Description
Motors - Abbreviation of "moment and vector" (to represent sums of spins)
M = RT
where:
- R = rotor
- T = translator
- M = motor
Rotor
A rotation can be represented by the following expression as shown on this page.
R a R†
where:
- R is known as a Rotor, since it is the geometric product of two pure vectors (multivectors containing only grade 1 factors) then the result is a multivector with only real and bivector (grade 0 and 2) parts. The real part comes from the inner product and the bivector part comes from the outer product.
- a is the vector being rotated
- † is the reversal operation.
Translator
We are looking for a similar operator to the Rotor which uses the same form to translate a vector by a given displacement:
a+d = T a T†
where:
- T is the Translator, it is a multivector.
- a is the input vector
- d is the vector displacement
- † is the reversal operation.
Motor
The expression:
M a M †
Should translate and rotate a vector in one expression.
To do this we combine the Rotor and Translator we get:
M = RT
Cliffords definitions
A vector is a quantity with magnitude and direction (e.g. linear velocity or moment).
A rotor is a quantity with magnitude, direction, and position (e.g. rotational velocity about a fixed axis or force along line of action).
A motor is the sum of two or more rotors, which can be represented as a wrench or twist about a certain screw. For example, the sum of arbitrary system of forces is, in general, not a force but a combination of force and moment.
Representing Motors
In 2 Dimensions
2D motors can be represented by either of the following algebras which are equivalent 'isomorphic' to each other:
Dual Complex Numbers
A dual whose elements are complex numbers or a complex numbers whose elements are duals, (its the same thing). Dual complex numbers are explained on this page.
A Geometric Algebra G 0,1,1.
That is a Geometric Algebra generated from a 2 dimensional vector space one of which squares to -ve and the other dimension which squares to zero. There is an example of how to use this on this page.
In 3 dimensions
3D motors can be represented by either of the following algebras which are equivalent 'isomorphic' to each other:
Dual Quaternions
A dual whose elements are quaternions or a quaternion whose elements are duals, (its the same thing). Dual Quaternions are explained on this page.
A Geometric Algebra G+3,0,1.
That is a Geometric Algebra generated from a 4 dimensional vector space with 3 dimensions which square to +ve and one dimension which squares to zero. We then take the subalgebra made up of the even grade elements: scalar + 6 dimensional bivector + pseudoscalar. There is an example of how to use this on this page.
Projective Space and Conformal Space G 3,1,0 and G 4,1,0
Projective Space and Conformal Space allow us to embed our 3D euclidean space into a higher dimensional space. Projective space adds one dimension and conformal space adds two dimensions, these new dimensions represent zero and infinity, this allows us to represent translation as a rotation around infinity.
