Least-square curve and surface localization for shape conformance checking

Verification of shape conformance for free-form curves and surfaces is comm only achieved by minimizing the sum of square deviations between measured p oints and a nominal curve/surface, thereby solving an optimal parameter est imation (OPE) problem. Finding the optimal rigid body (ORB) transformation between the measured points and nominal surface, an important step in the O PE problem, traditionally has involved iteratively solving a nonlinear opti mization problem in six variables. This paper demonstrates that the optimiz ation problem in six variables may be reduced to solving four degree-two im plicit equations in four variables, which can be regarded as an eigenvalue problem. This results in considerable savings in the number of computations . A thorough analysis of the savings in computations and several examples a re presented.