Hyperbola minimum-zone evaluation and fitting based on geometry-optimised search algorithm

To enhance the accuracy of the evaluation of profile error and precisely fit a hyperbola at an arbitrary location in a plane in conjunction with its geometric characteristics and the minimum zone principle, a novel minimum-zone evaluation and fitting algorithm, called the geometry-optimised search algorithm (GOSA), is proposed. First, the least squares method is employed to fit the measured hyperbola and obtain its geometric parameters of the fitted hyperbola. Second, two focal points of the least squares hyperbola are chosen as the initial reference points, and four auxiliary points are arranged (as square vertices) around each reference point. Based on these auxiliary points, a series of auxiliary hyperbolas are constructed using the geometric characteristics of the hyperbola. Third, the regions containing all the measuring points are calculated by using the normal distance between the all measurement points and each auxiliary hyperbola. Finally, by comparing, determining, and changing the reference points, followed by constructing new auxiliary points and auxiliary hyperbolas, a minimum zone evaluation and fitting for the measured hyperbola profile are accomplished. The simulation results verify the validity and practicability of the algorithm.