Circumscribed Circles in Integer Geometry

TL;DR

This paper explores the properties of circumscribed circles in integer geometry, focusing on conditions for their existence and the spectra of their radii for integer and rational circles.

Contribution

It introduces the concepts of integer and rational circumscribed circles and characterizes when finite integer sets admit such circles, including their radius spectra.

Findings

Conditions for finite integer sets to have integer circumscribed circles

Description of the spectra of radii for integer and rational circumscribed circles

Characterization of circumscribed circles in integer geometry

Abstract

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.