As indicated earlier, we can define similarity geometry using the
subgroup S(2) = {t ε I(2) : t(∞) =∞},
with elements restricted to
the complex plane.
Of course, with this definition, we have to prove that the geometry
has the familiar invariants.
We apply the inversive crossratio to this by considering
crossratios
(u,v,z,∞) for distinct complex u,v,z. If t is in S(2),
then it maps (u,v,z,∞) to
(t(u),t(v),t(z),∞), the corresponding
crossratio for the three image points.
Recall also that, in this case, we have (u,v,z,∞) = (uz)(vz),
the modulus of (uz)/(vz) is the ratio UZ/VZ, and
the argument of (uz)/(vz) is the signed angle VZU.
From our discussions above, we have
Similarity Invariants
Ratios of lengths and the size of angles are invariants
of similarity geometry.
The only aspect that requires much comment is invariance of
the ratio of lengths of any two segments. The invariance of
the modulus of the crossratio (u,v,z,∞) shows that UZ/VZ
is invariant. A general ratio UZ/VW can be expressed as
a product (UZ/VZ)(ZV/WV), so is invariant.

