note-math

使用二次型反演 $\left|𝑥\right|\frac{𝑥}{\left|𝑥\right|}⇝\frac{1}{\left|𝑥\right|}\frac{𝑥}{\left|𝑥\right|}$. 共形变换的一种

${\mathrm{𝕊}}^{𝑛-1}$ 南北极球极投影坐标. $\begin{array}{ccc}{\mathrm{𝕊}}^{𝑛-1}& ⟶& {\mathrm{ℝ}}^{𝑛-1}\\ 𝑥& ⟿& 𝜉\end{array}$. $𝑁\text{ or }𝑆=𝑝=\left(\mp 1,0\right)$, 共线 $\frac{𝑝-𝑥}{𝑝-\left(0,𝜉\right)}=𝜆$, 解给出映射

$$\begin{array}{ccc}{\mathrm{𝕊}}^{𝑛-1}& ⟶& {\mathrm{ℝ}}^{𝑛-1}\\ \left({𝑥}_1,\dots ,{𝑥}_{𝑛}\right)& ⟿& \frac{{𝑥}_2,\dots ,{𝑥}_{𝑛}}{1\pm {𝑥}_1}\end{array}$$

$|𝜉{|}^2=\frac{|{𝑥}_2,\dots ,{𝑥}_{𝑛}{|}^2}{{\left(1\pm {𝑥}_1\right)}^2}=\frac{1-{𝑥}_1^2}{{\left(1\pm {𝑥}_1\right)}^2}$

$1+|𝜉{|}^2=2\frac{\pm {𝑥}_1}{{\left(1\pm {𝑥}_1\right)}^2}$

$$\begin{array}{ccc}{\mathrm{ℝ}}^{𝑛-1}& ⟶& {\mathrm{𝕊}}^{𝑛-1}\\ 𝜉& ⟿& \frac{\pm \left(1-|𝜉{|}^2\right),𝜉}{1+|𝜉{|}^2}\end{array}$$

transition-function 是二次型反演

$$\begin{array}{ccc}{\mathrm{ℝ}}^{𝑛-1}& ⟶& {\mathrm{ℝ}}^{𝑛-1}\\ 𝜉& ⟿& \frac{1\pm {𝑥}_1}{1\mp {𝑥}_1}𝜉=\frac{𝜉}{|𝜉{|}^2}\end{array}$$