3
$\begingroup$

I'm reading through the first chapter of Ahlfors's Complex Analysis book, and during the section on stereographic projections, he says that we can map any $z = x+iy \in \mathbb{C}$ onto the unit sphere in three dimensions injectively using the equation $z= \frac{x_1+ix_2}{1-x_3}$.

He then says that $x:y:-1= x_1:x_2:x_3-1$, which implies that $(x,y,0)$, $(x_1,x_2,x_3)$, and $(0,0,1)$ all lie on the same line.

My question is what the colon symbol means in this context. I only remember it being used in the context of sets as a replacement for the "|" symbol.

  • 0
    could it mean ratio? (random guess, I did not verify)2011-01-19

3 Answers 3

6

It is a symbol that represents ratios; $A:B$ means "the ratio of $A$ to $B$". (Shows up all the time in Euclid; e.g., Book V.) When we write "$A:B=C:D$", we mean the ratio of $A$ to $B$ is the same as the ratio of $C$ to $D$.

Here he is talking about three ratios; the fact that $x:y:-1 = x_1:x_2:x_3$ means that the vectors determined by $(x,y,-1)$ and by $(x_1,x_2,x_3)$ are parallel. Note that $(x,y,-1) = (x,y,0) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and end point at $(x,y,0)$; and $(x_1,x_2,x_3-1) = (x_1,x_2,x_3) - (0,0,1)$ is the vector with starting point at $(0,0,1)$ and endpoint at $(x_1,x_2,x_3)$. Since $x:y:-1 = x_1:x_2:x_3$, they are parallel; since they both start at the same point, that means that the line through $(0,0,1)$ and $(x,y,0)$ (determined by the first vector) and the line through $(0,0,1)$ and $(x_1,x_2,x_3)$ (determined by the second vector) are the same, since they are parallel and they both go through $(0,0,1)$.

  • 0
    @user1736: If $(a,b,c)$ and $(x,y,z)$ have the same nonzero ratios between components, say $a:b = x:y$, and $b:c=y:z$, then if $a=\lambda b$, then $x=\lambda y$; i$; and if $b=\nu c$ then $y=\nu z$. This gives $a=\lambda\nu c$ and $x=\lambda\nu z$, so $(a,b,c)=(\lambda\nu c,\nu c,c) = c(\lambda,\nu,1)$ and $(x,y,z)=(\lambda\nu z,\nu z,z) = z(\lambda\nu,\nu,1)$, so they are both multiples of the same vector. Essentially, as Bill says, they define the same point in projective space.2011-01-19
2

The colon notation $\rm\ (x\::\:y\::\:z)\ $ denotes a point in projective space (vs. comma notation $\rm\ (x\:,\:y\:,\:z)\ $ for points in affine space).$\ $ The notation highlights the fact that $\rm\ (x\::\:y\::\:z)\ =\ (\lambda\:x\::\:\lambda\:y\::\:\lambda\:z)\:,\ $ which explains the equality stated by Ahlfors. See for example Kedlaya: Projective Geometry.

1

The notation $a:b = c:d$ usually means that the proportions between $a,b$ and $c,d$ are the same, e.g. $a/b=c/d$ or more generally $ad = bc$. In this case we have a triple proportion.