Circumscribed Circles in Integer Geometry

An integer affine transformation is an affine transformations that preserves the integer lattice \(\z ^2\). We denote the set of all integer affine transformations by \(\aff (2,\z )\). Similar to the Euclidean isometries, \(\aff (2,\z )\) contains integer translations, integer rotations and integer symmetries. They correspond to translations by integer vectors, multiplication by matrices in \(\SL (2,\z )\) and multiplication by matrices in \(\GL (2,\z ){\setminus }\SL (2,\z )\) respectively.

We say that two integer sets are integer congruent if there exists an integer affine transformation sending one set to another.

An angle in \(\r ^2\) with an integer point as its vertex is called an integer angle. An integer angle that contains an integer point other than its vertex on each of its sides is called a rational integer angle.

Let us recall some basic notions of integer geometry (see  [9] ). The integer length \(\il (AB)\) of a vector \(AB\) in \(\z ^2\) is defined as the number of lattice points that the vector passes through, minus one. Note that the integer length is given by the greatest common divisor of the differences of coordinates. The integer distance \(\id (A,B)\) between integer points \(A\) and \(B\) is the integer lengths of \(AB\). The integer distance \(\id (O,L)\) between an integer point \(O\) and an integer line \(L\) is the index of the sub-lattice generated by vectors \(OV\), where \(V\) runs through all integer points on the line \(L\).

The integer area \(\is (ABC)\) of a triangle \(ABC\) is the index of the sub-lattice generated by \(AB\) and \(AC\) in \(\z ^2\). In fact, the integer area is equal to the absolute value of the determinant \(\det (AB,AC)\), and therefore it is twice the Euclidean area of the triangle \(ABC\).

We define an integer circle with centre \(O\in \z ^2\) and radius \(r\in \z \), \(r>0\) as the locus of all points \(P\) such that \(\il (OP)=r.\)

Proposition 1.1.

The intersection of an integer line \(L\) with an integer circle is either empty, or an infinite periodic subset of integer points on \(L\), or two points.

The integer radial line of an integer circle \(C\) is an integer line passing through the centre of \(C\). An integer radial line of \(C\) intersects \(C\) in two points. An integer tangent line to an integer circle \(C\) of radius \(r\) with centre \(O\) is an integer line \(L\) such that \(\id (O,L)=r\).

Figure 1. An integer circle circumscribed about an integer quadrangle.

Figure  1 shows in bold those points of the integer unit circle \(S_0\) centred at the origin \(O\) whose coordinates do not exceed \(5\) in absolute value. The polygon in Figure  1Homeindex-autopage-1 is called a Farey starburst and is obtained by connecting these points by straight segments in the order of increasing argument. The vertices \(A,B,C,D\) belong to the integer circle \(S_0\), hence \(S_0\) is a circumscribed circle of the quadrangle \(ABCD\).

Let us discuss basic definitions of integer trigonometry introduced in  [7] , [6] (for the multi-dimensional trigonometry see  [4] ).

Note that any rational angle is integer congruent to exactly one integer arctangent. So the values of integer trigonometric functions form in fact a complete set of invariants of rational angles up to integer congruence.

The integer sine has a nice geometric definition:

\[ \isin \angle ABC =\frac {\is (ABC)}{\il (AB)\il (AC)} \]

which directly corresponds to the Euclidean formula for the area of a parallelogram in terms of the sine of its angle. The integer tangent is closely related to the geometry of numbers and their connections to continued fractions  [8] .

Proof.

By definition, \(\frac {p}{q}\in \LaQ (S)\) implies \(p\in \LaZ (q S)\), i.e. the set \(qS\) has an integer circumscribed circle of radius \(p\). Proposition  2.3 implies that \(p\) is a divisor of \(\id (qa,qb)=q\cdot \id (a,b)\) for any \(a,b\in S\). As \(p\) and \(q\) are co-prime, it follows that \(p\) is a divisor of \(\id (a,b)\).

This proposition implies that the rational spectrum is bounded.

Corollary 3.6.

Let \(S\) be an integer set, \(|S|\ge 2\). Then the rational spectrum \(\LaQ (S)\) of \(S\) and the set of numerators of irreducible fractions in \(\LaQ (S)\) are bounded.

Proof.

Proof.

Proof.

Let \(\frac {p}{q}\) be an irreducible fraction such that \(\max (\LaQ (S))=\frac {p}{q}\).

1. We will show that \(q\) is a divisor of \(\tau \): We know that \(S\) and hence \(\tau S\) is not a covering set of \(\calT _t\) for any prime \(t\not \in \{t_1,\ldots ,t_n\}\). For \(i=1,\dots ,n\), the set \(t_i S\) and hence \(\tau S\) is not a covering set of \(\calT _{t_i}\). In summary, the set \(\tau S\) is not a covering set of \(\calT _t\) for every prime \(t\), i.e. \(\tau S\) is tori-transparent. Theorem  2.9 implies \(1\in \LaZ (\tau S)\) and hence \(\frac {1}{\tau }\in \LaQ (S)\). Proposition  3.8 implies \(q\,|\,\tau \).

2. We will now show that \(q=\tau \): We have shown that \(q\) is a divisor of \(\tau \). Suppose that \(q\ne \tau \) then \(q\) is the product of some but not all of \(t_1,\dots ,t_n\). We can assume without loss of generality that \(t_1\) is not a divisor of \(q\). We know that \(S\) is a covering set of \(\calT _{t_1}\) and \(\gcd (t_1,q)=1\), hence \(q S\) is also a covering set of \(\calT _{t_1}\) and therefore not tori-transparent. Theorem  2.9 implies that \(\LaZ (qS)=\emptyset \). On the other hand, we know that \(\frac {p}{q}\in \LaQ (s)\), hence \(p\in \LaZ (q S)\) in contradiction to \(\LaZ (q S)=\emptyset \). Hence \(q=\tau \).

3. We will next show that

   \[\LaQ (S)\subset \left \{\frac {1}{c}\cdot \frac {p}{\tau }\,\bigg |\,c\in \z _{+}\right \}:\]

   Consider an irreducible fraction \(\frac {p'}{q'}\) in \(\LaQ (S)\). We know that the irreducible fraction \(\frac {p}{q}=\frac {p}{\tau }\) is the maximum of \(\LaQ (S)\). Proposition  3.8 implies that \(p'\,|\,p\) and  \(\tau \,|\,q'\), hence there exists \(c\in \z _{+}\) such that

   \[\frac {p'}{q'}=\frac {1}{c}\cdot \frac {p}{\tau }.\]

4. We will now show that

   \[\left \{\frac {1}{c}\cdot \frac {p}{\tau }\,\bigg |\,c\in \z _{+}\right \}\subset \LaQ (S):\]

   Let \(c\in \z _+\). We know that \(\frac {p}{\tau }\in \LaQ (S)\), hence \(p\in \LaZ (\tau S)\) and therefore \(\LaZ (\tau S)\ne \emptyset \). Theorem  2.9 implies that the set \(\tau S\) is tori-transparent. It follows that the set \(c(\tau S)\) is also tori-transparent. Theorem  2.9 implies that \(1\in \LaQ (c\tau S)\) and therefore \(\frac {1}{c}\in \LaQ (\tau S)\). We know that \(p,\frac {1}{c}\in \LaQ (\tau S)\), hence \(\frac {p}{c}\in \LaQ (\tau S)\) according to Proposition  3.7 . Therefore \(\frac {p}{c\tau }\in \LaQ (S)\).

5. Finally, we will show that the greatest common divisor of all integer distances between pairs of points in \(S\) is a multiple of \(p\): We know that \(\frac {p}{q}=\frac {p}{\tau }\in \LaQ (S)\), hence \(p\in \LaZ (\tau S)\) and therefore \(\tau S\) has a circumscribed circle of radius \(p\). It follows that the integer distance between any two points in \(\tau S\) is a multiple of \(p\). We know that \(\gcd (p,\tau )=\gcd (p,q)=1\), hence the integer distance between any two points in \(S\) is a multiple of \(p\).

Note that while Theorem  3.10 states that the greatest common divisor of all integer distances between pairs of points in \(S\) is a multiple of \(p\), it is not necessarily equal to \(p\) as can be seen in the following example:

To give a more precise description of circumscribed spectra, we will need the following definition:

Proof.

Proof.

An integer segment or triangle always admits a unit integer circumscribed circle.

Proof.

We obtain the following corollary:

Proof.

Consider \(A,B\in S\). Any integer circumscribed circle of \(S\) is in particular an integer circumscribed circle of the segment \(AB\), hence the integer length of the segment \(AB\) is divisible by \(r\).

Let us recall the Euclidean Extended Sine Rule: for a triangle \(ABC\) we have

\[ \frac {|AB|}{\sin \angle BCA }= \frac {|BC|}{\sin \angle CAB }= \frac {|CA|}{\sin \angle ABC }= 2R, \]

where \(R\) is the radius of the circumscribed circle.

As was shown in  [6] , the first two of these equalities hold in lattice geometry:

\[ \frac {\il (AB)}{\isin \angle BCA}= \frac {\il (BC)}{\isin \angle CAB}= \frac {\il (CA)}{\isin \angle ABC}. \]

Proposition  4.1 tells us that there is no natural generalisation for the last equality. Indeed, the circumscribed spectrum depends entirely on the integer length of the edges of the triangle and does not depend on the angles.

For instance consider two triangles, one with vertices \((0,0)\), \((1,0)\), \((0,1)\) and another with vertices \((0,0)\), \((1,2)\), \((2,1)\). For both triangles, all edges are of unit integer length. The sets of integer sines of the angles of these triangles are distinct, for the first triangle all integer sines are equal to \(1\) while for the second triangle all integer sines of the angles are equal to \(3\). Nevertheless the circumscribed spectra for both triangles coincide.

Proof.

Note that every empty integer strictly convex quadrilateral is integer congruent to the coordinate square \(S_1\) with vertices \((0,0)\), \((1,0)\), \((1,1)\) and \((0,1)\). The square \(S_1\) is a covering set of \(\calT _2\), hence it is not tori-transparent. Theorem  2.9 implies that \(S_1\) does not admit integer circumscribed circles of any radius.

However some quadrangles have integer circumscribed circles.

The situation is similar to the Euclidean geometry, where a quadrangle has a circumscribed circle if and only if its opposite angles add up to \(\pi \). The lattice version of this rule is as follows:

Proof.

Theorem  2.9 implies that a quadrangle admits an integer circumscribed circle if and only if its set of vertices \(V\) is tori-transparent, i.e. is not a covering set of any integer torus \(\calT _t\) for \(t\ge 2\). The set \(V\) cannot be a covering set of \(\calT _t\) for \(t>2\) since \(|V|=4<t^2=|\calT _t|\). Hence the set \(V\) is tori-transparent if and only if it is not a covering set of \(\calT _2\).

Remark 4.7.

The conditions for a quadrangle to admit a circumscribed circle can be stated in terms of the parity of the six integer distances between its pairs of vertices as follows: An integer quadrangle admits an integer circumscribed circle if and only if at least one of the integer distances between its vertices is even.

On the other hand, the existence of an integer circumscribed circle is not determined solely by the integer angles of the integer quadrangle. For example, the angles of the quadrangles with vertices \(A(0,0)\), \(B(0,1)\), \(C(1,1)\), \(D(1,0)\) and \(P(-1,0)\), \(Q(-1,1)\), \(R(0,1)\), \(S(1,0)\) are congruent to each other, however the latter one admits a circumscribed circle, for example one centred at the origin \(O(0,0)\), while the former one does not.

        

In fact, the argument used in the proof of Proposition  4.6 holds for all \(n\)-gons with \(n\le 8\):

Proposition 4.8.

An integer \(n\)-gon with \(n\le 8\) has an integer circumscribed circle if and only if the set of its vertices is not a covering set of \(\calT _2\).

In general, the following statement holds:

Proposition 4.9.

An integer \(n\)-gon admits an integer circumscribed circle if and only if its vertices are not a covering set of \(\calT _t\) for every \(t\le \sqrt n\).

Acknowledgments:
Grant support for R.S.: This work was supported by a Vacation Bursary of the London Mathematical Society.

We would like to thank the anonymous referee for their helpful suggestions.