972. Hyperbolic Plane

The hyperbolic plane can be represented by the open unit disc, namely the set of points $(x,y)$$(x, y)$ in ${\mathbb{R}}^2$$\Bbb R^2$ with $x^2+y^2<1$$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)$$\mathcal V(N)$ be the set of points $(x,y)$$(x, y)$ such that $x^2+y^2<1$$x^2 + y^2 \lt 1$ and $x,y$$x, y$ are both rational numbers with denominator not exceeding $N$$N$.

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

Find $T(12)$$T(12)$.

972. 双曲平面

双曲平面 可以用一个 开单位圆盘 表示，即 ${\mathbb{R}}^2$$\mathbb{R}^2$ 中全体满足 $x^2+y^2<1$$x^2 + y^2 < 1$ 的点 $(x,y)$$(x, y)$ 之集合。

该平面的 测地线 或是该圆盘的一条直径，或是一条垂直于该圆盘边界的圆弧。下图展示了一个双曲平面和它的两条测地线，其中一条是直径、一条是圆弧。

记 $\mathcal{V}(N)$$\mathcal V(N)$ 为全体满足 $x^2+y^2<1$$x^2 + y^2 < 1$ 且二者分母都不超过 $N$$N$ 的有理点 $(x,y)$$(x, y)$ 的集合。

记 $T(N)$$T(N)$ 为满足如下条件的有序三元组 $(P,Q,R)$$(P, Q, R)$ 数量：$P,Q,R$$P, Q, R$ 是 $\mathcal{V}(N)$$\mathcal V(N)$ 中三个不同的点，且存在一条测地线同时经过这三个点。例如 $T(2)=24$$T(2) = 24$、$T(3)=1296$$T(3) = 1296$。

求 $T(12)$$T(12)$。

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