## 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.