4.1 Set of circumscribed circles

Definition 4.1.1.

Let $S\subset\mathbb{Z}^{2}$ . The set of centres of circumscribed circles of $S$ with radius $r$ , $c_{r}(S)$ is the set of all points $(x,y)$ such that the integer circle $S_{r}(x,y)$ is a circumscribed circle of the set $S$ .

Remark 4.1.2.

For $S=\emptyset$ , we have $c_{r}(S)=\mathbb{Z}^{2}$ .

Definition 4.1.3.

Let $S\subset\mathbb{Z}^{2}$ . The set of circumscribed centres of $S$ , $C(S)$ is the union of sets:

C(S)=\bigcup_{r\in\mathbb{Z}}c_{r}(S).

The set of circumscribed centres is the set of all points which are the centre of a circumscribed circle of $S$ . In other terms, this is the set of all points that are equidistant to all vertices of $S$ .

Example 4.1.4.

Let $p$ be an integer point. Then

C(\{p\})=\mathbb{Z}^{2}\setminus\{p\}.

Remark 4.1.5.

Recall that in Euclidean geometry, a line is the locus of points equidistant from two points. In integer geometry, the locus of points equidistant from $p_{1}$ and $p_{2}$ is given by

L=C(\{p_{1},p_{2}\})).

In this chapter, we study properties of $c_{r}(S)$ and $C(S)$ . In particular, we compare the sets $S$ and $c_{r}(c_{r}(S))$ .

Proposition 4.1.6.

Let $S$ be a set of integer points, and let $r$ be an integer. Then $S\subset c_{r}(c_{r}(S))$ .

Proof.

Let $x\in S$ . Then for all points $p\in c_{r}(S)$ we have that $\operatorname{l\ell}(x,p)=r$ . Thus $x\in c_{r}(c_{r}(S))$ . Hence, $S\subset c_{r}(c_{r}(S))$ . ∎

Corollary 4.1.7.

Let $S\subset\mathbb{Z}^{2}$ . Then $C(C(S))\subset S$ .

Proof.

This follows from the fact $c_{r}(S)\subset C(S)$ ∎

Remark 4.1.8.

Since $c_{r}(\emptyset)=\mathbb{Z}^{2}$ , if $S$ does not admit a circumscribed circle of radius $r$ , we have that $c_{r}(c_{r}(S))=\mathbb{Z}^{2}$ .

Definition 4.1.9.

Let $S$ be an integer set with $(0,0)\in S$ and let $m=\max\Lambda_{\mathbb{Q}}(S)$ when it exists. Otherwise, let $m=1$ . Then the set

m\left(c_{1}\left(c_{1}\left(\frac{1}{m}S\right)\right)\right)

is the principle circular-congruent set to $S$ we denote it by $\overline{c}(S)$ .

Let $\hat{S}$ be an integer set. Then if $\overline{c}(\hat{S})=\overline{c}(S)$ , we say that $\hat{S}$ and $S$ are circular-congruent.

Remark 4.1.10.

In the case when $S$ has only one element, $\Lambda_{\mathbb{Q}}(S)$ is not bounded.

Remark 4.1.11.

We define $\overline{c}$ with $\max\Lambda_{\mathbb{Q}}(S)$ so that it is defined for sets $S$ which only admit rational circumscribed circles.

Remark 4.1.12.

If $S$ and $\hat{S}$ are integer-congruent subsets of $\mathbb{Z}^{2}$ then $S$ and $\hat{S}$ are circular-congruent.

Definition 4.1.13.

We say that $S$ is maximal if $S=\overline{c}(S)$ .

Example 4.1.14.

A one-point set is maximal.

For a two-point set we have a direct generalisation of the classical definition of a line as the set of points at the same distance from a given pair of points.

Problem 2.

Is it true that $c(c(S))=\overline{c}(S)$ in case of $c(S)$ is non-empty? What about $c_{r}(c_{r}(S))?$

Definition 4.1.15.

Let $S$ be a set. We say that $S^{\prime}$ is the toric complement to $S$ if:

•

$S^{\prime}$ is tori-transparent at all points in $S$ ,
•

$S^{\prime}$ includes all points in $\mathbb{Z}^{2}$ that are not tori-equivalent to a point in $S$ .

Remark 4.1.16.

If $\max(\Lambda_{\mathbb{Q}}(S))=1$ then the toric complement to $S$ is $c_{1}(S)$ .

Theorem 4.1.17.

Let $S$ be a finite set, then $\overline{c}(S)=S$ .

Proof.

Without loss of generality we suppose that $\max(\Lambda(S))=1$ .

Since set of centres of $S$ is the toric complement to $S$ , we write

c_{1}(S)=\bigcap_{a\in S}\mathbb{Z}^{2}\setminus\cup[a]_{p}.

Hence

\overline{c}(S)=\bigcup_{a\in S}\cap[a]_{p}.

Since there is only one point in $\mathbb{Z}^{2}$ which intersects each integer torus at the same point, we find that $\overline{c}(S)=S.$ ∎

Example 4.1.18.

Let $S$ be a finite set. Then the complement $\overline{S}$ has no circumscribed circles and hence $cc(\overline{S})=\mathbb{Z}^{2}$ . Therefore $\overline{S}\neq cc(\overline{S})$ . Hence the statement of Problem 4.1.17 is not necessarily true for an infinite set $S$ .

For the next proposition, we need the following definition:

Definition 4.1.19.

An exhaustive filtration $\mathcal{F}$ is affine-invariant if for any $S\subset\mathbb{Z}^{2}$ and $A\in\operatorname{Aff}(2,\mathbb{Z})$ we have the following. If the density of $S$ exists, then the density of $A(S)$ exists and

\#_{\mathcal{F}}S=\#_{\mathcal{F}}A(S).

Proposition 4.1.20.

Let $F$ be an affine-invariant exhaustive filtration. Let $C$ be a unit circle and let $C^{\prime}\subset C$ such that the density of $C^{\prime}$ is not less than the density of intersection of two integer unit circles with respect to some exhaustive filtration. Then $c_{1}(c_{1}(C^{\prime}))=C$ .

Proof.

Let $p$ be the centre of $C$ . Then $p\in c_{1}(C)$ and hence $p\in c_{1}(C^{\prime})$ . Assume that any $q\neq p$ is also in $c_{1}(C^{\prime})$ . Therefore $c_{1}(C^{\prime})\subset(S_{1}(p)\cap S_{1}(q))$ . This is impossible due to our assumption. Hence $c_{1}(C^{\prime})=\{p\}$ . Therefore $c_{1}(c_{1}(C^{\prime}))=C$ . ∎

Remark 4.1.21.

Let $C$ be a unit circle and let $C^{\prime}\subset C$ such that the density $\#_{\mathcal{I}}C^{\prime}\geq\#_{\mathcal{I}}(S)$ Then $c_{1}(c_{1}(C^{\prime}))=C$ .

Problem 3.

Is $cc(C)=uu(C)$ , here $uu(C)$ is the intersection of circles centered at the the intersection of circles centered at $S$ .