A new mathematical model is presented for the tolerances of planar surfaces. The model is compatible with the ASME Standard for geometric tolerances. Central to the new model is a Tolerance-Map, a hypothetical volume of points which corresponds to all possible locations and variations of a segment of a plane which can arise from tolerances on size, form, and orientation. Every Tolerance-Map®2 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. Stackup relations are developed for parts where the centers of faces are offset laterally. 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.