The Cross Ratio of a Mobius Transformation - Definition Clarification

Question

so the Mobius transformation $f(z)$ of an extended complex plane preserves the following cross ratio $$\frac{(w_1-w_4)(w_3-w_2)}{(w_1-w_2)(w_3-w_4)}=\frac{(z_1-z_4)(z_3-z_2)}{(z_1-z_2)(z_3-z_4)}$$ where $w_i$ and $z_i$ belongs to the extended complex plane. I have a silly question about the numeration. Let's say I want to recover that cross ratio in a couple of years, and I will know that I need to put $$(a-b)(c-d)$$ on the top. But, what goes on the bottom $$(a-c)(b-d)\text{ or }(a-d)(b-c)\text{ and two more options. }$$ So, to answer that question, I consider all 4 cases and simply did the renumerations. So, I got that the choice does not matter as I always will obtain the desired $$\frac{(z_1-z_4)(z_3-z_2)}{(z_1-z_2)(z_3-z_4)}.$$ Is there a shorter way to argue that? Or, is there another intuition that stands behind that definition?

Answer 1

The cross-ratio – as I learned it many years ago – is defined as $$ (z_1, z_2, z_3, z_4) = T(z_1) $$ where $T$ is the (unique) Möbius transformation which maps $(z_2, z_3, z_4)$ to $(1, 0, \infty)$, in that order. This definition holds for all $z_k$ in the extended complex plane, and it easy to remember.

For finite values $z_k$ that is $$ (z_1, z_2, z_3, z_4) = \frac{(z_1 −z_3)(z_2 −z_4)}{(z_1 − z_4)(z_2 − z_3)} $$ which coincides with the definition given in Wikipedia.

One could also map $(z_2, z_3, z_4)$ to any other permutation of $(1, 0, \infty)$, that gives $6$ possible definitions of the cross-ratio. I assume that the actual definition is just by convention. The important properties (it is invariant under Möbius transformations, and it is a real number exactly if the four points line on a circle or a line) are not affected by that choice.