I'm not really very happy with the method developed in this section to handle 3D isometries. It works but it uses the bivector for both rotation and displacement. I would be much happier if the displacement were represented by a vector. I think the above approach could cause problems if we start to use this terminology for physics.
We can use a 3D multivector to represent isometries by using this form:
multivector = scalar part of quaternion + vector representing translation + bivector part of quaternion.
but we would have to combine subsiquent transforms using the addition operation rather than the multipication operation. Also I'm not sure this would properly handle rotations which follow translations.
To do all the things that I would like I think we need to go to geometric algebras based on 5D vector spaces?