Maths - 3D Clifford / Geometric Algebra

Geometric Algebra of 3 Dimensional Vectors G 3,0,0

For 3 dimensions we can do the same thing, this time the results can be generated by 3 basis vectors, e1, e2and e3. Now multiplying out gives the following combinations: 3d clifford
basis grade
e scalar
e1,e2, e3 vector
e1^e2, e1^e3 , e2^e3 bivector
e1^e2^e3 tri-vector

In this case the bivector is 3D the same number of dimensions as the vector, the bivector has the same properties as the vector e1cept that a•a is a negative number for bivectors but it is a positive number for vectors (we will discuss multiplication rules later).

In this case the tri-vector has 1 dimension as for the scalar.

So in this case the multivector is made up of 8 scalar values.

This can lead to quantities that are quite tedious to write out therefore, in some cases, I will use the following abbreviations to make them easier to read and write:

  base value numerical value
full
shortened
unit scalar 1 e
unit length base vectors e1 e1
e2 e2
e3 e3
unit length base bivectors e1^ e2 e12 e12
e3^ e1 e31 e31
e2^ e3 e23 e23
unit length base tri-vector e1^ e2^ e3 e123 e123

In the above table the base values are all unit length, they are like the 'i' operator in complex numbers or quaternions, these base values can then be multiplied by the numerical value which gives the actual value. Therefore any 3D multivector can be specified with these 8 numerical values which are just real (scalar) numbers.

In the case of 3D vectors the base values e1, e2and e3 could be thought of as unit vectors in the x,y and z direction. We could have used the notation ex, ey and ez but this might not fit so well when we work with higher dimensions. For this geometrical interpretation e1, e2and e3 could be any 3, mutually perpendicular, unit length vectors. In fact they don't have to be mutually perpendicular provided they are not all in one plane.

There are 3 combinations of outer products which cant be further reduced: e1^ e2, e3^ e1 and e2^ e3 , we could have chosen other combinations as base values but these are consistent and make the multiplication tables, below, more symmetrical. We can think of these as unit length base bivectors.

The order, starting at the scalar at the top down to the tri-vector at the bottom, also keeps the tables as symmetrical as possible, if we start from the top we are adding e1, e2and e3 if we start from the bottom we are removing e1, e2, e3 and this is important as explained in further pages.

So, using this notation, the general case of a 3D multivector is written:

e + e1 e1 + e2 e2 + e3 e3 + e12 e12 + e31 e31 + e23 e23 + e123 e123

This is similar to a complex number a + i b where:

The multivector just has more parts to it.

So how do we choose which bivector bases that we use? For instance, why don't we use e13 instead of e31? We could have done this but, by using the values that we have then we get a correspondence with cross multiplication as shown here and also the multiplication table is symmetrical. Also all the e123 terms on the bottom left to top right diagonal are positive.

The following table is described and derived on this page. I have colour coded it as follows:

a*b
b.e b.e1 b.e2 b.e3 b.e12 b.e31 b.e23 b.e123
a.e 1 e1 e2 e3 e12 e31 e23 e123
a.e1 e1 1 e12 -e31 e2 -e3 e123 e23
a.e2 e2 -e12 1 e23 -e1 e123 e3 e31
a.e3 e3 e31 -e23 1 e123 e1 -e2 e12
a.e12 e12 -e2 e1 e123 -1 e23 -e31 -e3
a.e31 e31 e3 e123 -e1 -e23 -1 e12 -e2
a.e23 e23 e123 -e3 e2 e31 -e12 -1 -e1
a.e123 e123 e23 e31 e12 -e3 -e2 -e1 -1

I have colour coded the elements to indicate which products commute and anticommute as follows:

commutes
anticommutes

You can check this by reflecting the cell you want to check in the leading diagonal (top left to bottom right) this is equivalent to swapping rows and columns, i.e. reversing the order of multiplication If the cell and the reflected cell are equal then it commute, if the cell is minus the reflected cell then it anticommutes.

If we changed any of the bivector bases, say by using e13 instead of e31, then we would loose the symmetry and also the correspondence with cross multiplication.

 

Cayley Table and Graph

If the basis vectors square to +ve then the Cayley table and graph are as follows:

Cayley Table
a*b
b.e b.e1 b.e2 b.e3 b.e12 b.e31 b.e23 b.e123
a.e 1 e1 e2 e3 e12 e31 e23 e123
a.e1 e1 1 e12 -e31 e2 -e3 e123 e23
a.e2 e2 -e12 1 e23 -e1 e123 e3 e31
a.e3 e3 e31 -e23 1 e123 e1 -e2 e12
a.e12 e12 -e2 e1 e123 -1 e23 -e31 -e3
a.e31 e31 e3 e123 -e1 -e23 -1 e12 -e2
a.e23 e23 e123 -e3 e2 e31 -e12 -1 -e1
a.e123 e123 e23 e31 e12 -e3 -e2 -e1 -1
Cayley Graph quaternion cayley digraph

There are 3 generators since e1, e2 and e3 are the generators.

Even subalgebra

If we take the even subalgebra (scalar and bivector) then the Cayley table and graph are as follows:

Cayley Table Cayley Graph
a*b
b.e b.e12 b.e31 b.e23
a.e 1 e12 e31 e23
a.e12 e12 -1 e23 -e31
a.e31 e31 -e23 -1 e12
a.e23 e23 e31 -e12 -1
caley 3d digraph even

We only need two generators for the even sub-algebra, in this case we have chosen e12 and e23 but any 2 of the 3 bivectors.

Relationship to Cayley-Dickson Algebras

The relationship between the 8D Clifford algebras generated from 3D vectors and the 8D algebras (such as octonions) generated by the Cayley-Dickson doubling process is explained on this page.

  8D Clifford algebras octonions
division algebra (inverse exists) not always yes
commutes no no
associative yes no

Representing 3D solid bodies

So a geometric algebra, based on three dimensional vectors, has multivectors which have 8 scalar values (1 scalar, 3 vector, 3 bivector, 1 tri-vector). This would appear ideal for representing 3D solid bodies which have 6 degrees of freedom (3 for translation, 3 for orientation). Unfortunately however, the arithmetic of 3d multivectors can't be used to represent isometries (transforms representing translation and rotation). I guess that we could invent new operations to represent isometries with 3d multivectors but this would loose the properties described on this page. In order to represent 3D solid bodies using geometric algebra we need to use 4D multivectors as will be described on further pages. This means that we need 16 scalar variables to represent isometries, the same as a 4x4 matrix, which can also represent isometries and most people find matrices easier to understand. A number of people have told me that geometric algebra can be more powerful so I am investigation this on these pages. Perhaps a taste of the power of geometric algebra in this area can be found when we include the time variable as the scalar variable, this fits very well as the space we experience has 3 dimensions of distance and one dimension of time, using this we can derive the constants including energy and momentum, in one equation, as shown here.

Addition Rules

We just add corresponding terms. When adding blades of different grade then we cant reduce it further and we leave the + in the number.

For example:

3 + 4 e1 + 5 e23

added to

5 + 4 e1 + 3 e23

gives

8 + 8 e1 + 8 e23

Multiplication Rules

The main type of multiplication is geometric multiplication as follows,

Given a multivector with numerical values: a.e, a.e1, a.e2, a.e3, a.e12, a.e31, a.e23 and a.e123 multiplied by a second multivector with numerical values: b.e, b.e1, b.e2, b.e3, b.e12, b.e31, b.e23 and b.e123 then what are the resulting numerical values. Multiplying out each term gives the following result:

e = a.e * b.e + a.e1 * b.e1 + a.e2 * b.e2 + a.e3 * b.e3 - a.e12 * b.e12 - a.e31 * b.e31 - a.e23 * b.e23 - a.e123 * b.e123
e1 = a.e1 * b.e + a.e * b.e1 - a.e12 * b.e2 + a.e31 * b.e3 + a.e2 * b.e12 - a.e3 * b.e31 - a.e123 * b.e23 - a.e23 * b.e123
e2 = a.e2 * b.e + a.e12 * b.e1 + a.e * b.e2 - a.e23 * b.e3 - a.e1 * b.e12 - a.e123 * b.e31 + a.e3 * b.e23 - a.e31 * b.e123
e3 = a.e3 * b.e - a.e31 * b.e1 + a.e23 * b.e2 + a.e * b.e3 - a.e123 * b.e12 + a.e1 * b.e31 - a.e2 * b.e23 - a.e12 * b.e123
e12 = a.e12 * b.e + a.e2 * b.e1 - a.e1 * b.e2 + a.e123 * b.e3 + a.e * b.e12 + a.e23 * b.e31 - a.e31 * b.e23 + a.e3 * b.e123
e31 = a.e31 * b.e - a.e3 * b.e1 + a.e123 * b.e2 + a.e1 * b.e3 - a.e23 * b.e12 + a.e * b.e31 + a.e12 * b.e23 + a.e2 * b.e123
e23 = a.e23 * b.e + a.e123 * b.e1 + a.e3 * b.e2 - a.e2 * b.e3 + a.e31 * b.e12 - a.e12 * b.e31 + a.e * b.e23 + a.e1 * b.e123
e123 = a.e123 * b.e + a.e23 * b.e1 + a.e31 * b.e2 + a.e12 * b.e3 + a.e3 * b.e12 + a.e2 * b.e31 + a.e1 * b.e23 + a.e * b.e123

This result is derived here. It also explains the other types of multiplication such as inner and outer products.

Representing Transforms

One way to use multivectors is to represent transforms, to do this we need to show how to use it to map one point (vector) onto another point (vector), this table shows different ways to

Translation type possible representation how to transform point
translation vector p + t
orthogonal (rotation)

quaternion

3*3 matrix

q p q'

[m]v

isometries (rotation + translation) multivector A p A -1
linear (rotation, translation, scale (different for each dimension), reflection, shear, etc.) 4*4 matrix [m]v

The various types of transformation are described on this page.

So in the case of multivectors we are proposing to translate the vector pin into the vector pout by using the equation:

pout = A pin A -1

where:

Not every multivector has an inverse so, in order for this to work, we need to work with a subset of A that A -1 always exists (and solid body movement will always have an inverse). One condition that will ensure that an inverse always has an inverse is:

A A†= 1

where:

This is similar to unit length quaternions, where q conj(q) = 1 or orthogonal matrices where [m][m]t = 1

Another requirement is that if p is a vector then A p A -1 will also be a vector.

Are these two requirements related? Can we use this identity to relate them:

(A B)† = B† A†


Further Reading

You may be interested in other means to represent orientation and rotational quantities such as:

Or you may be interested in how these quantities are used to simulate physical objects:


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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.

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.

 

Other Math Books

Terminology and Notation

Specific to this page here:

 

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