# 庞加莱圆盘模型

## 距离函数

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

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

## 度量形式

${\displaystyle ds^{2}=4{\frac {\sum _{i}dx_{i}^{2}}{(1-\sum _{i}x_{i}^{2})^{2}}}.\,}$

## 过两点的直线

${\displaystyle x^{2}+y^{2}+ax+by+1=0,\,}$

${\displaystyle x^{2}+y^{2}+{\frac {u_{2}(v_{1}^{2}+v_{2}^{2})-v_{2}(u_{1}^{2}+u_{2}^{2})+u_{2}-v_{2}}{u_{1}v_{2}-u_{2}v_{1}}}x+{}}$
${\displaystyle {\frac {v_{1}(u_{1}^{2}+u_{2}^{2})-u_{1}(v_{1}^{2}+v_{2}^{2})+v_{1}-u_{1}}{u_{1}v_{2}-u_{2}v_{1}}}y+1=0.}$

${\displaystyle x^{2}+y^{2}+{\frac {2(u_{2}-v_{2})}{u_{1}v_{2}-u_{2}v_{1}}}x-{\frac {2(u_{1}-v_{1})}{u_{1}v_{2}-u_{2}v_{1}}}y+1=0.\,}$

## 庞加莱圆盘模型中的角

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

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

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

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

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

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

## 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