A new mathematical model for representing geometric tolerances is applied to rectangular faces, is extended to show its sensitivity to the precedence (ordering) of datum reference frames, and is adapted to include material modifiers. The model is compatible with the ASME/ISO Standards for geometric tolerances. Central to the new model is a Tolerance-Map, a hypothetical volume of points that corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, position, form, and orientation. Every Tolerance-Map is a convex set. This model is one part of a bi-level model that we are developing for geometric tolerances. The new model makes stackup relations apparent in an assembly, and these can be used to allocate size and orientational tolerances; the same relations also can be used to identify sensitivities for these tolerances. All stackup relations can be met for 100% interchangeability or for a specified probability. Methods are introduced whereby designers can identify trade-offs and optimize the allocation of tolerances. Examples are presented that illustrate important features of the new model.
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