Def/Range topograph



=Water= Suppose that $Q(x,y)$ is a binary quadratic form. Label the faces of the domain topograph by the values of the quadratic form, to obtain the range topograph; namely, every face corresponding to a primitive lax vector $\pm (x,y)$ is labelled by $Q(x,y)$.

We define lakes and rivers in the range topograph.

Lakes
Suppose that $\pm (x,y)$ is a primitive lax vector. We call the corresponding face of the topograph a defines::lake (of $Q$) if $Q(x,y) = 0$.

If $\{ \pm \vec v, \pm \vec w \}$ is a lax basis, and the faces $\pm \vec v$ and $\pm \vec w$ are both lakes of $Q$, we say that the pair of faces is a double-lake.

River segments
Suppose that $ \{ \pm \vec v, \pm \vec w \}$ is a lax basis; thus $\pm \vec v$ and $\pm \vec w$ correspond to faces in the topograph, which share an edge. We call this edge a river segment (of $Q$) if one of the following conditions holds:
 * $Q(\vec v) > 0$ and $Q(\vec w) < 0$.
 * $Q(\vec v) < 0$ and $Q(\vec w) > 0$.

In other words, a river segment is an edge which separates faces of opposite sign in the range topogrpah of $Q$.

Rivers
Suppose that $R$ is a set of edges in the topograph. We say that $R$ is a defines::river (of $Q$) if the following conditions hold:
 * Every edge in $R$ is a river segment.
 * If $p$ and $q$ are endpoints of edges of $R$, then there is a path from $p$ to $q$ using only edges from $R$. In other words, $R$ is connected.
 * $R$ is maximal among sets of edges with the previous two properties.

Riverbends
Suppose that $r = \{ \pm \vec v, \pm \vec w \}$ is a river segment. We say that $r$ is a riverbend if the following two conditions hold:
 * $\{ \pm \vec v, \pm (\vec v - \vec w) \}$ is a river segment.
 * $\{ \pm \vec w, \pm (\vec v + \vec w) \}$ is a river segment.

=Climbing= Suppose that $Q(x,y)$ is a binary quadratic form. Label the faces of the domain topograph by the values of the quadratic form, to obtain the range topograph; namely, every face corresponding to a primitive lax vector $\pm (x,y)$ is labelled by $Q(x,y)$.

We metaphorically think of the value $Q(x,y)$ as the "altitude" of the face corresponding to $\pm (x,y)$. With this metaphor, we consider "climbing" edges. Suppose that $\{ \pm \vec v, \pm \vec w \}$ is a lax basis, corresponding to an edge $h$ in the domain topograph.

The edge $e$ is next to four regions in the domain topograph, which are labelled by four integers in the range topograph. Besides the two faces corresponding to $\pm \vec v$, and $\pm \vec w$, the edge $e$ is next to $\pm (\vec v + \vec w)$ and $\pm (\vec v - \vec w)$. We label the edge $e$ with an arrow as follows: Thus, the arrow should point from the smaller integer to the larger integer, in the range topograph.
 * An arrow from $\pm (\vec v - \vec w)$ to $\pm (\vec v + \vec w)$ if $\pm (\vec v - \vec w) < \pm (\vec v + \vec w)$.
 * An arrow from $\pm (\vec v + \vec w)$ to $\pm (\vec v - \vec w)$ if $\pm (\vec v + \vec w) < \pm (\vec v - \vec w)$.
 * We label the edge with a "zero" if $\pm (\vec v - \vec w) = \pm (\vec v + \vec w)$. We call this a defines::neutral edge.

Travelling along edges, in the direction of the arrows, is called climbing (in the range topograph of $Q$).



Wells
If $Q$ is a binary quadratic form, and $Q(x,y) > 0$ for all nonzero vectors $(x,y)$, then one can find a well or a double-well in the range topograph of $Q$. A defines::well in a range topograph is a vertex, from which all three edges point away.

A defines::double-well in a range topograph is a pair of vertices joined by a neutral edge, from which all four other edges point away.

=Observations=


 * If $\Delta < 0$, then $Q$ takes only positive or only negative values.
 * Two lakes are always joined, either as a double-lake or with a river.
 * Around a lake, you find an arithmetic progression with common difference equal to the square root of the discriminant.
 * At most one river sticks out of a lake.
 * If $\Delta \neq 0$, then every lake has a river sticking out, or there is a double-lake.
 * If $\Delta = 0$, you have one lake surrounded by a constant progression.
 * If $\Delta > 0$, and not a perfect square, then there is an endless river.
 * Along an endless river, the pattern of numbers repeats.
 * There is at most one river.
 * An endless river is endless in both directions.