Theorem II1
The inversive cross-ratio is invariant under the Mobius group M(2).
Proof
Suppose that a Mobius function m is defined by
m(z) = (az+b)/(cz+d), with ad-bc ≠ 0.
Then we have
m(u,v,w,z) = (m(u)-m(w))(m(v)-m(z))/(m(u)-m(z))(m(v)-m(w)).
Considering each factor in turn:
| m(u)-m(w) |
= ((au+b)(cw+d)-(cu+d)(aw+b))/(cu+d)(cw+d) |
|
= (ad-bc)(u-w)/(cu+d)(cw+d), |
m(v)-m(z) = (ad-bc)(v-z)/(cv+d)(cz+d),
m(u)-m(z) = (ad-bc)(u-z)/(cu+d)(cz+d),
m(v)-m(w) = (ad-bc)(v-w)/(cv+d)(cw+d).
When we combine these, the (ad-bc) and the (c?+d) all cancel,
and the result follows at once.
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