the fixed points is the point at infinity. When a ≠ d the second fixed point is finite and is given by In this case the transformation will be a simple transformation composed of translations, rotations, and dilations: If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: Topological proof Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: Firstly, the projective linear group PGL(2,K) is sharply 3-transitive – for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially dimension counting, as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity. Next, one can see by identifying the Möbius group with that any Möbius function is homotopic to the identity. Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The Lefschetz–Hopf theorem states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic. By contrast, the projective linear group of the real projective line, PGL(2,R) need not fix any points – for example has no (real) fixed points: as a complex transformation it fixes ±i – while the map 2x fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more. Normal form Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form. We first treat the non-parabolic case, for which there are two distinct fixed points. Non-parabolic case: Every non-parabolic transformation is conjugate to a dilation/rotation, i.e., a transformation of the form (k ∈ C) with fixed points at 0 and ∞. To see this define a map