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I have the plane H and its equation is x+y-z = 0. I want to find the image of H$\cap\mathbb{S}^{2}$ under stereographic projection.

I basically said that if the point (x,y,z) gets mapped to say (p,q,0) then the pre-image of (p,q,0), which is ($\frac{2p}{p^{2}+q^{2}+1}$,$\frac{2q}{p^{2}+q^{2}+1}$,$\frac{p^{2}+q^{2}-1}{p^{2}+q^{2}+1}$), must satisfy the equation of the plane.

That gives us 2p+2q = p$^{2}$+q$^{2}$-1

$\Rightarrow$ (p-1)$^{2}$ + (q-1)$^{2}$ = 3 , which is equation of a circle with center at (1,1) and radius $\sqrt{3}$.

Any comment on my work would be appreciated.

BCLC
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HumbleStudent
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1 Answers1

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A way to check your answer is that the image of the intersection of $x+y=z, x^2+y^2+z^2=1$ is $(\frac{x}{1-z},\frac{y}{1-z},0) = (\frac{x}{1-(x+y)},\frac{y}{1-(x+y)},0)$ and should be still $C[(1,1),\sqrt{3}]$. I think it's kinda hard to see that $(\frac{x}{1-(x+y)},\frac{y}{1-(x+y)},0)$ is a circle in $\mathbb C$ (well, circle, cline, ellipse, whatever it is, it's obviously in $\mathbb C$) w/c is why using the inverse formula (which I did too), is better I guess. Anyhoo, we can check that

$$(\frac{x}{1-(x+y)}-1)^2+(\frac{y}{1-(x+y)}-1)^2=3,$$

that is, if you somehow think of subtracting 1 and then squaring. So yeah, it's indeed a circle.

BCLC
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