Mobius Transformation Decomposition

By Xah Lee. Date:

We can rewrite this expression (A*Z+B)/(C*Z+D) into partial fractions.

If C=={0,0}, then we know D != {0,0} because the requirement that with (A*D-B*C) != {0,0}.

So, with C=={0,0}, the expression becomes A/D * Z + B/D

As we can see, a MT is simply a dilation, rotation, and a translation.

Now suppose if C!={0,0}. Then, (A*Z+B)/(C*Z+D) can be rewritten as:

A/C -(A D - B C)/C * 1/(C Z + D)

From this, we can see that MT is a sequence of:

C*Z                 dilation and rotation
Z+D                 translation
1/Z                 geometric inversion and reflecton
-(A D - B C)/C * Z  dilation and rotation
Z+A/C               translation

If we multiply A/C -(A D - B C)/C * 1/(C Z + D) by (1/C)/(1/C), then we get this form:

A/C -(A D - B C)/C^2 * 1/(Z + D/C)

From this, we see that MT is a sequence of the following transformations: translation, inversion and reflection, dilation and rotations, another translation.

f1[Z]:= Z+D/C     (translation)
f2[Z]:= 1/Z       (inversion and reflection)
f3[Z]:= - (A D-B C)/C^2 * Z      (dilation and rotation)
f4[Z]:= Z+A/C         (translation)

MT is then f4[f3[f2[f1[Z]]]]. A simple simplification of the expression shows this to be so.

(Note that in f2, the Z will not be {0,0} because Z is required to not be -D/C in MT.)

Notice the function f3 has (A D-B C) as its numerator. If that value is 0, than the transformation maps every point to {0,0}. Transformations that maps every image to a single point is not interesting. Therefore in the definition of mobius transformation, we require that A D - B C != {0,0}. This will make discussion of mobius transformation much simpler without mentioning this exceptional case.

Just curious, what does A D-B C mean in terms of their coordinate values? It means: {-b1 c1 + b2 c2 + a1 d1 - a2 d2, -b2 c1 - b1 c2 + a2 d1 + a1 d2}

If we now find the inverse transformation of each of f1, f2, f3, f4, and call them g1, g2, g3, g4, then sequence them together g1[g2[g3[g4[z]]]], then what we have is a inverse mobius transformation.

g1[Z_] := Z - D/C
g2[Z_] := 1/Z
g3[Z_] := -C^2/(A D - B C)*Z
g4[Z_] := (Z - A/C)

The value for g1@g2@g3@g4@z is: (B - D*z)/(-A + C*z)

Here, the expression is again a MT with the roles of A and D exchanged and both made negative. The requirement that A D - B C != {0,0} has the same meaning here.