We present in this report a novel kind of geometrical transformations, which we have named polyrigid. Within their framework, it is possible to define local rigid deformations in a given number of simple regions, while simultanously guaranteeing the smoothness and invertibility of the global transformation. Entirely parametric, this new type of tool is highly suitable for inference, and it is successfully applied to the non-rigid registration of histological slices. These general transformations are a nice alternative to classical B-Spline transformations (which do not guaranty invertibility). In future work, other applications will be considered, for instance in 3D registration.