Abstract: Local fairing techniques are extensively used in the geometry processing of curves and surfaces. They also play an important role in the multiresolution shape editing and synthesis applications. However, due to the inter-dependency of the vertices after applying the current fairing techniques, their inverses are not local. Finding a local fairing operation with local inverse provides a well-defined relationship between the smooth vertices and the initial vertices. This paper introduces a new fairing operation for curves and surfaces that is smoothing and local but with a local inverse. In the curve domain, we find a class of banded smoothing matrices with banded inverses. Then, using the geometric interpretation of the corresponding local operation, this class is extended to surfaces. We discuss the advantages of using this new fairing operation in different applications. Also, the resulting operation is used to find novel subdivision schemes with well-defined reverse subdivisions.