Maths - 4D Rotations and Translations using multivectors

A 4D rotation is represented by the product of two 4D vectors (a e1 + b e2 + c e3 + d e4)(e e1 + f e2 + g e3 + h e4) which gives one scalar term and six bivector terms. However if we then go on to combine that with another rotation, that will be equivalent to 4 reflections, this means we may get an e1234 (quadvector term).

In general in 'n' dimensions we need to use the complete even subalgebra to represent any combination of rotations.

Note that this is not the same as octonons, as we explained here the algebras generated by the Cayley-Dickson doubling process diverge from the Clifford algebras in 4D and above.

Representing 3D transforms

To represent isometry transforms in 3D we need 3 or 4 scalar values (quaternion or 3D vector) to represent rotations and we need 3 scalar values (a 3D vector) to represent translation. A multivector based on 3D vectors therefore has enough values to represent 3D isometries , for instance (e12,e31,e23) might hold the rotation value and (e1,e2,e3) hold the translation value, but does the algebra work out? Can we do useful operations on the multivector using such a notation? I cant find a way to represent a combined translation and rotation with these.

We need to use a Geometric Algebra based on a 4 dimentional vector space G 3,0,1 to do what we want.

3D Rotation and Translation Using G 3,0,1 (Dual Quaternion)

This can be used to represent the group of spatial displacements in three dimensions SE(3) which can be used to represent a combination of 3D rotation and displacement. If you would prefer to do this using Dual Quaternion Notation see this page.

The G 3,0,1 or Dual Quaternion algebra can model the movement of a solid object in 3D, which can rotate and translate without changing shape, that is distances and angles between points on the object are preseved. In mathematical terms it can be used to represent the group of spatial displacements SE(3). The advantage of this is that we can represent a combined rotation with displacement operation with one multiplication operation.

There are 8 elements, the 4 quaternion elements (e, e12, e31 and e23) and their duals (e1234, e43, e42 and e41). This gives G 3,0,1 algebra represented by an 8x8 multiplication table as shown here:

a*b
b.e b.e12 b.e31 b.e23 b.e41 b.e42 b.e43 b.e1234
a.e e e12 e31 e23 e41 e42 e43 e1234
a.e12 e12 -1 e23 -e31 -e42 e41 -e1234 e43
a.e31 e31 -e23 -1 e12 e43 -e1234 -e41 e42
a.e23 e23 e31 -e12 -1 -e1234 -e43 e42 e41
a.e41 e41 e42 -e43 e1234 0 0 0 0
a.e42 e42 -e41 -e1234 e43 0 0 0 0
a.e43 e43 -e1234 e41 -e42 0 0 0 0
a.e1234 e1234 e43 e42 e41 0 0 0 0

Representing Transforms

The advantage of this is that we can represent a combined rotation with displacement operation with one multiplication operation.

We will do the transform using the same equation as we used for quaternion rotation:

P2=M * P1 * M'

where:

Pure Displacement (no rotation)

Let

The transform can be done by:

Pout = M * Pin * M †
(e + p2) = (e + h)(e + p1)(e + h)
= (e + h + p1)(e + h)
since (x e41 + y e42 + z e43)²=0 we can ignore these terms
= (e + h + h + p1)
= (e + 2 h + p1)

So this adds the dispacement (2*half) to the original point which is what we want.

If the rotation is 0 degrees, the rotation part is represented by:
(e + 0 e41 + 0 e42 + 0 e43), in other words e.

Example

So if we are initially at point (x=3, y=4, z=5) this will be represented by the multivector:

P1 = e + 3 e41 + 4 e42 + 5 e43

A pure displacement will be represented by the multivector:

(e + x/2 e41 + y/2 e42 + z/2 e43)

We now want to displace the initial point by (x=4, y=2, z=6) this will be represented by the multivector:

q = (e + 2 e41 + 1 e42 + 3 e43)

pure translation diagram

So to combine these, to give the resulting position, we use P2 = q * P1 * q' to give:

p2 = (e + 2 e41 + 1 e42 + 3 e43)*(e + 3 e41 + 4 e42 + 5 e43)* (e + 2 e41 + 1 e42 + 3 e43)

multiplying out the first two terms using the above multiplication table we get:

P2 = (e + 5 e41 + 5 e42 + 8 e43)*(e + 2 e41 + 1 e42 + 3 e43)
P2 = (e + 7 e41 + 6 e42 + 11 e43)

Pure Rotation (no displacement)

If the displacement is zero then e41 = e42 = e43 = 0 and the rotation is represented by the normalised quaternion as explained on this page.

pure rotation

Example

Applying a rotation of point (3,4,5) by 180° around the x axis is given by:

P2 = (0 e+ e12)*(e + 3 e41 + 4 e42 + 5 e43)*(0 e - e12)
P2 = ( e12 - 3e42 -5e1234 + 4e41)*( - e12)
P2 = e - 3e41 - 4e42 + 5e43

Combined Displacement and Rotation (displace then rotation)

Example

Starting from the previous position: (e + 3 e41 + 4 e42 + 5 e43)

and both displace by (x=4, y=2, z=6) and applying a rotation of 180° around the x axis represented by: (e12)

Therefore:

Q = (0 e + e12) (e + 2 e41 + 1 e42 + 3 e43)
Q = (e12 - 2 e41 -3 e42 + 1 e43)

So applying the transform gives:

P2 = (e12 - 2 e41 -3 e42 + 1 e43)*(e + 3 e41 + 4 e42 + 5 e43)*(e12 - 2 e41 -3 e42 + 1 e43)
P2 = (e12 -5 e41 -8 e42 + 5 e43)*(-e12 + 2 e1234 -3 e42 + 1 e43)
P2 = (e + 7e41 -6 e42 -11 e43)

combined rotation and translation

Rotation about a point

In the above we are rotating about the origin. This is sometimes what we want to do, for instance, if we are observing the world from the frame of reference of a car and we assume it to be at the origin then, when it turns, the other cars will appear to rotate round it. If we want a car to rotate about its own centre then we can just multiply the complex number part (e and e12) and ignore the e and ie part.

If we want to rotate about an arbitrary point then we first translate this point to the origin, then rotate about the origin, then translate the origin back to the point (as discussed on this page).

about a point

Combining Transforms

Pure Rotation

M = r

Pure Translation

M = e + ½t

Rotation then Translation

M = (e + ½t)*r
= r + ½t r

Translation then Rotation

Q = r*(e + ½t)
= r + ½r t

Rotation about a point

Q = (e + ½c)*r*(e - ½c)
= (e + ½c)*(e - ½r c)
= r - ½r c + ½c r

Inverse of Transform

To invert the multivector 'r' we use its reversal 'r†', to invert the translation 't' we use '-t'. With combined transforms we must also reverse the order that they are combined (don't forget that order is important for translations and their corresponding multivector).

The inverse of common transforms are:

  M M-1
Pure Rotation r r†
Pure Translation 1 + ε½t 1 - ε½t
Rotation then Translation r + ε½t r r† - ε½r† t
Translation then Rotation r + ε½r t r† - ε½t r†
Rotation about a point 'c' r - ε½r c + ε½c r r† + ε½r† c - ε½c r†

Equivalence to Dual Quaternion

The dimensions are related as follows:

G+ 3,0,1
dual quaternion
e 1
e12 i
e31 j
e23 k
e41
e42
e43
e1234

Use for Physics

The reason for taking this approach is to combine linear and rotational movement into a single transform and to automatically handle the interaction between them. It would be good to be able to use this algebra to work with physics equations. For instance, imagine we have a set of forces acting on a solid body, can we determine the linear and rotational movement by using dual quaternions?

G+3,0,1 represents both the linear and rotational movement as bivectors which are associated with planes, or quantities associated with them like rotations, we would expect a linier displacement to be represented by a vector. I think this issue could be a problem if we try to formulate the physics equations in terms of dual quaternions.

However linier movement can be represented as a rotation... If we rotate around a point at infinity! One way to handle this type of situation would be to use projective space as explained on this page. Is it possible to use this to do what we want? I would appreciate any help.


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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.

cover Geometric Computing for Perception Action Systems: Concepts, Algorithms, and Scientific Applications (Hardcover). This is the only book I have so far come across that has a reasonable explanation of 'motors' and why it is useful to use 4D Geometric algebra to represent kinematics of solid bodies (in chapter 2). The book is quite a slim volume considering that it covers both fundamental concepts and practical applications. Therefore I think you will need to have a good understanding of Geometric Algebra before starting on this book.

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