Evaluating the Form Errors of spheres, cylinders and cones using the primal-dual interior point method Xiangchao Zhang ∗1,3 , Xiangqian Jiang 3 , Alistair B Forbes 2 , Hoang D Minh 2 , and Paul J Scott 3 1 Shanghai Ultra-Precision Optical Manufacturing Engineering Centre, Fudan University, China 2 National Physical Laboratory, UK 3 Centre for Precision Technologies, University of Huddersfield, UK December 23, 2013 Abstract In precision metrology, the form errors between the measured data and reference nominal surfaces are usually evaluated in four approaches: least squares elements (LSE), minimum zone elements (MZE), maxi- mum inscribed elements (MIE) and minimum circumscribed elements (MCE). The calculation of MZE, MIE and MCE is not smoothly dif- ferentiable, thus very difficult to be solved. In this paper a unified method is presented to evaluate the form errors of spheres, cylinders and cones in the sense of MZE, MIE and MCE. The primal-dual inte- rior point method is adopted to solve this nonlinearly constrained optimisation problem. The solution is recursively updated by arc search until the Karush-Kuhn-Tucker conditions are satisfied. This method is computationally efficient and its global convergence can be guaranteed. Some benchmark data are employed to demonstrate the * Corresponding author. Room 405, Building of Advanced Materials, Fudan Univer- sity, 2005 Songhu Road, Shanghai, 200438, China. Tel.: +86-21-51630347, Fax: +86-21- 65641344, Email: zxchao@fudan.edu.cn 1