# Maths - Motors

## 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.

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.

 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.