In fact the homeomorphisms of "'S "'1 induced by quasiconformal self-maps of "'D "'are precisely the quasisymmetric homeomorphisms of "'S "'1; these are exactly homeomorphisms which do not send four points with cross ratio 1 / 2 to points with cross ratio near 1 or 0.
22.
I certainly don't want to try all 6 permutations of the 4 points and see what pops up on the cross ratio, but at the same time I can't see intuitively where the-\ tan ^ 2 ( \ frac { d } { 2 } ) could have come from in order to try and work out how to take the cross ratio to get the desired result.
23.
I certainly don't want to try all 6 permutations of the 4 points and see what pops up on the cross ratio, but at the same time I can't see intuitively where the-\ tan ^ 2 ( \ frac { d } { 2 } ) could have come from in order to try and work out how to take the cross ratio to get the desired result.
24.
While the projective linear group of the plane is 3-transitive ( any three distinct points can be mapped to any other three points ), and indeed simply 3-transitive ( there is a " unique " projective map taking any triple to another triple ), with the cross ratio thus being the unique projective invariant of a set of four points, there are basic geometric invariants in higher dimension.
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