Fitting and error evaluation for rotating paraboloid in arbitrary position using geometric iterative optimization algorithm

Abstract

To realize a more accurate fitting and error evaluation for a rotating paraboloid in an arbitrary position, a new fitting and error evaluation method based on the geometrical characteristics of the paraboloid is presented for rotating paraboloid in arbitrary positions. The proposed method is named the ‘geometric iterative optimisation algorithm’. First, the initial reference vertex, focus and initial profile error of the measured rotating paraboloid are obtained based on coordinate transformations and the least-squares method, and a reference paraboloid is established by using the reference vertex and focus. Second, two regular hexahedrons are collocated by taking the reference vertex and focus as datum points, and the initial profile error as the side length. Eight auxiliary vertexes and focuses (that is, the vertexes of two regular hexahedrons) are obtained. Using the auxiliary vertexes and focuses, a series of auxiliary paraboloids defined by the auxiliary vertexes and focuses are established. Third, reference profile error and auxiliary profile errors of the measured rotating paraboloid are calculated by considering the reference paraboloid and the auxiliary paraboloids as ideal paraboloids. Fourth, the reference points are changed or the side lengths of the hexahedron are decreased by comparing the the reference profile error and the auxiliary profile errors. The minimum zone fitting and evaluation for the paraboloid profile are realized by repeating the above process. The principle and steps involved when using the proposed method to solve the rotating paraboloid profile error are described in detail, and mathematical formulas are presented. The simulation results show that this method can obtain not only accurate results but it is also stable.