The mathematical group structure of the set of displacements

Abstract

With the aim of representing physical rigid body motions, this article deals with the Euclidean displacement operators. Remaining as elementary as possible, an intrinsic mathematical formalism is presented with the use of affine geometry together with Lie's theory of groups. We start by defining a differential operator associated with the velocity field. Then, thanks to the Lie acceptance of the exponential function, we derive the formal Lie expression of finite displacements which are shown to be equivalent to affine direct orthonormal transformations. Lie sub-algebras of screw velocity fields lead to the description of the displacement sub-groups. The geometric modelling of small displacements is also evoked. This paper unifies elements previously published in [2] and [5].