Circumscribed Circles in Integer Geometry

Integer geometry on a plane deals with objects whose vertices are points in ${\mathbb{Z}}^2$. The congruence relation is provided by all affine transformations preserving the lattice ${\mathbb{Z}}^2$. In this paper we study circumscribed circles in integer geometry. We introduce the notions of integer and rational circumscribed circles of integer sets. We determine the conditions for a finite integer set to admit an integer circumscribed circle and describe the spectra of radii for integer and rational circumscribed circles.