庞加莱圆盘模型中，过两点的直线

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Analytic geometry constructions in the hyperbolic plane
解析几何中一个基本构造是寻找过两个定点的一条直线。在庞加莱圆盘模型中，平面上的直线定义为与单位圆正交的圆弧
$$x^2 + y^2 + a x + b y + 1 = 0$$
这是与单位圆正交的圆的一般形式，可以证明这是连接这两点双曲距离最短的曲线（测地线）。给定圆盘中不在同一直径上两点 $u$ 和 $v$，我们可以求出 $a$ 和 $b$，得到
\begin{multline*}
x^2 + y^2 + \frac{u_2(v_1^2+v_2^2)-v_2(u_1^2+u_2^2)+u_2-v_2}{u_1v_2-u_2v_1}x \\
+\frac{v_1(u_1^2+u_2^2)-u_1(v_1^2+v_2^2)+v_1-u_1}{u_1v_2-u_2v_1}y + 1 = 0\end{multline*}如果点 $u$ 和 $v$ 是 ideal points（在单位圆周上且不是对径点），上面的公式简化成
\[x^2+y^2+\frac{2(u_2-v_2)}{u_1v_2-u_2v_1}x - \frac{2(u_1-v_1)}{u_1v_2-u_2v_1}y + 1 = 0.\]

红色圆弧是 Poincaré 圆盘模型中的测地线； 它投影到绿色双曲面上的棕色测地线。
ideal points（单位圆周上的点）投影到无穷远。

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几何画板官网: Advanced Sketch Gallery
Poincaré Disk Model of Hyperbolic Geometry
This sketch from Sketchpad's Samples collection depicts the hyperbolic plane using the Poincaré Disk model, in which a line through two points is defined as the Euclidean arc passing through the points and perpendicular to the circle. Custom tools allow you to create constructions in the hyperbolic plane.

Sketchpad Samples
Download: Poincare_Disk.gsp
Poincare_Disk.gsp (45.42 KB, Downloads: 55)

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Geometry Explorer
Providing an environment for the study of Euclidean, Hyperbolic, Elliptic, Spherical, Analytic, Transformational, Fractal, and Turtle Geometries.
Exploring Geometry, Michael Hvidsten
Geometry with Geometry Explorer, Michael Hvidsten