Computation of spatial displacements from geometric features

This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes and their combinations. Using an oriented projective three space with a Clifford Algebra, all three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimal number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines (each pair being non-planar) are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimal number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.