2.3 Integer Angles ‣ Chapter 2 Notions and Definitions ‣ Density of Intersecting Integer Circles and Bisection of Integer Angles

Definition 2.3.1.

An angle in integer geometry is rational if it is integer congruent to an angle $ABC$ with $A,B,C\in\mathbb{Z}^{n}$ .

Definition 2.3.2.

Let $\triangle ABC$ be an integer triangle, that is, let $A,B,C$ be integer points. Then the integer area, $\text{IS}(\triangle ABC)$ is the index of the sublattice generated by the vectors $AB$ and $AC$ .

Remark 2.3.3.

This is the number of copies of $AB$ and $AC$ needed to generate the lattice $\mathbb{Z}^{2}$ .

Example 2.3.4.

Let $ABC$ be an integer triangle with $A=(0,0),B=(3,2)$ and $C=(2,5)$ .

Here, the integer area is given by

\text{IS}(\triangle ABC)=\begin{Vmatrix}3&2\\ 2&5\end{Vmatrix}=|3\cdot 5\ -\ 2\cdot 2|=|15-4|=11.

Definition 2.3.5.

Let $AOB$ be a rational angle. Then the integer sine of $AOB$ is given by the formula

\operatorname{lsin}(AOB)=\frac{\text{IS}(\triangle AOB)}{\operatorname{l\ell}(% A,O)\operatorname{l\ell}(O,B)}.

Proposition 2.3.6.

Let $\alpha$ be a rational angle. Then $\operatorname{lsin}(\alpha)$ is equal to the index of the angle $\alpha$ .

Definition 2.3.7.

Let $\alpha=AOB$ be an integer angle. The sail of $\alpha$ is the boundary of the convex hull of all points included in $\alpha$ other than $O$ .

Definition 2.3.8.

Let $\alpha$ be an integer angle with sail $A$ a broken line with vertices $A_{i}$ . The lattice length sine sequence (LLS sequence for short) $(a_{i})$ of $\alpha$ is defined by the rule

a_{2k}=\operatorname{l\ell}(A_{k},A_{k+1});

a_{2k+1}=\operatorname{lsin}(\angle A_{k-1}A_{k}A_{k+1}).

Definition 2.3.9.

Let $\alpha$ be an integer angle with LLS-sequence $(a_{1},a_{2},\ldots)$ . Then the integer tangent of $\alpha$ is given by the regular continued fraction

\operatorname{ltan}(\alpha)=[a_{1};a_{2},\ldots].

Definition 2.3.10.

Let $\alpha$ be an integer angle. Then the integer cosine of $\alpha$ is given:

\operatorname{lcos}(\alpha)=\frac{\operatorname{lsin}(\alpha)}{\operatorname{% ltan}(\alpha)}

Proposition 2.3.11.

Let $\alpha$ be an integer angle. Recall that there exists a co-prime pair of integers $m,n$ such that $\alpha$ is integer congruent to some angle $\overline{\alpha}=AOB$ with $A=(1,0)$ , $O=(0,0)$ and $B=(m,n)$ .

Then the LLS sequence for $\alpha$ is a continued fraction sequence for $\frac{m}{n}.$

Corollary 2.3.12.

Let $\alpha$ be integer congruent to $AOB$ with $A=(1,0)$ , $O=(0,0)$ and $B=(m,n)$ . Then

\operatorname{ltan}(\alpha)=\frac{m}{n}.