Abstract: Manufacture errors and deformation of a radio telescope or radar antenna reflector have a direct effect on the system performance. To precisely figure out the actual surface error of a reflector, a best paraboloid fitting for the deformed reflector is necessary, Based on the space motion principle of a rigid body, a novel algorithm on paraboloid fitting is developed, which introduces 7 generalized degrees of freedom (DOF) defined as the 6 rigid body DOFs and the focal length of the paraboloid. The orientation and vertex position as well as the best focal length of the required best-fit paraboloid are explicitly computed by the least square procedure. Traditional fitting strategies based on quadric surface eigenvalues extraction and on normal deviation minimization with small deformation are reviewed and afterwards compared with the provided new method with examples. The application demonstrates the stability and accuracy of the proposed algorithm. Moreover, a simple error estimation method based on iteration concept is also included.

Key words: Coordinate transformation, Degrees of freedom, Errors, Least squares, Paraboloid surface fitting, Rigid body displacement