Study’s kinematic mapping – a tool for motion design

Via Study’s kinematic mapping S the 6-parametric Lie group SE(3) of direct Euclidean displacements can be identified with a certain hyperquadric M6 in 7-dimensional real projective space. The mapping has nice geometric properties; for instance one parametric rotation groups are represented by straight lines on M6, coordinate transformations in Euclidean 3-space are represented by special automorphisms of M6. With the help of S Euclidean kinematics can be considered as a point-geometry in the sense of Felix Klein’s Erlangen program. We give an application in the field of motion design: The problem of constructing a motion interpolating a sequence of given positions can be solved by constructing an appropriate curve interpolating the corresponding points on Study’s quadric.