Ngô Quốc Anh

October 9, 2017

Jacobian of the stereographic projection not at the North pole

Filed under: Uncategorized — Ngô Quốc Anh @ 18:57

Denote $\pi_P : \mathbb S^n \to \mathbb R^n$ the stereographic projection performed with $P$ as the north pole to the equatorial plane of $\mathbb S^n$. Clearly when $P$ is the north pole $N$, i.e. $N = (0,...,0,1)$, then $\pi_N$ is the usual stereographic projection.

As we have already known that, for arbitrary $\xi \in \mathbb S^n$, the image of $\xi$ is

$\displaystyle \pi_P : \xi \mapsto x = P+\frac{\xi-P}{1-\xi \cdot P}.$

Here the point $x \in \mathbb R^n$ is being understood as a point in $\mathbb R^{n+1}$ by adding zero in the last coordinate. For the inverse map, it is not hard to see that

$\displaystyle \pi_P^{-1} : x \mapsto \xi =\frac{|x|^2-1}{|x|^2+1}P+\frac 2{|x|^2+1}x.$

The purpose of this entry is to compute the Jacobian of the, for example, $\pi_P^{-1}$ by comparing the ratio of volumes.

First pick two arbitrary points $x, y \in \mathbb R^n$ and denote $\xi = \pi_P^{-1}(x)$ and $\eta = \pi_P^{-1}(y)$. The Euclidean distance between $\xi$ and $\eta$ is

$\displaystyle |\xi -\eta|^2 = \sum_{i=1}^{n+1} |\xi_i - \eta_i|^2 =\sum_{i=1}^n |\xi_i - \eta_i|^2+|\xi_{n+1} - \eta_{n+1}|^2.$

October 3, 2017

Stereographic projection not at the North pole and an example of conformal transformation on S^n

Filed under: Riemannian geometry — Tags: , — Ngô Quốc Anh @ 12:09

Denote $\pi_P : \mathbb S^n \to \mathbb R^n$ the stereographic projection performed with $P$ as the north pole to the equatorial plane of $\mathbb S^n$. Clearly when $P$ is the north pole $N$, i.e. $N = (0,...,0,1)$, then $\pi_N$ is the usual stereographic projection.

Clearly, for arbitrary $x \in \mathbb S^n$, the image of $x$ is

$\displaystyle \pi_P : x \mapsto y = P+\frac{x-P}{1-x \cdot P}.$

For the inverse map, it is not hard to see that

$\displaystyle \pi_P^{-1} : y \mapsto x =\frac{|y|^2-1}{|y|^2+1}P+\frac 2{|y|^2+1}y.$

Derivation of $\pi_P$ and $\pi_P^{-1}$ are easy, for interested reader, I refer to an answer in . Let us now define the usual conformal transformation $\varphi_{P,t} : \mathbb S^n \to \mathbb S^n$ given by

$\displaystyle \varphi_{P,t} : x \mapsto \pi_P^{-1} \big( t \pi_P ( x) \big)$