invariants of similarity geometry

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 cross-ratio to this by considering
cross-ratios (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
cross-ratio for the three image points.

Recall also that, in this case, we have (u,v,z,∞) = (u-z)(v-z),
the modulus of (u-z)/(v-z) is the ratio |UZ|/|VZ|, and
the argument of (u-z)/(v-z) 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 cross-ratio (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.

inversive invariant page