Rigid transformations in the continuous space of R² are well known as topologically and geometrically preserved operations. However, these properties are usually lost when we consider these transformations in the discrete space Z² due to the required discretization process. In this context, we study conditions and verifications allowing the preservation of the topology and geometry of discrete objects by arbitrary rigid transformations. These studies allow us to present a rigid transformation method on Z² to preserve these properties. The approach is based in particular on a polygonal representation of the discrete object, the rigid transformation is applied to the polygon then followed by a process of discretization to obtain a result in Z².