The cross ratio of 4 factors A, B, C, D is outlined by
the place XY denotes the size of the road phase from X to Y.
The concept of a cross ratio goes again a minimum of so far as Pappus of Alexandria (c. 290 – c. 350 AD). Quite a few theorems from geometry are said when it comes to the cross ratio. For instance, the cross ratio of 4 factors is unchanged underneath a projective transformation.
Advanced numbers
The cross ratio of 4 (prolonged [1]) advanced numbers is outlined by
Absolutely the worth of the advanced cross ratio is the cross ratio of the 4 numbers as factors in a airplane.
The cross ratio is invariant underneath Möbius transformations, i.e. if T is any Möbius transformation, then
That is linked to the invariance of the cross ratio in geometry: Möbius transformations are projective transformations on a posh projective line. (Extra on that right here.)
If we repair the primary three arguments however depart the final argument variable, then
is the distinctive Möbius transformation mapping z1, z2, and z3 to ∞, 0, and 1 respectively.
The anharmonic group
Suppose (a, b; c, d) = λ ≠ 1. Then there are 4! = 24 permutations of the arguments and 6 corresponding cross ratios:
Seen as features of λ, these six features type a gaggle, generated by
This group known as the anharmonic group. 4 numbers are stated to be in harmonic relation if their cross ratio is 1, so the requirement that λ ≠ 1 says that the 4 numbers are anharmonic.
The six components of the group will be written as
Hypergeometric transformations
Once I was wanting on the six doable cross ratios for permutations of the arguments, I thought of the place I’d seen them earlier than: the linear transformation formulation for hypergeometric features. These are, for instance, equations 15.3.3 by 15.3.9 in A&S. They relate the hypergeometric perform F(a, b; c; z) to comparable features the place the argument z is changed with one of many components of the anharmonic group.
I’ve written about these transformations earlier than right here. For instance,
There are deep relationships between hypergeometric features and projective geometry, so I assume there’s a sublime clarification for the similarity between the transformation formulation and the anharmonic group, although I can’t say proper now what it’s.
Associated posts
[1] For completeness we have to embody a degree at infinity. If one of many z equals ∞ then the phrases involving ∞ are dropped from the definition of the cross ratio.
