Chebyshev fitting of complex surfaces for precision metrology

The form qualities of precision components are essential for their functionalities. The Peak-to-Valley parameters are widely adopted to assess the form accuracies of optical components. The commonly used least squares method is prone to over-estimation, thus the Chebyshev fitting should in turn be implemented. In this paper the original minimax optimisation problem is converted into an unconstrained differentiable minimisation problem by the exponential penalty functions. The fitting accuracy and numerical stability are balanced by employing an active-set strategy and adjusting the configuration parameters adaptively. Finally some benchmark data sets are applied to demonstrate the validity and efficiency of this method.