Computational Investigations for a New, Constrained Least-Squares Datum Definition for Circles, Cylinders, and Spheres

Abstract

For engineering drawings and CAD definitions, the problem of a suitable datum definition for datum features of circles, spheres, and cylinders has been sought by standards writers over decades. The maximum-inscribed and minimum-circumscribed definitions that have often been used have known problems relating to stability in many common, industrial cases. Examples of these problem cases include cylindrical datum features having an hourglass shape, barrel shape, or the shape of a tapered shaft and circular or spherical datum features that are dimpled. For this cause, many resort to using a least-squares fit whose diameter is scaled to be just inside (or just outside) the datum feature. However, we show this shifted least-squares solution has its own drawbacks. This paper investigates a new datum definition based on a constrained least-squares criterion. The use of this definition for datum planes has already elegantly solved the problem of providing a full contact solution when that solution is stable, while providing a balanced, stable solution in the case of rocker conditions. With that success as motivation, we now investigate using this definition for circles, spheres, and cylinders. We demonstrate that the constrained least-squares is an excellent choice for several known problematic cases. This datum definition maintains stability in cases where the maximum-inscribed fits are not unique and thus not stable. Yet they also maintain close adherence to the maximum-inscribed solution when such solutions are stable. We also show that the constrained least-squares solution has clear benefits over the shifted least squares solution. This is the first computational investigation into the behavior of the constrained least-squares as a possible datum definition for these features. While not being fully comprehensive, these initial findings indicate that the constrained least-squares appears to be a safe and advantageous datum definition choice and provide substantial optimism that results in future investigated cases will be pleasing as well.