The paper develops a simple and effective method for adaptive smoothing tangential direction fields defined on piecewise smooth surfaces approximated by triangle meshes. The method consists of simultaneous and iterative updating each vertex direction by a weighted sum of the directions at the neighboring vertices. The weights themselves depend on directions: if the direction at a given vertex differs strongly from the direction at a neighboring vertex, then the weight associated with that neighboring direction is chosen small. Choosing such nonlinear weights allows us to preserve important direction field discontinuities during the smoothing process.Applications of the developed smoothing technique to curvature extrema detection and non-photorealistic rendering with curvature lines are given.