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Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

Published

Author(s)

Craig M. Shakarji, Vijay Srinivasan

Abstract

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of input points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the input points are in general position: (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.
Proceedings Title
Proceedings of the ASME 2017 International Mechanical Engineering Congress & Exposition
Conference Dates
November 3-9, 2017
Conference Location
Tampa, FL
Conference Title
The ASME 2017 International Mechanical Engineering Congress & Exposition

Keywords

constrained least-squares, coordinate metrology, datum, dimensional metrology, fitting, geometric dimensioning and tolerancing, least-squares, optimization, parallel planes, planar datum, plane, standards

Citation

Shakarji, C. and Srinivasan, V. (2017), Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums, Proceedings of the ASME 2017 International Mechanical Engineering Congress & Exposition, Tampa, FL, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=924372 (Accessed February 27, 2024)
Created November 9, 2017, Updated April 9, 2018