Systems and methods for preventing interference between two physical tooth models in a physical dental arch model includes digitally representing the surfaces of each of the two physical tooth models by a mesh of points in three dimensions, interpolating the meshes of points to produce interpolated surfaces to represent the boundaries of the two physical tooth models, wherein the interpolated surfaces representing the boundaries of the two physical tooth models intersect at least at one point to form an overlapping portion. The method also includes specifying a straight line running through the overlapping portion and intersecting the two interpolated surfaces representing the boundaries of the two physical tooth models and calculating the length of the straight line in the overlapping portion to quantify the interference between the two physical tooth models.