Ted Mobius Transformations

Hey all, it's been a little over a week since my last post, but unlike this functor, I haven't forgotten about you. This week I'm going to introduce the titular Mobius Transformation. This bad boy, also known as a Linear Fractional Transformation, twist, turn, and stretch the sphere based on where 0, 1, and ∞ are mapped. These transformations also allow us to construct the Klienian group fractals.

As you can see the Ted Mobius Transformation adds age and a tux.

I want this post to be manageable and concise, but I also want to discuss a bunch of topics, like fixed and periodic points, and whether they're attracting or repelling, as those let us construct the Julia and Fatou set which are really awesome for building fractals. I think I'll just discuss what a Mobius Transformation is here, and then later I'll post about those points and sets.

A Mobius Transformation is a transformation from the Riemannian Sphere to the Riemmanian Sphere T(z) = (az+ b)/(cz+ d), where a,b,c,d are complex numbers such that ad-bc !=0. That last condition may remind a mathematical reader of a condition of an invertible 2x2 matrix.

Let's look at some inputs for z to see how our sphere changes. Some of the following text may be hard to parse over blogger (I'll try to format it in a way that's not too bad), but if you spend a second, you can easily convince yourself it's true. Maybe one day I'll get Latex and rewrite some of these lines.

T(∞)= (a∞ +b)/(c∞ +d) = a/c. That makes sense, as if a and c are infinitely large, then adding or subtracting a finite number won't change either of them. This also shows why we need to be on the Riemannian Sphere, because on the complex plane (without infinity) then there's no number we can plug into T(z) to get a/c.

T(0) = (a0 +b)/(c0 +d)= b/d. 

T(1) = (a1 +b)/(c1 +d)= (a+b)/(c+d).

These are some of the intuitive numbers to plug in, to see what we'd get. But now let's try some less initially intuitive but geometrically significant numbers.

T(-d/c) = (-ad/c +b)/(-cd/c +d) = (-ad +bc)/(-cd +dc) = (-ad +bc)/0) = ∞. But a tricky reader might want to test this function is well defined. "How do you know you'll never get 0/0?" Well, tricky reader, the only way we'd get 0/0 would be if -ad +bc =0, but as you'll remember by our definition of a Mobius Transformation, -ad +bc =0 can't be the case.

T(-b/a) = (-ab/a +b)/(-cb/a +d) = (-b +b)/(ad - bc) = 0/(-ad +bc) = 0. Remember, ad-bc can't equal 0, so it remains well defined.

This next one will be a bit more complicated looking, so bear with me. I'm going to color code fractions:
T((d-b)/(a-c)) = (a ((d-b)/(a-c)) +b)/(c ((d-b)/(a-c)) +d) = ((ad-ba)/(a-c) + (ba-bc)/(a-c))/((cd-cd)/(a-c) + (da-da)/(a-c)) = ((ad-bc)/(a-c))/(ad-bc)/(a-c)) = 1.

You can convince yourself now that these transformations work under composition and can be inverted.

Mobius Transformations can be thought of as a series of composition of different transformations. First take T(z) = z and scale it by c and translate it by d. Now you have T(z) = cz +d. Second, invert it so that T(z) = 1/(cz + d). Third, scale it by -1/c (almost like conjugating with the first step), and get T(z) = -1/c(cz + d). Finally, translate that by a/c: a/c - 1/c(cz + d). I'll do the algebraic manipulation below, or you can trust me that after some finagling, it comes out to T(z) = (az +b)/(cz +d). The key to the algebraic manipulation, which took me frustratingly way, way too long to see is realizing that ad-bc =1.

a/c - 1/c(cz + d) = a/c - (bc -ad)/c(cz + d)= (a(cz +d) + bc - ad)/ (c(cz+d)) = (caz + da + bc -ad)/(c(cz+d) = (caz +bc)/(c(cz+d)) = (az+b)/(cz+d).

That's really cool for several reasons. First, it makes finding the inverse very easy, because instead of composing T₄∘T₃∘T₂∘T₁(z), you can just do T^-1₁∘T^-1₂∘T^-1₃∘T^-1₄(z). Also if we take all of our Mobius Transformations to create a group under composition, with easily seen generators.

Pretty much, Mobius Transformations send three points to any other three points, and by doing so, determine the entire transformation of the sphere. This means we can determine exactly how the sphere acts based on where 0, 1, and ∞ are sent. But we can also look at other points. You can see yourself that given points a,b,c, T(z)= (z-a)/(z-b) * (c-b)/(c-a) sends a to 0, b to ∞, and c to 1.

Or going a step further, you can see that given poitns a,b,c,d,e,f, if T(z) = w is a Mobius transformation, and (z-a)/(z-b) * (c-b)/(c-a) = (w-d)/(w-e) * (f-e)/(f-d), then the transformation maps a to d, b to e, and c to f.

Take a look at this picture I drew representing a Mobius Transformation abstractly:

I'll post more later about fixing points, specific Mobius Transformations (like ones that preserve symmetries, ie, which one maps the unit circle to itself, etc), and their relation to fractals. This post should just introduce Mobius Transformations, there's still much more to go!