One way to study (mixed signature) quadratic forms in two variables is to study the topograph. Looks like the signature doesn't matter: here is (1,1) and (1-,1).

The name is derived from τοποσ (Greek: "place") and γραφή ("writing"). I read that if you're really good at reading topographs you can extract information like the genus, class number, solve the Pell equation, and more.

There are two resources I found for topographs:

**The Sensual Quadratic Form**, John H Conway**Topology of Numbers**, Allen Hatcher

They are basically drawing the dual tree of the Farey Tesselation, which is a tiling of $\mathbb{H}$ or $\mathbb{D}$ by hyperbolic triangles. Is there a more serious name for putting trees on $\mathbb{H}$?

This question emerges, for example, trying to draw these things with a computer and I needed to decide a natural place to put the interior vertices, and I couldn't think of one. The "outer" vertices are indexed by $\text{P}\mathbb{Q}^1$ and the interior vertices could be in any reasonable place.

There could be a serious name for this structure, like the Bruhat-Tits building or maybe it's in Serre's book on Trees. Any guidance?

A figure similar to the topograph also appears in a discussion of the Bruhat-Tits tree for $\text{SL}(2, \mathbb{Q}_2)$. [notes]

An illustrated theory of numbersby Martin H. Weissman (AMS). $\endgroup$ – David G. Stork Nov 13 '17 at 17:27