Mobius Transformation
Definition
Mobius transform is defined as:
f[Z]:=(A*Z+B)/(C*Z+D)
with (A*D-B*C) != {0,0}.
If (A*D-B*C) == {0,0}, then the whole thing collaps to a point at A/C. (more about this later)
Note that the fact (A*D-B*C) != {0,0} also means that the denominator will not be {0,0}. That is, C and D will not both be {0,0}.
This function is defined for all points except when the input z is equat to -D/C, because if z==-D/C, the denominator is Zero, and is not defined.
Mobius Transformation as real 2D Vectors
Since complex numbers are just a pair of numbers with a multiplication defined. We can rewrite the Mobius Transformation as a function of 2D real vectors. Recall, that we define the operatitons for addition, multiplication, and inversion as:
complexPlus[{a_, b_}, {c_, d_}] := {a + c, b + d}
complexTimes[{a_, b_}, {c_, d_}] := {a*c - b*d, b*c + a*d}
complexInversion[{a_, b_}] := 1/(a^2 + b^2)*{a, -b}
If we write the complex number A as {a1,a1}, and B as {b1,b2} and so on, then, the expression
(A*Z+B)/(C*Z+D)
can be written in a functional notation as
complexTimes[
complexPlus[complexTimes[{a1, a2}, {z1, z2}], {b1, b2}],
complexInversion[
complexPlus[complexTimes[{c1, c2}, {z1, z2}], {d1, d2}]
]
]
or in operator notation as
({a1, a2}⊗{z1, z2}⊕{b1, b2}) ⊗ {1,0} ⊘ ({c1,c2}⊗{z1, z2}⊕{d1,d2})
If we actually compute it, we get the formula for Mobius Transformation as a 2D vector in terms of their components:
mobiusTransform[{z1,z2}]:=
{-(((b2 + a2*z1 + a1*z2)*(-d2 - c2*z1 - c1*z2))/((d2 + c2*z1 + c1*z2)^2 +
(d1 + c1*z1 - c2*z2)^2)) + ((b1 + a1*z1 - a2*z2)*(d1 + c1*z1 - c2*z2))/
((d2 + c2*z1 + c1*z2)^2 + (d1 + c1*z1 - c2*z2)^2),
((b1 + a1*z1 - a2*z2)*(-d2 - c2*z1 - c1*z2))/((d2 + c2*z1 + c1*z2)^2 +
(d1 + c1*z1 - c2*z2)^2) + ((b2 + a2*z1 + a1*z2)*(d1 + c1*z1 - c2*z2))/
((d2 + c2*z1 + c1*z2)^2 + (d1 + c1*z1 - c2*z2)^2)}
This is a function of 2 arguments z1 and z2, and 8 real constants a1, a2, b1, b2, c1, c2, d1, d2.
Mathematica notebook for the computation mobius_transform.nb
Mobius Transformation Decomposition
Circle Maps to Circles
Theorem: Under mobius transform, any line in the preimage becomes a circle in the postimage, and any circle in the preimage remains a cicle in the postimage.
Proof: Note that mobius transform is composed of a sequence of isometries plus a inversion. The isometries preserves distances, while the inversion does exactly what the theorem states.
Angle Preseservation
Theorem: If two lines makes a angle theta, then after the transformation, the image of these two lines's tangents will still make a angle theta.
Since a mobius transform is composed of a sequence of translation, dilation, reflection, and geometric inversion, and all these preserve angles, therefore a mobius transform also preserves angles.
Symmetry Principle
If two points are symmetric with respect to a circle. (as in Geometric Inversion) Then, after a MT, the two points still be symmetric with respect to the image circle.
This fact can easily be seen from the decomposition of MT, and the fact that two geometric inversions over the same circle will maintain two symmetric points over the circle.
Circle Orientation
A MT maps an oriented circle C to an oriented circle C' in such a way that the region to the left of C is mapped to the region to the left of C' [Tristan Needham]
(needs to rephrase and expand this)
3 Points Defines a Mobius Transformation
Normalization
If (a d - b c)==1, then we say that the mobius transformation is normalized. Normalized MT makes some theorization simpler…
The Cross Ratio
Stereographic Projection
Point at Infinity
To study what happens to a mapping such as f[Z]:=1/Z that maps origin to Infinity, we can study it by first map it to 1/Z, which maps points at infinity to the Origin, then study what happens at the Origin. For example, if f is a function that maps Origin to Infinity, then we study f[1/Z] at the Origin.
Mobius Transformation as a Group
Two mobius transformations in sequence is equivalent to a single mobius transformation.
If we denote the first mobius transformation M1 is [a1,b1,c1,d1] and second M2 is [a2,b2,c2,d2], then their composition M2@m1 [a2 a1 + b2 c1, a2 b1 + b2 d1, c2 a1 + d2 c1, c2 b1 + d2 d1]
From the MT decomposition, this means, a sequence of translation, inversion, reflection thu real axis, dilation and rotation, translation, and another sequence of these, is equivalent to a single such sequence. (the fact of this or its details is in the works…)