors. This is similar to a union operation 'U' in set theory except that the plane contains not only the vectors A and B but also all the other vectors in that plane.

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

More about join on this page.

Meet

If we have two planes then they define a vector (assuming the planes are different from each other). That is along the line where the planes intersect so this is similar to the intersection operation in set theory.

Meet Example

Lets start with the simple case of the basis bivectors e3^e1 and e1^e2. The meet is e1 we can see that this is the vector common to both planes.

meet

Transforms

There are many types of translation that we can apply to our elements such as: rotation,scaling,translation,reflection, projection (parallel), projection (perpendicular) discussed in more detail on this page.

Perhaps the most useful is rotation, this is done using two geometric multiplications, 'sandwiching' the element being rotated.

We often need to transform points which, unlike the other types of elements discussed on this page, are displaced from the origin and then we want to rotate them around the origin.

 


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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.

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.

 

Terminology and Notation

Specific to this page here:

 

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