The Relationship between Minimum Zone Sphere and Minimum Circumscribed Sphere and Maximum Inscribed Sphere

Abstract

Among the four methods (minimum zone sphere, minimum circumscribed sphere, maximum inscribed sphere, and least square sphere), only the minimum zone sphere complies with ANSI and ISO standards and has the minimum sphericity error value. Evaluation of sphericity error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The minimum circumscribed sphere and the maximum inscribed sphere are all easily solved by iterative comparisons, so the relationship between the minimum zone sphere and the minimum circumscribed sphere, the maximum inscribed sphere is proposed to solve efficiently the minimum zone problem. The relationship is implemented and validated with the data available in the literature.

Method for sphericity error evaluation using geometry optimization searching algorithm

“…The core of sphericity evaluation methods (LSS, MIS, MCS and MZS) is to resolve the parameters of the centre of the two concentric sphere surfaces containing all measured points [1][2][3][4][5][6][7][8][9][10][11][12][13]. Upon this, the principle of the GOSA is as follows.…”

“…Experiment shows that the obtained sphericity error is smaller than the least square solution. Meng et al [11][12][13] thought the evaluation of sphericity error is formulated as a non-differentiable unconstrained optimization problem and hard to handle. The minimum circumscribed sphere and the maximum inscribed sphere are all easily solved by iterative comparisons, so the relationship between the minimum zone sphere, the minimum circumscribed sphere and the maximum inscribed sphere is proposed to efficiently solve the minimum zone problem.…”

A Novel Approach for High-Precision Evaluation of Sphericity Errors Using Computational Geometric Method and Differential Evolution Algorithm

“…The aforementioned criterion enables the acquisition of a unique minimum sphericity error, and it is also the arbitration method when there is inconsistency in the evaluation of the sphericity error. The minimum zone criterion is an unconstrained and non-differentiable optimization problem, which poses significant challenges in terms of finding a viable solution [11][12][13].…”