A geometric theory of form, profile, and orientation tolerances

Abstract

This article develops a geometric theory which unifies the formulation and computation of form (straightness, flatness, cylindricity, and circularity), profile and orientation tolerances stipulated in ANSI Y14.5M standard. The theory rests on an important observation that a toleranced feature exhibits a symmetry subgroup G 0 under the action of the Euclidean group, SE(3). Thus, the configuration space of a toleranced (or a symmetric) feature can be identified with the homogeneous space SE(3)/ G 0 of the Euclidean group. Geometric properties of SE(3)/ G 0, especially its exponential coordinates carried over from that of SE(3), are analyzed. We show that all cases of form, profile and orientation tolerances can be formulated as a minimization or constrained minimization problem on the space SE(3)/ G 0, with G 0 being the symmetry subgroup of the underlying feature. We develop a simple geometric algorithm, called the Symmetric Minimum Zone (SMZ) algorithm, to unify the computation of form, profile, and orientation tolerances. Finally, we use numerical simulation results comparing the performances of the SMZ algorithm against the best known algorithms in the literature.