The hyperbolic plane can be represented by the open unit disc, namely the set of points $(x,y)$ in ${\mathbb{R}}^2$ with $x^2+y^2<1$.

A geodesic is defined as either a diameter of the open unit disc or a circular arc contained within the disc that is orthogonal to the boundary of the disc.
The following diagram shows the hyperbolic plane with two geodesics; one is a diameter and the other is a circular arc.

Let $\mathcal{V}(N)$ be the set of points $(x,y)$ such that $x^2+y^2<1$ and $x,y$ are both rational numbers with denominator not exceeding $N$.

Let $T(N)$ be the number of ordered triples $(P,Q,R)$ such that $P,Q,R$ are three different points in $\mathcal{V}(N)$ and there is a hyperbolic line passing through all of them.
For example, $T(2)=24$ and $T(3)=1296$.

Find $T(12)$.