Tolerance-Maps Applied to the Straightness and Orientation of an Axis

Among the least developed capabilities in well-developed mathematical models for geometric tolerances are the representation of tolerances on form, orientation, and of Rule {\#}1 in the Standards, i.e. the coupling between form and allowable variations for either size or position of a feature. This paper uses Tolerance-Maps¹ (T-Maps¹) to describe these aspects of geometric tolerances for the straightness and orientation of an axis within its tolerance-zone on position. A Tolerance-Map is a hypothetical point-space, the size and shape of which reflect all variational possibilities for a target feature; for an axis, it is constructed in four-dimensional space. The Tolerance-Map for straightness is modeled with a geometrically similar, but smaller-sized, four-dimensional shape to the 4D shape for position; it is a subset within the T-Map for position. Another internal subset describes the displacement possibilities for the subset T-Maps that limits form. The T-Map for orientation and position together is formed most reliably by truncating the T-Map for position alone.