Source: https://computingengineering.asmedigitalcollection.asme.org/article.aspx?articleid=2675175
Timestamp: 2019-04-19 19:05:45+00:00

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Manuscript received September 27, 2017; final manuscript received February 22, 2018; published online June 12, 2018. Assoc. Editor: Jitesh H. Panchal. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Approved for public release; distribution is unlimited.
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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A simple example of: (a) specification during design of a part and (b) establishment of a system of primary and secondary datum planes on a manufactured instance of the part. The secondary datum plane B is required to be perpendicular to the primary datum plane A.
An example set of points and concentric circles for Lemmas 1a, b. The value of the objective function can be improved if the inscribed circle CI (or the circumscribed circle CC) does not contact a data point.
Construction for the proof of Theorem 1a. Growing the inscribing circle from C1 to C2 causes the distance from a data point pi to C2 to be less than the distance from pi to C1.
Construction for the proof of Theorem 1b. Shrinking the circumscribing circle from C1 to C2 causes the distance from a data point pi to C2 to be less than the distance from pi to C1.
Construction of an example case with only two contact points for an optimal circle. Shifting the center from the origin results in a greater value for the objective function for the data points shown.
Construction of an example case with only two contact points for an optimal sphere. Moving the center from the origin results in a greater objective function for the data points shown.
A one-parameter family of inscribing circles having a common chord. The search space for an optimal solution could be reduced by searching along the perpendicular bisector of a common chord.
Shakarji CM, Srinivasan V. Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums. ASME. J. Comput. Inf. Sci. Eng. 2018;18(3):031008-031008-8. doi:10.1115/1.4039583.

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