I am attempting to classify (convex rational) cones in $\Bbb{R}^2$.
We say here that $\sigma\subset\Bbb{R}^2$ is a cone if there exist $u,v\in\Bbb{Z}^2$ which $\Bbb{R}$-span the whole plane $\Bbb{R}^2$ and $$ \sigma=\mathop{Cone}(u,v)=\{tu+sv: \ t,s\in \Bbb{R}_{\geq0}\}$$ We can here wlog assume that $u$ lies on the positive $x$-axis and $v$ on the upper half-plane.
We say that two cones $\sigma,\sigma'$ are isomorphic if there exists a group isomorphism $$A\colon \Bbb{Z}^2\longrightarrow\Bbb{Z}^2$$ (hence $\det A=\pm1$) such that $A(\sigma)=\sigma'$. Now classification takes the following form
Claim: any cone $\sigma$ as above is isomorphic to $$\sigma'=\mathop{Cone}\{(1,0), (q,p)\}$$ where $p,q\in\Bbb{N}$ are relatively prime and either $p=q=1$ or $1\leq q<p$. Moreover $p,q$ are uniquely determined.
any hint for attacking this? thanks