Kinematic interpretation of the study Quadric’s ambient space

Abstract

It is well known that real points of the Study quadric (sliced along a 3-dimensional generator space) correspond to displacements of the Euclidean 3-space. But we still lack of a kinematic meaning for the points of the ambient 7-dimensional projective space P7. This paper gives one possible interpretation in terms of displacements of the Euclidean 4-space. From this point of view we also discuss the extended inverse kinematic map and motions corresponding to straight lines in P7. Moreover we present an application of this interpretation in the context of interactive motion design.