In the present paper, we investigate discretization of curves based on polynomials in the 2-dimensional space.
Under some assumptions, we propose an arithmetic characterization of thin and connected discrete approximations
of such curves. In fact, we reach usual discretization models, that is, <i>GIQ, OBQ </i>and <i>BBQ</i> but with a
generic arithmetic definition.
In the present paper, we introduce an arithmetical definition of discrete circles with a non-constant thickness and we exhibit different classes of them depending on the arithmetical discrete lines. On the one hand, it results in the characterization of regular discrete circles with integer parameters as well as J. Bresenham's circles. As far as we know, it is the first arithmetical definition of the latter one. On the other hand, we introduce new discrete circles, actually the thinnest ones for the usual discrete connectedness relations.