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Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

[+] Author and Article Information
Craig Shakarji

ASME Member, Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A.
craig.shakarji@nist.gov

Vijay Srinivasan

ASME Fellow, Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, U.S.A.
vijay.srinivasan@nist.gov

1Corresponding author.

ASME doi:10.1115/1.4039583 History: Received September 27, 2017; Revised February 22, 2018

Abstract

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

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