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:
We want to be able to represent 3D solid body movements (rotations and translations) in one operation.
Initially it would seem that multivectors based on 3D vectors would be ideal for this because such a multivector contains a 3D bivector (which could represent rotations) and a 3D vector (which could represent translations). However there are problems with this approach, one problem is that multivectors are not always invertible, whereas 3D isometry translations do always have an inverse.
There are subsets of multivectors that do always have an inverse (such as a * a†=1) but this restriction means that the vector part is no longer independent of the bivector. this means we have to go to higher dimensional multivectors to represent independent rotation and translation.
In order to explore this subject I have calculated the condition a * a†=1 for multivectors based on various dimensional vectors on these pages:
- Isometry properties of multivectors based on 2D vectors.
- Isometry properties of multivectors based on 3D vectors.
- Isometry properties of multivectors based on 4D vectors.
- Isometry properties of multivectors based on 5D vectors.