庞加莱圆盘模型

距离函数

$\delta (u, v) = 2 \frac{||u-v||^2}{(1-||u||^2)(1-||v||^2)},\,$

$d(u, v) = \operatorname{arccosh} (1+\delta (u,v)).\,$

度量形式

$ds^2 = 4 \frac{\sum_i dx_i^2}{(1-\sum_i x_i^2)^2}.\,$

过两点的直线

$x^2 + y^2 + a x + b y + 1 = 0,\,$

$x^2 + y^2 + \frac{u_2(v_1^2+v_2^2)-v_2(u_1^2+u_2^2)+u_2-v_2}{u_1v_2-u_2v_1}x + {}$
$\frac{v_1(u_1^2+u_2^2)-u_1(v_1^2+v_2^2)+v_1-u_1}{u_1v_2-u_2v_1}y + 1 = 0.$

$x^2+y^2+\frac{2(u_2-v_2)}{u_1v_2-u_2v_1}x - \frac{2(u_1-v_1)}{u_1v_2-u_2v_1}y + 1 = 0.\,$

庞加莱圆盘模型中的角

$\cos(\theta) = u \cdot s.\,$

$\cos^2(\theta) = \frac{P^2}{QR},$

$P = u \cdot (s-t),\,$
$Q = u \cdot u,\,$
$R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).\,$

$\cos^2(\theta) = \frac{P^2}{QR},$

$P = (u-v) \cdot (s-t) - (u \wedge v) \cdot (s \wedge t),\,$
$Q = (u-v) \cdot (u-v) - (u \wedge v) \cdot (u \wedge v),\,$
$R = (s-t) \cdot (s-t) - (s \wedge t) \cdot (s \wedge t).\,$

$P = (u-v) \cdot (s-t) + (u \cdot t)(v \cdot s) - (u \cdot s)(v \cdot t),\,$
$Q = (1 - u \cdot v)^2,\,$
$R = (1 - s \cdot t)^2.\,$

参考文献

• James W. Anderson, Hyperbolic Geometry, second edition, Springer, 2005
• Eugenio Beltrami, Theoria fondamentale delgi spazil di curvatura constanta, Annali. di Mat., ser II 2 (1868), 232-255
• Saul Stahl, The Poincaré Half-Plane, Jones and Bartlett, 1993