The Fundamental Theorem of Weird Geometry
If L = (A,B,C) and L' = (A',B',C') are lists of distinct points of C,
then there is a unique element t of W(2) mapping L to L'.
Proof
L and L' are lists of distinct points of the extended complex plane.
By the Fundamental Theorem of Inversive Geometry, there is a
unique element s of I+(2) mapping L to L'.
As s is inversive, it maps i-lines to i-lines. Now, three distinct points
determine a unique i-line. Since all the points lie on C, each list must
determine C. Thus s maps C to C.
By the Interior-Exterior Theorem, s maps D0
either to D0
or to D1.
In the first case, s ε H(2).
In the second case, as h0 (inversion in C) maps D1
to D0, so that
s* = h0os ε H(2).
As h0 fixes all points of C, s* maps L to L'.
The required t is obtained by restricting s or s* to C.
The element is unique since any element of W(2) is the restriction of
an element of H(2), i.e. of an element of I(2).
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