We study patterns that appear in discrete circles with integer center and radius. As the radius goes to infinity, the patterns get closer to digital straight segments: the notion of tangent words (described in Monteil DGCI 2011) allows to grasp their shape. Unexpectedly, some tangent convex words do not appear infinitely often due to deep arithmetical reasons related to an underlying Pell-Fermat equation. The aim of this paper is to provide a complete characterization of the patterns that appear in integer discrete circles for infinitely many radii.