Continued fractions and lines across the Stern--Brocot diagram

This paper concerns the relationships between continued fractions and the geometry of the Stern-Brocot diagram. Each rational number can be expressed as a continued fraction $[a_0;a_1,\dots ,a_n]$ whose terms $a_i$ are integers and are positive if $i\ge 1$. Select an index $i\in \{1,\dots ,n\}$ and replace $a_i$ with an integer $m$ to obtain a continued fraction expansion for an extended rational ${\alpha }_m\in \mathbb{Q}\cup \{\mathrm{\infty }\}$. This paper shows that the vertices of the Stern-Brocot diagram corresponding to the numbers $\{{\alpha }_m{\}}_{m\in \mathbb{Z}}$ lie on a pair of (extended) Euclidean lines across the diagram. The slopes of these two lines differ only by a sign change and they meet at the point $L=([a_0;a_1,\dots ,a_{i-1}],0)\in {\mathbb{R}}^2$. Moreover, as $\mid m\mid \to \mathrm{\infty }$, the associated vertices move down these lines and converge to $L$. This paper concludes with a discussion which interprets this result in the context of 2-bridge link complements and Thurston’s work on hyperbolic Dehn surgery.