I am reading through Ahlfor's Complex Analysis and I came upon an item from his book which confused me (page 19). As a background, the Riemann Sphere is the unit sphere in $R^3$: $x_1^2 + x_2^2 + x_3^2 = 1$. We can associate with each 3-tuple $(x_1,x_2,x_3)$ obeying this condition (except $(0,0,1)$) a complex number $z = \frac{x_1^2+x_2^2}{(1-x_3)^2} = \frac{1+x_3}{1-x_3}$. Straightforward calculation shows how $x_1$, $x_2$, and $x_3$ can be written in terms of $z$ : $x_1 = \frac{z+\bar{z}}{1+|z|^2}$, $x_2 = \frac{z - \bar{z}}{i(1+|z|^2)}$ and $x_3 = \frac{|z|^2-1}{|z|^2+1}$. We associate the point $(0,0,1)$ with the "point at infinity," which along with the complex numbers forms the extended complex numbers.
All of this makes sense to me. What confuses me is Ahlfors' proof that there is a straight line connecting the point of infinity, any other point $p$ on the Riemann sphere, and the stereographic projection of $p$ in the complex plane. This is how Ahlfors describes it (summarizing but sticking very closely to the text):
"If the complex plane is identified with the $(x_1,x_2)$-plane with the $x_1$-axis and $x_2$-axis corresponding to the real and imaginary axis, respectively, the stereographic transformation takes on a simple geometric meaning. Writing $z = x+ iy$ we can verify that $x:y:-1$ = $x_1:x_2:x_3-1$, and this means that the points $(x,y,0)$, $(x_1,x_2,x_3)$, and $(0,0,1)$ are in a straight line."
It seems that what he is doing here is subtracting the respective coordinates of $(0,0,1)$ from the terms of both of the ratios to get an equality between the ratios. But when I try to calculate the ratio $x:y:-1$, I get $-\frac{x_1}{x_2}$, as expected, and when I try to calculate the ratio $x_1:x_2:x_3-1$, I obtain $-\frac{x_1(|z|^2 + 1)}{2x_2}$, which is equal to the first ratio only when $|z_1| = 1$, but I'm not sure why (or if) we can assume that.
So, my questions are:
- Did I do something wrong when calculating the ratios $x:y:-1$, and $x_1:x_2:x_3-1$?
- Even if the ratios are equal, why in the world would that imply that there is a straight line between the point at infinity, a point $p$ on the Riemann sphere, and $p$'s stereographic projection?
Brevan Ellefsen's comment is completely correct. The only further remark I'd make is that the complex numbers can be embedded into the complex projective line the same way we embed the affine line over any field into the projective line over that field. So we define the projective complex line as the set of all ratios $(z_1:z_2)$ where $z_1,z_2 \in \mathbb C$ and $z_1$ and $z_2$ are not both 0.Then we have a map $f: \mathbb C \to$ the complex projective line defined by $f(z)=(z:1)$. The only point on the complex projective line that is not the image of any point in $\mathbb C$ is (1:0), the so-called "point at infinity." Thus if $g: \text {sphere except north pole} \to \mathbb {R^2=C}$ is stereographic projection then $f \circ g$ maps the sphere except the north pole to the complex projective line and we extend this map to a bijection from from the entire sphere onto the complex projective line by sending the north pole to (1:0).