We present a new curve skeleton model designed for surface modeling and processing. This skeleton is defined as the geometrical integration of a piecewise harmonic parameterization defined over a disk-cylinder surface decomposition. This decomposition is computed using a progressive Region Graph reduction based on both geometric and topological criteria which can be iteratively optimized to improve region boundaries. The skeleton has an analytical form with regularity inherited from the surface one. Such a form offers well-defined surface-skeleton and skeleton-surface projections. The resulting skeleton satisfies quality criteria which are relevant for skeleton-based modeling and processing. We propose applications that benefit from our skeleton model, including local thickness editing, inset surface creation for shell mapping, as well as a new mid-scale feature preserving smoothing.