Help with unknown notation In Ahlfors' Complex Analysis, page 19 it says (in relation with the Riemann sphere):
"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."
My question is: what do the colon mean? And how does it follow that the points lie on a line?
Thank you.
 A: The colons are ordinarily used for homogeneous coordinates. For example, consider the real plane. We have Cartesian coordinates $(x,y)$. The homogeneous coordinates $(x:y)$ are equivalence classes of points. We say that two, non-zero points $(x_1,y_1)$ and $(x_2,y_2)$ are equivalent if and only if there exists a non-zero real number, say $\lambda$, for which $(x_1,y_1) = \lambda(x_2,y_2) = (\lambda x_2, \lambda y_2)$. Geometrically, we take the set of all lines through the origin and identity them as a single point. So the point $(1,1)$ is the same as $(2,2)$ and $(-3,-3)$, etc. The set of all of these lines, or the set of all equivalence classes, is called the real projective line, and we denote it by $\mathbb{RP}^1$.
In general, the projective space $\mathbb{RP}^n$ consists of all equivalence classes $(x_0:x_1:\ldots:x_n)$ where $x_i \in \mathbb{R}$ and at least one $x_j \neq 0$. Again, geometrically, this is the space of lines through the origin. 
I suspect the notation you have found is the same, but over the complex numbers. Take a look at this:


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*https://en.wikipedia.org/wiki/Complex_projective_plane
A: The colon is a shorthand for ratio. In your example, it means
$$\frac xy = \frac {x_1}{x_2} \qquad \text{and} \qquad \frac{y}{-1} = \frac{x_2}{x_3-1}.$$
From there, it's a bit of work to see that they're on a line. Suppose that when $t=0$ we are at $(0,0,1)$, and when $t=1$ we are at $(x,y,0)$. Then our line is $(0,0,1) + t(x,y,-1)$. Well, now if the $x$-coordinate is $x_1$, then $t=\frac{x_1}{x}$.
So we are at the point $(0,0,1)+(x_1, \frac{x_1y}{x}, \frac{-x_1}{x})$. Since $\frac xy = \frac {x_1}{x_2}$, the $y$-coordinate is $x_2$. Similarly, since $\frac{x}{-1}=\frac{x_1}{x_3-1}$, the $z$-coordinate is $x_3$.
