I'm studying differentiable manifolds and I'm trying to work out the details of the stereographic atlas for $S^n$. I'm trying to deduce the expressions for the stereographic projection. I did a sketch:
The imagem of the point $x$ through the (north) stereographic projection would the point $(y_1, \ldots, y_n)\in \mathbb R^n$ where the coordinates $y_1, \ldots, y_n$ are determined as in the picture.
For finding the explicit expression for this map I'm trying to use the metric relations of triangle rectangles. Notice $$y=|y|\frac{(x_1, \ldots, x_n, 0)}{\sqrt{x_1^2+\ldots+x_n^2}}=|y|\frac{(x_1, \ldots, x_n, 0)}{\sqrt{1-x_{n+1}^2}}$$ hence it suffices finding $|y|$. Using the relation $$|x|^2=|(x_1, \ldots, x_n, 0)| |y|$$ we find $$|y|=\frac{1}{\sqrt{1-x_{n+1}^2}}$$ from what $$y=\frac{1}{1-x_{n+1}^2}(x_1, \ldots, x_n, 0),$$ so that the map I'm looking for would be $$(x_1, \ldots, x_{n+1})\in S^n\longmapsto \frac{1}{1-x_{n+1}^2}(x_1, \ldots, x_n).$$ However this is wrong, its should be: $$(x_1, \ldots, x_{n+1})\in S^n\longmapsto \frac{1}{1-x_{n+1}} (x_1, \ldots, x_n).$$ Can anyone tell me where I've gone wrong?
I think it must be a silly mistake but I can't see where it is.
Thanks
Obs: In general, in the triangle:
we have $b^2=ma$.
An answer after 6 years ... so what was your mistake?
As Dejan Govc wrote in his comment, the equation $b^2 = ma$ is not true for general triangles.
What is a correct geometric argument to determine $\lvert y \rvert$?
In your sketch, look at the two similar right triangles with vertices $N, 0, y$ and $x,x',y$ with $x' = (x_1,\ldots,x_n,0)$. The ratio of their catheti is the same, thus we get $$\frac{\lvert y \rvert}{1} = \frac{\lvert y \rvert - \lvert x' \rvert}{x_{n+1}} . \tag{1}$$ We have $\lvert x' \rvert = \sqrt{x_1^2+\ldots+x_n^2} = \sqrt{1-x_{n+1}^2}$ so that $(1)$ gives the solution $$\lvert y \rvert = \frac{\sqrt{1-x_{n+1}^2}}{1-x_{n+1}} . \tag{2}$$ Inserting $(2)$ in your equation $$y =|y|\frac{x'}{\sqrt{1-x_{n+1}^2}}$$ produces $$y = \frac{x'}{1-x_{n+1}} . \tag{3}$$ which is that what it should be. By the way, note that your incorrect solution is not defined at the south pole $x = (0,\ldots, 0,1)$.