Ostrovsky-Berman, Y.Joskowicz, L.Gershon Elber and Nicholas Patrikalakis and Pere Brunet2016-02-172016-02-1720043-905673-55-X1811-7783https://doi.org/10.2312/sm.20041384We present a framework for the systematic study of parametric variation in planar mechanical parts and for ef ciently computing approximations of their tolerance envelopes. Part features are speci ed by explicit functions de ning 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). Tolerance envelopes are useful in many design tasks such as quantifying functional errors, identifying unexpected part collisions, and determining device assemblability. We derive geometric properties of the tolerance envelopes and describe four ef cient algorithms for computing rst-order linear approximations with increasing accuracy. Our experimental results on three realistic examples show that the implemented algorithms produce better results in terms of accuracy and running time than the commonly used Monte Carlo method.I.3.3 [Computational Geometry and Object Modeling]Curvesurfacesolidand object representationsJ.6 [Computeraided design (CAD)]Computer Aided Tolerancing10.2312/sm.20041384135-143