Quaternionic approach to equiform kinematics and line-elements of Euclidean 4-space and 3-space

We extend the quaternionic kinematic mapping of Euclidean displacements of Euclidean 4-space E4 to the group of equiform transformations S(4). As a consequence the equiform motions of basic elements (points, oriented lines, oriented planes, oriented hyperplanes) of E4 can be written compactly in terms of 2×2 quaternionic matrices. This representation is extended to oriented line-elements of E4 and to instantaneous screws of S(4), for which a classification (incl. corresponding normal forms) is given. Based on this preparatory work we study the relation between instantaneous equiform motions and the geometry of line-elements (path normal-elements, path tangent-elements) in E4. Finally, we show that the line-elements of projective 3-space can be mapped bijectively on the Segre variety Σ3,2.