When we are transforming something we use the 'sandwich' product as follows:
Pout = M Pin M-1
where:
- M = a multivector which defines the transform
- Pin = an object to be transformed, for example, a point.
- M-1 = the multiplicative inverse
- Pout = the result of the transform
The multiplicative inverse may not exist for all multivectors and can be hard to calculate, for algebras based on 3D (and less) vectors then the multiplicative inverse is equivalent to the reverse (described on this page). Other words M-1 = M† can only be guaranteed in 3D or less.
Pout = M Pin M†
where:
- M† = the reversal of the transform, that is we reverse the order of basis vectors in each term separately.
On this page we will look at the algebra (as opposed to the geometric interpretation) of this product in the hope of finding the most efficient way to implement it and to see if it will give us a better intuitive understanding.
The Sandwich Product
One way to look at the product:
M Pin M-1
Is that it is equal to:
Pout * M M-1
Which is just 'Pout' (the result we want) because a quaternion times its inverse is unity (M M-1 = 1).
So by just reversing the order of the multiplication we change Pin into Pout. Now if M and Pin have common factors, for instance if,
M = e12
Pin = e2
then reversing the order will change the sign, so,
Pout = - e2
But for independent terms, for instance if,
M = e12
Pin = e3
then reversing the order will not effect the sign, so,
Pout = e3
In this case 'e3' is the axis of rotation so its not effected by the rotation. For other values of M we can start to see that the component along the axis of rotation will not be effected by the transform but the component in the plane of rotation will be rotated.
Using a single element
If a and b are single elements then the product of the form a b a† will produce either +b or -b, for example,
e12 e3 e21
gives
e3
We can easily calculate this by merging the suffixes together:
e12321
then applying the rules on this page.
If we do this for different elements we can start to see a pattern emerging:
| number of terms in a | number of terms in b | number of common terms | sign | example |
|---|---|---|---|---|
| 1 | 1 | 0 | -ve | e1 e2 e1 = -e2 |
| 1 | 1 | 1 | +ve | e1 e1 e1 = -e1 |
| 2 | 1 | 0 | +ve | e12 e3 e21 = e3 |
| 2 | 1 | 1 | -ve | e12 e1 e21 = -e1 |
| 2 | 2 | 0 | +ve | e12 e34 e21 = e34 |
| 2 | 2 | 1 | -ve | e12 e13 e21 = -e13 |
| 2 | 2 | 2 | +ve | e12 e12 e21 = e12 |
| 3 | 1 | 0 | -ve | e123 e4 e321 = -e4 |
| 3 | 1 | 1 | +ve | e123 e1 e321 = e1 |
| 3 | 2 | 0 | +ve | e123 e45 e321 = e45 |
| 3 | 2 | 1 | -ve | e123 e14 e321 = -e14 |
| 3 | 2 | 2 | +ve | e123 e12 e321 = e12 |
| 4 | 1 | 0 | +ve | e1234 e5 e4321 = e5 |
| 4 | 1 | 1 | -ve | e1234 e1 e4321 = -e1 |
So we can compress this down to:
| number of terms in a | number of terms in b | number of common terms | sign |
|---|---|---|---|
| even | even | even | +ve |
| even | even | odd | -ve |
| even | odd | even | +ve |
| even | odd | odd | -ve |
| odd | even | even | +ve |
| odd | even | odd | -ve |
| odd | odd | even | -ve |
| odd | odd | odd | +ve |
so if:
- a = true if a = odd
- b = true if a = odd
- c = true number of common terms =odd
then the expression for the sign is
(a & b) ^ c
where
- result = true represents -ve and false represents +ve
- & = logical and
- ^ = exclusive or
Using Multivectors (multiple elements)
Instead of 'a' being a single element it may now be a sum such as ax+ay so the sandwich product will be:
(ax + ay) b (ax† + ay†)
multiplying out all the terms gives:
ax b ax† + ay b ax† + ax b ay† + ay b ay†
So now we have terms which have different leading and trailing terms, so this transform can change b into a different basis. but can it change it to a different grade? If so under what circumstances?
Transforms in Different Dimensions
As we have seen, the results depend on the dimension of the vectors that the algebra is based on. For more details see the appropriate page for each dimension:
