On calculating the finite centre of rotation for rigid planar motion

In this technical note, a simpler least squares derivation for calculating the angle of rotation and finite centre of rotation of a set of marker points undergoing rigid planar rotation and translation is shown. The major advantage of the approach, other than the simple derivation, is the automatic inclusion of the calculation of a scaling factor between the two point sets - the calculation of which was not obvious in previous approaches [Challis J. Estimation of the finite center of rotation in planar movements. Med Eng Phys 2001;23(3):227–33, Spoor C, Veldpaus F. Rigid body motion calculated from spatial coordinates of markers. J Biomech 1980;13:391–3]. The final numerical calculations are similar to those of [Challis J. Estimation of the finite center of rotation in planar movements. Med Eng Phys 2001;23(3):227–33] and are trivial to implement. A matlab routine for computing the two quantities, and the scaling factor, is included. We demonstrate the method on a clinical example using lateral radiographs of the lumbar spine.