Rigid motions (i.e. transformations based on translations and rotations) are simple, yet important, transformations in image processing. In Euclidean spaces, namely R^n, they are both topology and geometry preserving. Unfortunately, these properties are generally lost in Z^n. In particular, when applying a rigid motion on a digital object, one generally alters its structure but also the global shape of its boundary. These alterations are mainly caused by (re)digitization during the transformation process. In this specific context of digitization, some solutions for the handling of topological issues were proposed in Z^2 and/or Z^3. In this article, we also focus on geometric issues, in the case of Z^2. More precisely, we propose a rigid motion algorithmic scheme that relies on an initial polygonization and a final digitization step. The intermediate modeling of a digital object of Z^2 as a piecewise affine object of R^2 allows us to avoid the geometric alterations generally induced by standard pointwise rigid motions. The crucial step is then related to the final (re)digitization of the polygon back to Z^2. To tackle this issue, we propose a new notion of quasi-regularity that provides sufficient conditions to be fulfilled by an object for guaranteeing both topology and geometry preservation, in particular the preservation of the convex/concave parts of its boundary.