Prerequisites
This page compares quaternion multiplication and orthogonal matrix multiplication as a means to represent rotation.
If you are not familiar with this subject you may like to look at the following pages first:
Description
The transformations that we want to model, such as rotation and translation, are reversible so we need a mathematical algebra that is also reversible, that is we need to always find the inverse, so:
if ,
multivector a translates b into c
then we need to be able to find the inverse a-1 which translates c into b
one condition which meets this requirement is:
a a†=scalar
as explained here.
So using the multiplication table here and multiplying out the terms of a a† gives:
e = +e*e+e1*e1+e2*e2+e3*e3+e12*e12+e31*e31+e23*e23+e123*e123
e1 = +e*e1+e1*e+e2*e12-e3*e31+e12*e2-e31*e3+e23*e123+e123*e23
e2 = +e*e2-e1*e12+e2*e+e3*e23-e12*e1+e31*e123+e23*e3+e123*e31
e3 = +e*e3+e1*e31-e2*e23+e3*e+e12*e123+e31*e1-e23*e2+e123*e12
e12 = -e*e12+e1*e2-e2*e1-e3*e123+e12*e-e31*e23+e23*e31+e123*e3
e31 = -e*e31-e1*e3-e2*e123+e3*e1+e12*e23+e31*e-e23*e12+e123*e2
e23 = -e*e23-e1*e123+e2*e3-e3*e2-e12*e31+e31*e12+e23*e+e123*e1
e123 = -e*e123-e1*e23-e2*e31-e3*e12+e12*e3+e31*e2+e23*e1+e123*e
canceling out positive and negative terms and adding equal terms (and drop the a. prefix sice all terms refer to the same multivector) gives:
scalar = e*e+e1*e1+e2*e2+e3*e3+e12*e12+e31*e31+e23*e23+e123*e123
0 = 2*e*e1+2*e2*e12-2*e3*e31+2*e23*e123
0 = 2*e*e2-2*e1*e12+2*e3*e23+2*e31*e123
0 = 2*e*e3+2*e1*e31-2*e2*e23+2*e12*e123
0 = 0
0 = 0
0 = 0
0 = 0
Luckily the last 4 equations cancel out completely and we can simplift the other equations as follows:
scalar = e2+e12+e22+e32+e122+e312+e232+e1232
0 = e*e1+e2*e12-e3*e31+e23*e123
0 = e*e2-e1*e12+e3*e23+e31*e123
0 = e*e3+e1*e31-e2*e23+e12*e123
One solution to these equations would be to let:
- e= -e123
- e23= e1
- e31= e2
- e12= e3
This gives:
scalar = 2*e2+2*e12+2*e22+2*e32
0 = e*e1+e2*e3-e3*e2-e1*e
0 = e*e2-e1*e3+e3*e1-e2*e
0 = e*e3+e1*e2-e2*e1-e3*e
so 3 more equations cancel out and we are left with:
scalar = 2*e2+2*e12+2*e22+2*e32
If we let scalar = 2 then e, e1, e2 and e3 have a unit length.
If turns out that this quantity (now consisting of 4 scalar values) has the same properties as a quaternion and so if we make it unit length as suggested it can represent a rotation transformation. However, if we want to represent an isometry (rotation and translation), we need to goto 4D multivectors as described here.
Pure Rotation
If all ex, ey, ez and exyz input terms set to zero gives:
| e = | a.e * b.e | 0 | 0 | 0 | - a.exy * b.exy | - a.ezx * b.ezx | - a.eyz * b.eyz | 0 |
| ex = | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| ey = | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| ez = | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| exy = | a.exy * b.e | 0 | 0 | 0 | + a.e * b.exy | + a.eyz * b.ezx | - a.ezx * b.eyz | 0 |
| ezx = | a.ezx * b.e | 0 | 0 | 0 | - a.eyz * b.exy | + a.e * b.ezx | + a.exy * b.eyz | 0 |
| eyz = | a.eyz * b.e | 0 | 0 | 0 | + a.ezx * b.exy | - a.exy * b.ezx | + a.e * b.eyz | 0 |
| exyz = | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
If we include exyz back in we get:
