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Using quadratic form inversion $\left|𝑥\right|\frac{𝑥}{\left|𝑥\right|}⇝\frac{1}{\left|𝑥\right|}\frac{𝑥}{\left|𝑥\right|}$. A type of conformal transformation

${\mathrm{𝕊}}^{𝑛-1}$ Stereographic projection coordinates of the North and South poles. $\begin{array}{ccc}{\mathrm{𝕊}}^{𝑛-1}& ⟶& {\mathrm{ℝ}}^{𝑛-1}\\ 𝑥& ⟿& 𝜉\end{array}$. $𝑁\text{ or }𝑆=𝑝=\left(\mp 1,0\right)$, collinear $\frac{𝑝-𝑥}{𝑝-\left(0,𝜉\right)}=𝜆$, the solution give the map

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

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

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

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

transition-function is quadratic form inversion

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