Recently, a sufficient condition, namely quasi-regularity, has been proposed for preserving the connectivity during the process of digitization of a continuous object whose boundary is not necessarily differentiable. Under this condition, a rigid motion scheme for digital objects of $\mathbb Z^2$ is proposed to guarantee that a well-composed object will remain well-composed, and its global geometry will be approximately preserved. In this paper, we are interested in polygons generated from digital objects and their rigid motions in $\mathbb Z^2$. For this, we introduce a notion of discrete regularity which is a restriction of quasi-regularity for polygons. This notion provides a simple geometric verification, based on the measure of lengths and angles, of quasi-regularity which is originally defined with morphological operators. Furthermore, we present a method for geometry-preserving rigid motions based on convex decomposition of polygons. This paper focuses on the implementation and on the reproduction of the method linking to an online demonstration. The way of using the C++ code source in other contexts is shown as well.