# 双曲空间模型

## 彭加莱平面

### 跟踪曲线 Tractrix

$\displaystyle{ dy/dx=-\frac{y}{\sqrt{a^2-y^2}} }$

$\displaystyle{ x=\mathrm{arccosh}\frac{1}{y}-\sqrt{1-y^2}. }$

\displaystyle{ \begin{align} &x=\mathrm{arccosh}\,e^{s}-\sqrt{1-e^{-2s}}\\ &y=e^{-s} \end{align} }

## 旋转跟踪曲面

\displaystyle{ \begin{align} &x=\mathrm{arccosh}\,e^s-\sqrt{1-e^{-2s}}\\ &y=e^{-s}\mathrm{cos\,}\theta\\ &z=e^{-s}\mathrm{sin\,}\theta \end{align} }

$\displaystyle{ dl^2=ds^2+e^{-2s}d\theta^2 }$

$\displaystyle{ dl^2=\frac{d\gamma^2+d\theta^2}{\gamma^2}=\begin{pmatrix}d\gamma&d\theta\end{pmatrix}\begin{pmatrix}\frac{1}{\gamma^2} & 0 \\ 0 & \frac{1}{\gamma^2}\end{pmatrix}\begin{pmatrix}d\gamma\\d\theta\end{pmatrix} }$

$\displaystyle{ dl=\frac{\sqrt{d\gamma^2+d\theta^2}}{\gamma} }$

$\displaystyle{ \left( \begin{matrix} \frac{1}{\gamma^2} & 0\\ 0&\frac{1}{\gamma^2}\\ \end{matrix} \right) }$

## 彭加莱平面

### 定义

$\displaystyle{ g(\textbf{w}_1,\textbf{w}_2)=\frac{\textbf{w}_1\cdot \textbf{w}_2}{v^2} }$

$\displaystyle{ g(\textbf{x},\textbf{y})=\textbf{x}^TA\textbf{y} }$

$\displaystyle{ \left( \begin{matrix} \frac{1}{v^2} & 0\\ 0&\frac{1}{v^2}\\ \end{matrix} \right) }$

### 曲线长度

$\displaystyle{ l_{hyp}=\int_a^b{\frac{\sqrt{x'(t)^2+y'(t)^2}}{y(t)}}dt }$

### 测地线

$\displaystyle{ (u-d)^2+v^2=1/c^2 }$

### 双曲距离

$\displaystyle{ x=x_0+r \mathrm{cos\,}\theta,y=r \mathrm{sin\,}\theta }$

$\displaystyle{ d_{hyp}(P,Q)=\int_{\theta_1}^{\theta_2}\frac{\sqrt{x'(\theta)^2+y'(\theta)^2}}{y}d\theta=\int_{\theta_1}^{\theta_2}\frac{1}{\mathrm{sin\,}\theta}d\theta=\log\frac{\tan{\theta_2/2}}{\tan{\theta_1/2}} }$

$\displaystyle{ d_{hyp}(P,Q)=\log\frac{|z-\bar{z}'|+|z-z'|}{|z-\bar{z}'|-|z-z'|} }$ 其中，$\displaystyle{ z=x_1+y_1i,z'=x_2+y_2i,\bar{z}'=x_2-y_2i }$

### 等距变换

$\displaystyle{ d(P,Q)=d(\Gamma(P),\Gamma(Q)) }$

#### 同位变换 Homotheties

$\displaystyle{ l_{hyp}(\phi(\gamma))=\int_a^b\frac{\sqrt{\lambda^2x'(t)^2+\lambda^2y'(t)^2}}{\lambda y(t)}dt=\int_a^b\frac{\sqrt{x'(t)^2+y'(t)^2}}{y(t)}dt=l_{hyp}(\gamma) }$

#### 标准倒置变换

$\displaystyle{ \phi(x,y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right) }$

$\displaystyle{ x_1(t)=\frac{x(t)}{x(t)^2+y(t)^2}, y_1(t)=\frac{y(t)}{x(t)^2+y(t)^2} }$

$\displaystyle{ x'_1(t)=\frac{(y(t)^2-x(t)^2)x'(t)-2x(t)y(t)y'(t)}{(x(t)^2+y(t)^2)^2}, y'_1(t)=\frac{(y(t)^2-x(t)^2)y'(t)-2x(t)y(t)x'(t)}{(x(t)^2+y(t)^2)^2} }$

$\displaystyle{ x'_1(t)^2+y'_1(t)^2=\frac{x'(t)^2+y'(t)^2}{(x(t)^2+y(t)^2)^2} }$

$\displaystyle{ l_{hyp}(\phi(\gamma))=\int_a^b\frac{\sqrt{x'_1(t)^2+y'_1(t)^2}}{y_1(t)}dt=\int_a^b\frac{\sqrt{x'(t)^2+y'(t)^2}}{y(t)}dt=l_{hyp}(\gamma) }$

#### 水平平移

$\displaystyle{ \phi(x,y)=(x+x_0,y) }$

#### 一般形式

$\displaystyle{ z\mapsto \frac{az+b}{cz+d}, }$

$\displaystyle{ z\mapsto \frac{c\bar{z}+d}{a\bar{z}+b} }$

$\displaystyle{ ad-bc=1 }$

### 圆

$\displaystyle{ \frac{1+\frac{|z-z'|}{|z-\bar{z'}|}}{1-\frac{|z-z'|}{|z-\bar{z'}|}}=e^R }$

$\displaystyle{ \left(x-\frac{a-ac^2}{1-c^2}\right)^2+\left(y-\frac{b+bc^2}{1-c^2}\right)^2=\left(\frac{2bc}{1-c^2}\right)^2 }$

## 彭加莱圆盘

$\displaystyle{ J(z)=\frac{i z+1}{z+i} }$

### 度规

$\displaystyle{ g(\textbf{u},\textbf{v})=\frac{2\textbf{u}\cdot \textbf{v}}{1-x^2-y^2} }$ 其中g为P点处的度规，d(u,v)为欧氏空间下的两向量的内积。

$\displaystyle{ g(\textbf{u},\textbf{v})=\frac{2\textbf{u}\cdot \textbf{v}}{1-|z|^2} }$

$\displaystyle{ ds^2=\frac{4dx^2+4dy^2}{(1-x^2-y^2)^2}=\begin{pmatrix}dx&dy\end{pmatrix}\begin{pmatrix}\frac{1}{(1-x^2-y^2)^2}&0\\0&\frac{1}{(1-x^2-y^2)^2}\end{pmatrix}\begin{pmatrix}dx\\dy\end{pmatrix} }$

$\displaystyle{ ds=\frac{2\sqrt{dx^2+dy^2}}{1-x^2-y^2} }$

$\displaystyle{ ds^2=\frac{d\theta^2+r^2dr^2}{(1-r^2)^2}=\begin{pmatrix}d\theta&dr\end{pmatrix}\begin{pmatrix}\frac{1}{(1-r^2)^2}&0\\0&\frac{1}{(1-r^2)^2}\end{pmatrix}\begin{pmatrix}d\theta\\dr\end{pmatrix} }$

### 距离计算

$\displaystyle{ d_{plane}(P,Q)=\log \frac{|z-\bar{z}'|+|z-z'|}{|z-\bar{z}'|-|z-z'|} }$

$\displaystyle{ z=\frac{1-wi}{w-i},z'=\frac{1-w'i}{w'-i},\bar{z}'=\frac{1+\bar{w}'i}{\bar{w}'+i} }$

$\displaystyle{ d_{disk}(J(P),J(Q))=d_{plane}(P,Q)=\log{\frac{1+s}{1-s}} }$

$\displaystyle{ d_{disk}(0,w)=\log{\frac{1+|w|}{1-|w|}} }$

### 等距变换

$\displaystyle{ \phi(z)=\frac{\alpha z+\beta}{\bar{\beta} z+\bar{\alpha}}, or, \phi(z)=\frac{\alpha \bar{z}+\beta}{\bar{\beta}\bar{z}+\bar{\alpha}} }$

$\displaystyle{ |\alpha|^2-|\beta|^2=1 }$

### 圆

$\displaystyle{ \log{\frac{1+|w'-w|}{|1-w\bar{w'}|}}=r }$

$\displaystyle{ 1-a^2 c^2-b^2 c^2+\left(-2 a+2 a c^2\right) x+\left(a^2+b^2-c^2\right) x^2+\left(-2 b+2 b c^2\right) y+\left(a^2+b^2-c^2\right) y^2=0 }$

$\displaystyle{ \left(x-\frac{ac^2-a}{c^2-a^2-b^2}\right)^2+\left(y-\frac{bc^2-b}{c^2-a^2-b^2}\right)^2=\left(\frac{\left(-1+a^2+b^2\right) c}{\left(a^2+b^2-c^2\right)}\right)^2 }$

#### 圆的周长和面积

$\displaystyle{ C((0,0),R)=\int_0^{2\pi}\frac{2r}{1-r^2}d\theta=2\pi \sinh(R) }$

$\displaystyle{ A((0,0),R)=\int\int_{|z|\lt r}\frac{4}{(1-|z|^2)^2}d\sigma=\int_0^{R}dr\int_0^{2\pi}d\theta \frac{4r}{(1-r^2)^2}=2\pi(\cosh{R}-1) }$

## 扩展的彭加莱圆盘

$\displaystyle{ r_e=\tanh{r_h/2} }$

$\displaystyle{ r'=2\tanh^{-1}{r},\theta'=\theta }$

### 度规

$\displaystyle{ ds^2=4\frac{dr^2+r^2d\theta^2}{(1-r^2)^2}=4(dr,d\theta)\begin{pmatrix}1 & 0 \\ 0 & \frac{r^2}{(1-r^2)^2}\end{pmatrix} \begin{pmatrix} dr\\ d\theta\end{pmatrix} }$

$\displaystyle{ ds^2=4\frac{\frac{1}{4\cosh(r'/2)} dr'^2+\tanh^2(r'/2)d\theta'^2}{(1-\tanh^2(r'/2))^2}=dr'^2+\sinh^2(r')d\theta'^2 }$

$\displaystyle{ ds=\sqrt{dr'^2+\sinh^2(r')d\theta'^2} }$

### 任意两点距离公式

$\displaystyle{ \cosh{x}=\cosh(r)\cosh(r')-\sinh(r)\sinh(r')\cos(\Delta \theta) }$

$\displaystyle{ x=\log\frac{1+s}{1-s}, s=\frac{|w-w'|}{1-w\bar{w'}} }$ ，并设w,w'在彭加莱圆盘中的极坐标分别为：$\displaystyle{ (R,\theta),(R',\theta') }$

$\displaystyle{ s=\sqrt{\frac{R^2+R'^2-2RR'\cos\Delta\theta}{1+R^2R'^2-2RR'\cos\Delta\theta}} }$

$\displaystyle{ \cosh(x)=\frac{1+R^2+R'^2+R^2R'^2-4RR'\cos\theta}{(1-R^2)(1-R'^2)} }$

$\displaystyle{ \cosh(x)=\cosh(r)\cosh(r')-\sinh(r)\sinh(r')\cos(\Delta \theta)=\frac{1+R^2}{1-R^2}\cdot \frac{1+R'^2}{1-R'^2}-4 \frac{RR'\cos\Delta\theta}{(1-R^2)(1-R'^2)} }$

#### 近似表达式

$\displaystyle{ x\approx r+r'+2\log{\sin(\Delta\theta/2)} }$

$\displaystyle{ \Delta\theta/2) }$很小的时候，

$\displaystyle{ x\approx r+r'+2\log{\Delta\theta/2} }$

#### 圆

$\displaystyle{ r_e=\tanh(r_h/2) }$

## 参考文献

1. Francis Bonahon: Low-dimensional geometry - From Euclidean Surfaces to Hyperbolic Knots, American Mathematical Society, 1995
2. John Stillwell: Geometry of Surfaces, Springer-Verlag New York, 1992
3. Riccardo Benedetti, Carlo Petronio: Lectures on Hyperbolic Geometry, Springer-Verlag Berlin Hamberg, 1992
4. Wikipedia: https://en.wikipedia.org/wiki/Tractrix