The relationship between minimum zone sphere and minimum circumscribed sphere and maximum inscribed sphere

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.