# 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.