The finite displacements vector's method: an application to the scoliotic spine

This paper presents a vectorial method to directly obtain the components of the screw displacement between two positions of a body in a three-dimensional space (position of the helical axis of motion, rotation around this axis and translation along it). This method can be applied either to the case of a bone, moving with respect to the reference frame, or to the case of the relative motion of a joint; it gives exact formulae even if the displacements are finite; it generalizes the results (already published) obtained for finite displacements in the plane. The involved computation is easy, and the use of this method deals with only a small magnification of experimental errors. The technique of a screw displacement is applied to the vertebral segments of a scoliolic spine. The necessary data is taken from a couple of in-vivo X-rays. The goals of this study are: first, to describe the shape of the spine at each step of its evolution and second, to quantify the evolution in time of any segment of the spine between two states.