Projective View on Motion Groups I: Kinematics and Relativity

The paper provides a consistent study on the projective construction of low-dimensional motion groups starting with SO ( 3 ) and then gradually extending to the Galilean and Lorentzian settings. In the case of spatial rotations one simply needs to consider R P 3 with an additional group structure inherited from quaternion multiplication, which allows for associating particular types of curves in E 3 with rigid body kinematics, based on the corresponding Maurer–Cartan form. A similar construction in complex projective space yields the relativistic version of the above approach, while a dual extension leads respectively to the study of nonhomogeneous isometries. The text provides also plenty of examples as well as a brief discussion on possible further generalizations.