Moebius Transformations Revealed

Wow! Thats so sick.

riht... another thing im sure to think about before going to bed... in other words ill be up for a couple of extra hours lol

1:03.. tweety bird?? anyone? anyone?

This is a perfect way to torture you're brain O_O

Excellent. This video has become my main example of what to aim for with some I'm trying to make myself. A great book for this sort of thing is "Visual complex analysis" by Tristan Needham.
Moebius transforms are precisely the analytic bijections of the extended complex plane, and can also be characterised by their vanishing Schwarzian derivatives. THE way to think about rotations, naturally leading to quaternions.

Absolutely, "Visual Complex Analysis" is a GREAT book, shows how higher mathematics can be made exciting and understandable without any dumbing down.

to give you a bit background information, the transformations presented here do indeed include translation and rotation in 2 dimensions, as you can see in the video.
the formula is
f(z) = (az+b) / (cz+d)
for {a=c=1, d=0} you get the translation
f(z) = z+b
in the same way, for |a|=1 (that is a is a phase)
f(z) = a z
is a rotation

for a=real
f(z) = a z
is a scale transformation
and the actually significent part about the möbius transformation, the inversion for {a=b=0,c=1,d=0} is
f(z) = 1/z

and a second mistake. I confused c and d in the translation. It obviously should be {a=d=1,c=0}.

z is a complex (2-dimensional) number z = x+iy.

the formula is

x'+iy' = z' = f(z) = (az+b) / (cz+d) = (ax+iay+b) / (cx+icy+d)

that is {a,b,c,d}+{x,y} --> {x',y'}

so you could intepret it as a mapping from R6 to R2.
But you usually won't because a,b,c,d are just parameters of the transformation. Like in the real world R3 you can rotate around 3 axes and translate in 3 dimensions, but you don't consider that a mapping from R9 to R3.