Fractals/moebius

Möbius transformation[1][2][3][4]

Riemann sphere

definitionEdit

A Möbius transformation of extended complex plane   is a rational function f of the form

 

of one complex variable z.

Here the coefficients a, b, c, d and the result w are complex numbers satisfying


 
 

Representation or formEdit

  • function
  • matrix

FunctionEdit

 


 


 

MatrixEdit

In matrix form by using homogeneous coordinates:[5]

 


 


Matrix M is a square 2x2 invertible matrix[6]

ExamplesEdit

simpleEdit

The following simple transformations are also Möbius transformations:

  •   is an identity
  •   is a translation
  •   is a combination of a homothety and a rotation.
    • If   then it is a rotation
    • if   then it is a homothety
  •   inversion and reflection with respect to the real axis)

How to ...?Edit

eigenvalue and eigenvectorEdit

A number   and a non-zero vector   satisfying

 

are called an eigenvalue and an eigenvector of matrix M, respectively.

For dimensions 2 formulas involving radicals exist that can be used to find the eigenvalues. The eigenvalues can be found by using the quadratic formula:

 

diagonalizationEdit

a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. In other words all off-diagonal elements are zero in a diagonal matrix.

the main diagonal of a matrix   is the list of entries   where  , here  

the diagonalization of a matrix M gives a pair of matrices: D, P such that:[8]

  • D is diagonal (all elements not on the diagonal are 0)
  •  

For 2x2 matrices there is a simple closed form solution[9]

Product with a scalarEdit

If A is a matrix and c a scalar, then the matrices   and   are obtained by left or right multiplying all entries of A by c.

traceEdit

The trace   of a square 2x2 matrix  


 

is the sum of its diagonal entries

 


So  

determinantEdit

determinant   of matrix  

 

inverseEdit

Inverse Möbius transformation[10]

 


 

 .

interpolationEdit

How to smootly interpolate between möbius transformations?[11][12]


If you have two Möbius transformations represented as:

 

 


where coefficients are complex numbers

 

Is it possible to derive a third function  , where   and  , which "smoothly" interpolates between the transformations represented by   and  ?

The solution:

 


Specifying a transformation by three pointsEdit

Given a set of three distinct points z1, z2, z3 on the one Riemann sphere ( let's call it z-sphere) and a second set of distinct points w1, w2, w3 on the second sphere ( w-sphere) , there exists precisely one Möbius transformation f(z) with :

  •  
  •  

for i=1,2,3

Mapping to 0, 1, infinityEdit

The Möbius transformation with an explicit formula :[13]

 

maps :

  • z1 to w1= 0
  • z2 to w2= 1
  • z3 to w3= ∞

the unit circle to the real axis - first methodEdit

Let's choose 3 z points on a circle :

  • z1= -1
  • z2= i
  • z3= 1

then the Möbius transformation will be :

 

Knowing that :[14]

 

one can simplify this to :

 

In Maxima CAS one can do it :

(%i1) rectform((z+1)*(%i-1)/((z-1)*(%i+1)));
(%o1) (%i*(z+1))/(z−1)

where coefficients of the general form are :

 
 
 
 

so inverse function can be computed using general form :

 

Lets check it using Maxima CAS :

(%i3) fi(w):=(-%i-w)/(%i-w);
(%o3) fi(w):=−%i−w/%i−w
(%i4) fi(0);
(%o4) −1
(%i5) fi(1);
(%o5) −%i−1/%i−1
(%i6) rectform(%);
(%o6) %i

Find how to compute it without symbolic computation program (CAS)  :

(%i3) fi(w):=(-%i-w)/(%i-w);
(%o3) fi(w):=−%i−w/%i−w
(%i8) z:x+y*%i;
(%o8) %i*y+x
(%i9) z1:fi(w);
(%o9) (−%i*y−x−%i)/(−%i*y−x+%i)
(%i10) realpart(z1);
(%o10) ((−y−1)*(1−y))/((1−y)^2+x^2)+x^2/((1−y)^2+x^2)
(%i11) imagpart(z1);
(%o11) (x*(1−y))/((1−y)^2+x^2)−(x*(−y−1))/((1−y)^2+x^2)
(%i13) ratsimp(realpart(z1));
(%o13) (y^2+x^2−1)/(y^2−2*y+x^2+1)
(%i14) ratsimp(imagpart(z1));
(%o14) (2*x)/(y^2−2*y+x^2+1)
 
2 steps of unrolling the main cardioid of Mandelbrot set: Moebius map and conformal map
 
Unrolled main cardioid of Mandelbrot set for periods 7-13

So using notation :

 

one gets :

 

 

It can be used for unrolling the Mandelbrot set components [15]

the unit circle to the real axis - second methodEdit

Function :

 

sends the unit circle to the real axis :

  • z=1 to w=0
  • z=i to w=1
  • z=-1 to  

Mapping to the imaginary axisEdit

Function   sends the unit circle to the imaginary axis.[16]

ReferencesEdit

  1. Möbius transformation in wikipedia
  2. Moebius transformation animated GIFs by Fritz Mueller
  3. Möbius Transforms App by (c) Robert Woodley, 2016-2017.
  4. Transformations of the projective line
  5. oeis.org : Moebius transformation
  6. Matrix in wikipedia
  7. Squares that Look Round: Transforming Spherical Images by Saul Schleimer and Henry Segerman
  8. How to diagonalize a matrix ? in wikipedia
  9. interpolating moebius transformations
  10. inverse of a matrix by Bruce Simmons
  11. mathoverflow question: ow-to-smootly-interpolate-between-moebius-transformations
  12. interpolating moebius transformations by Claude Heiland-Allen
  13. Triple transitivity by David J Wright 2004-12-04
  14. math.stackexchange questions : how-to-do-this-transformation-of-complex-rational-function
  15. Stretching cusps by Claude Heiland-Allen
  16. math.stackexchange questions: what-mobius-transformation-maps-the-unit-circle-z-z-1-to-the-real-axis/335061#335061