This paper studies continuous computational geometry of motion and dev
elops a method for Computer Aided Geometric Design (CAGD) of motion in
terpolants. The approach uses a mapping of spatial kinematics to conve
rt the problem of interpolating displacements to that of interpolating
points in the space of the mapping. To facilitate the point interpola
tion, the previously unorientable mapping space is made orientable. Me
thods are then developed for designing spline curves in the mapping sp
ace with tangent, curvature and torsion continuities. The results have
application in computer animation of three-dimensional objects used i
n computer graphics, computer vision and simulation of mechanical syst
ems.