Conformal geometric algebra by extended Vahlen matrices

The classical Vahlen matrix representation of conformal transformations on R(n) is directly related to the versor representation of conformal geometric algebra (CGA) using R(n+1;1). This paper spells out the relationship, which enriches both fields with insights and techniques. We extend the Vahlen matrices to include the representation of blades in CGA, and then use a decomposition in terms of eigenlines to derive Chasles’ theorem for representation of Euclidean rigid body motions. This naturally leads to the logarithm of a Vahlen matrix of such a motion. We also derive the table of commutation relationships between the basic even conformal transformations (translation, rotation, uniform scaling and transversion), in which the rather involved translation-transversion result may be new.