What is the definition of straight line in $\hat{\mathbb{C}}$?
Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$?
($a,b$ are complex numbers and $b\neq 0$)
I'm currently reading "Ahlfors - Complex analysis", and this terminology 'line' appears in p.79 Thm. 13:
The cross ratio $(z_1,z_2,z_3,z_4)$ is real iff $z_1,z_2,z_3,z_4$ lie on a circle or on a straight line.
What's the definition of line?
(a) In ${\mathbb C}\sim{\mathbb R}^2$ a line is a line and is given by an equation of the form $ax+by=c$ with $a^2+b^2\ne0$. A parametric representation of such a line would be $$\ell:\quad t\mapsto z(t)= p t+ q\qquad(-\infty<t<\infty)$$ with $p\in{\mathbb C}\setminus\{0\}$.
(b) When ${\mathbb C}$ is extended to the Riemann sphere $\hat{\mathbb C}$ by adjunction of the point $\infty$ then it make sense to let the lines from (a) to go through this point $\infty$ as well.
(c) The "symmetry group" of $\hat{\mathbb C}$ is no longer the group of euclidean similarities $z\mapsto c\>z+ d$, $\>c\ne0$, but is the larger group ${\cal M}$ of Moebius transformations. In the world of ${\cal M}$ euclidean lines are no longer recognizable, but "circles" are. Any three different points $z_k\in\hat{\mathbb C}$ determine a unique "circle" $$C:=\{z\in\hat{\mathbb C}\>|\>{\rm cross\ ratio}(z_1,z_2,z_3,z)\in{\mathbb R}\cup\{\infty\}\}\ .$$ The stereographic image of $C$ will be either a circle or straight line in ${\mathbb C}$. In other words: When restricting such a "circle" to the subset ${\mathbb C}\subset\hat{\mathbb C}$ that is modeled by a sheet of paper we see either an ordinary euclidean circle in ${\mathbb C}$, or if the given "circle" happens to pass through the point $\infty$, we see a euclidean line.