# Möbius transform families

Möbius transformations are an important class of complex-valued functions that are connected to many areas of mathematics, including complex analysis, and hyperbolic geometry. They are given by the formula: \[ f(z) = \frac{az + b}{cz + d},\] where \(a,b,c,d\) are complex numbers. We must also insist that \(ad - bc \neq 0 \), or else the transformation collapses the whole plane to a single point.

## Fixed points

We can begin to better understand Möbius transformations by studying their fixed points, that is, the points satisfying the equation \[f(z) = z.\] This equation seems to be easy to solve, and if we do so, we discover that every Möbius transform has either one or two fixed points. The parabolic transformations have one fixed point, while the hyperbolic, elliptic, and loxodromic transformations each have two fixed points.

Each of the transformations in the opening animation has infinity as a fixed point. The hyperbolic, elliptic and loxodromic transformations also have zero as their second fixed point.

By looking at the animation, we can see the geometric significance of the various categories. Parabolic transformations are translations along the plane. Hyperbolic transformations send points gravitationally from one fixed point to another. Elliptic transformations rotate points, moving them neutrally with respect to the fixed points. Loxodromic transformations are a combination of rotation and gravitation.

## Lie groups

The set of Möbius transforms are an example of the continuous groups known as Lie groups.

If you're not familiar with the math jargon, the word group here is a technical term which means that Möbius transforms composed together define Möbius transforms, and that Möbius transforms have inverse transforms that are also Möbius transforms. The formula for the inverse transform is \[ f^{{ -1 }}(z)={\frac {dz-b}{-cz+a}}\], where the inverse transform is the function satisfying \[f^{{ -1 }}(f(z)) = z\]

By continuous, I mean that the set of Möbius transforms itself is a continuous manifold with tangent vectors and continuous paths that run inside it, just as does the surface of a sphere embedded inside three-dimensional space. For the sphere and the projective plane, continuous paths and tangent vectors are constructed by varying the angular coordinates. In the case of Möbius transforms, we create paths by varying the four parameters \(a, b, c, d\).

Tangent vectors passing through the identity element of the Lie group naturally define a continuous path of elements of the Lie group, via something called the exponential map \[ t \mapsto \exp(tX).\] For each element of the tangent space, the path through the Lie group starts at the identity transformation for \( t= 0 \) and ends at some other transformation\( \exp(X) \) when \( t= 1 \).

In general, it's hard to see what this continuous path of Möbius transforms might be. However, in the special cases \[\begin{aligned} f(z) &=& kz, \\ f(z) &=& z + b \end{aligned}\] we can correctly guess that the continuous one-parameter family of transformations are \[\begin{aligned} f(z) &=& k^t z, \\ f(z) &=& z + tb \end{aligned}\].

Fortunately, it turns out that all Möbius transformations are equivalent to one of these special cases!

## Conjugate transformations

First of all, what does it mean for two Möbius transformations to be equivalent? We say that two Möbius transformations \(f, g\) are equivalent if there is a third Möbius transformation \(h\) that relates them via \[f = h^{-1}gh \]

In other words, the transformation \(f\) is the same as \(g\) when viewed through the lens of the connecting transformation \(h\).

So we would like to see that any Möbius transformation is equivalent to one of our special cases for an appropriate connecting transform \(h\). But how do we find \(h\)? Clearly, the transformation \(h\) should send fixed points to zero and infinity, the fixed points of the special case transformations.

But we can see that the Möbius transformation \[h(z)={\frac {z-\gamma _{1}}{z-\gamma _{2}}}\] sends the two points \( \gamma_{1}, \gamma_{2} \) to zero and infinity. We can use this transformation to relate any transformation with two fixed points to the special case.

For the parabolic case, we only have one fixed point, so we use \[h(z)={\frac {1}{z-\gamma}}\]

## Transformations with finite fixed points

The special cases are easy to analyze because they are so simple algebraically. However, they all have infinity as a fixed point, and since infinity is not a point we can actually see, it is hard to imagine what is happening near this fixed point.

In the visualization above, you can see each of the special cases by selecting the transformations labelled "fixed point at infinity". Each of these transformations has a straightforward visualization, but in each case, the fixed point at infinity is very far off the page indeed.

We can fix this by conjugating the special case transformations with a transformation that maps infinity to a finite point and looking at the resulting transformation. This is what is shown when selecting the transformations labelled "finite fixed points".

For the non-parabolic transformations the fixed points at zero and infinity have been moved to \(\pm 1\). For the parabolic transformation, the single fixed point at infinity has been moved to zero.

This view of the transformations was less satisfying than I thought it would be. It's nice to be able to put the fixed points into view, but the transformations become much more complicated looking, and it's not that easy to imagine what is happening further out in the plane.

Another way to bring the fixed point at infinity into view, is to conjugate our transformations with stereographic projection. You can view these by selecting the transformations labelled "sphere".

This view is much nicer, because you can see the action in the entire space all at once. It also brings into view a very nice symmetry between the fixed points at zero and infinity.

The origin of the terminology loxodromic transformation comes from loxodromic navigation whereby a ship would sail with a constant angle to the north-south meridian lines, just as with the paths drawn in the spherical view of the loxodromic family of transformations.