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

Abstract

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of 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 set of points (i.e., 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.