Tolerance envelopes of parametric planar part models

We present a framework for modeling parametric variation in planar parts and for efficiently computing approximations of their tolerance envelopes. Part features are specified by algebraic equations defining their position and shape as a function of parameters whose nominal values vary along tolerance intervals. Their tolerance envelopes model perfect form Least and Most Material Conditions (LMC/MMC). We derive geometric properties of the tolerance envelopes and describe efficient algorithms for computing first-order linear approximations with successive accuracy. We show that the tolerance envelope of a parametric arc-line polygonal part with features has segments and can be computed in time, where is the maximum number of non-zero partial feature functions derivatives evaluated at nominal parameter values. Our implementation shows that the algorithms are practical on part models with tens of parameters.