Analyzing either high-frequency shape detail or any other 2D fields (scalar or vector) embedded over a 3D geometry is a complex task, since detaching the detail from the overall shape can be tricky. An alternative approach is to move to the 2D space, resolving shape reasoning to easier image processing techniques. In this paper we propose a novel framework for the analysis of 2D information distributed over 3D geometry, based on a locally smooth parametrization technique that allows us to treat local 3D data in terms of image content. The proposed approach has been implemented as a sketch-based system that allows to design with a few gestures a set of (possibly overlapping) parameterizations of rectangular portions of the surface. We demonstrate that, due to the locality of the parametrization, the distortion is under an acceptable threshold, while discontinuities can be avoided since the parametrized geometry is always homeomorphic to a disk. We show the effectiveness of the proposed technique to solve specific Cultural Heritage (CH) tasks: the analysis of chisel marks over the surface of a unfinished sculpture and the local comparison of multiple photographs mapped over the surface of an artwork. For this very difficult task, we believe that our framework and the corresponding tool are the first steps toward a computer-based shape reasoning system, able to support CH scholars with a medium they are more used to.