2.4 Integer Circles

Let $O$ be a point in $\mathbb{Z}^{2}$, and $r$ a positive integer number. An integer circle $\mathbb{Z}S_{r}(O)$ with centre $O$ and radius $r$ is the locus of all points $P$ such that

Consider an integer circle $S$ of radius $p$ centred at $O$. Then for $q\in\mathbb{Z}_{+}$ the set $\frac{1}{q}S$ is a rational circle.

Its radius is $p/q$ and its centre is $\frac{1}{q}O$.

Let $X\subset\mathbb{Z}^{2}$ be a set of integer points and $S_{r}(a,b)=S$ be an integer circle. Then $S$ is a circumscribed circle of $X$ if and only if for each $(x,y)\in X$ we have $(x,y)\in S$.

Let $S\subset\mathbb{Z}^{2}$. The integer circumscribed spectrum of $S$, $\Lambda_{\mathbb{Z}}(S)\subset\mathbb{Z}_{>0}$ is the set of radii of circumscribed integer circles of $S$.

Let $S\subset\mathbb{Z}^{2}$. The rational circumscribed spectrum of $S$, $\Lambda_{\mathbb{Q}}(S)\subset\mathbb{Q}_{>0}$ is the set of radii of circumscribed rational circles of $S$.

Let $m\in\mathbb{Z}$. The integer torus $\mod m$ is

The projection $\pi_{m}:\mathbb{Z}^{2}\to\mathcal{T}_{m}$ is given by $(x,y)\to(x\mod m,y\mod m)$.
We say that two integer points $v_{1}$ and $v_{2}$ in $\mathbb{Z}^{2}$ are equivalent mod $m$ if $\pi_{m}(v_{1})=\pi_{n}(v_{2})$, denoted by $v_{1}\equiv v_{2}\mod m$.

Let $L=\{l_{1},l_{2},\ldots,l_{n}\}$ be a set of lines, and $S_{r}(a,b)=S$ be an integer circle. We say that $S$ is an inscribed circle of $L$ if $\operatorname{ld}((a,b),l_{i})=r$ for all $l_{i}$, $i\in\{1,\ldots,n\}$.

Let $P$ be a polygon with its edges contained in lines $l_{1},l_{2},\ldots,l_{n}$, and $S_{r}(a,b)=S$ be an integer circle. We say that $S$ is an inscribed circle of $P$ if $S$ is an inscribed circle of $L$ and $(a,b)$ is on the interior of $P$.