A simple unified branch-and-bound algorithm for minimum zone circularity and sphericity errors

Abstract

This paper presents a simple branch-and-bound (B&B) algorithm to compute the minimum zone circularity/sphericity error, which is formulated as the problem of finding the centers of two concentric circles/spheres that contain given points between them and have minimal difference in their radii. For any square domain of the center, a lower bound on the minimal radius difference attainable over the domain is proposed, and this monotonically increases as the domain is divided into square subdomains. Along with the branching operation, a domain will become infinitely small and provide only a center of concentric circles/spheres such that its lower bound is equal to the radius difference. Using such a lower bound, therefore, the B&B algorithm guarantees the computing of the exact circularity/sphericity error and it is proven that the accuracy of the computed result can be easily specified by the termination tolerance of the algorithm as needed. In addition, the algorithm is very efficient and easy to implement.