This paper studies continuous computational geometry of motions and develops a method for Computer Aided Geometric Design (CAGD) of motion interpolants. The approach uses a mapping of spatial kinematics to convert the problem of interpolating displacements to point interpolation in the space of the mapping. To facilitate the point interpolation, the previously non-oriented mapping space is made orientable. Methods are then developed for designing spline curves in the mapping space with tangent, curvature and torsion continuities. The results have application in computer animation of three dimensional objects used in computer graphics, computer vision and simulation of mechanical systems.